ThesisPDF Available

Room Reverberation Prediction and Synthesis

October 2022

Thesis for: Doctor of Science (Technology)
Advisor: Vesa Välimäki, Sebastian J. Schlecht

Authors:

University of York

In this dissertation, the discussion is centered around the sound energy decay in enclosed spaces. The work starts with the methods to predict the reverberation parameters, followed by the room impulse response measurement procedures, and ends with an analysis of techniques to digitally reproduce the sound decay. The research on the reverberation in physical spaces was initiated when the first formula to calculate room's reverberation time emerged. Since then, finding an accurate and reliable method to predict reverberation has been an important area of acoustic research. This thesis presents a comprehensive comparison of the most commonly used reverberation time formulas, describes their applicability in various scenarios, and discusses their accuracy when compared to results of measurements. The common sources of uncertainty in reverberation time calculations, such as bias introduced by air absorption and error in sound absorption coefficient, are analyzed as well. The thesis shows that decreasing such uncertainties leads to a good prediction accuracy of Sabine and Eyring equations in diverse conditions regarding sound absorption distribution. The measurement of the sound energy decay plays a crucial part in understanding the propagation of sound in physical spaces. Nowadays, numerous techniques to capture room impulse responses are available, each having its advantages and drawbacks. In this dissertation, the majority of commonly used measurement techniques are listed, whereas the exponential swept-sine is described in more detail. This work elaborates on the external factors that may impair the measurements and introduce error to their results, such as stationary and non-stationary noise, as well as time variance. The dissertation introduces Rule of Two, a method of detecting nonstationary disturbances in sweep measurements. It also shows the importance of using median as a robust estimator in non-stationary noise detection. Artificial reverberation is a popular sound effect, used to synthesize sound energy decay for the purpose of audio production. This dissertation offers an insight into artificial reverberation algorithms based on recursive structures. The filter design proposed in this work offers precise control over the decay rate while being efficient enough for real-time implementation. The thesis discusses the role of the delay lines and feedback matrix in achieving high echo density in feedback delay networks. It also shows that four velvet-noise sequences are sufficient to obtain smooth output in interleaved velvet noise reverberator. The thesis shows that the accuracy of reproduction increases the perceptual similarity between measured and synthesised impulse responses. The insights collected in this dissertation offer insights into the intricacies of reverberation prediction, measurement and synthesis. The results allow for reliable estimation of parameters related to sound energy decay, and offer an improvement in the field of artificial reverberation.

Variable acoustics laboratory Arni and the equipment used in room impulse response measurements.

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Layout of the variable acoustics laboratory Arni showing the panels and the sound sources and receiver locations. The arrows show the order of panels closing on the walls and the ceiling.

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Air-absorption coefficient values obtained with different methods for Ta ¼ 298:15 K, RH of 50%, and pa ¼ 101:325 kPa. The vertical dashed lines mark the octave bands. w.r.t., with respect to.

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(Color online) Spectrogram of a measurement consisting of five consecutive swept-sine signals. The arrow points to an impulsive noise event appearing in sweep #3 at about 14 s.

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+27

Relationship between (top) gain-per-sample and (bottom) resulting reverberation time for the delay-line length of L = 1000 samples. Red markers indicate the octave bands. The horizontal dashed line in the top figure is the unit-gain limit, reaching which would lead to an infinite reverberation time.

…

Figures - uploaded by Karolina Prawda

Content may be subject to copyright.

Content uploaded by Karolina Prawda

Content may be subject to copyright.

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Preface

The research work presented in this doctoral dissertation was conducted

at the Acoustics Laboratory of the Department of Signal Processing and

Acoustics at Aalto University in Espoo, Finland, between August 2018 and

August 2022. The research was funded by the “Nordic Sound and Music

Computing Network—NordicSMC”, NordForsk project number 86892. A

part of the research was conducted at the Multisensory Experience Lab of

the Aalborg University Copenhagen in October and November of 2019.

Firstly, I would like to thank my supervisor, Prof. Vesa Välimäki, for

sharing his knowledge and expertise, as well as for providing support and

guidance that not only enabled me to pursue my research objectives, but

also to do so with a great amount of independence and creativity. I would

also like to express my gratitude to my thesis advisor, Prof. Sebastian J.

Schlecht, whose help has been truly invaluable in my scientiﬁc ventures,

starting from sharing research ideas and ending with the polishing of the

ﬁnal versions of my publications. I also thank the pre-examiners, Ph.D.

Lily Wang and Ph.D. Elliot Kermit Canﬁeld-Daﬁlou, for their valuable

feedback that helped to improve this thesis, and Prof. Peter Svensson for

agreeing to act as my opponent.

I am deeply indebted to Prof. Stefania Seraﬁn and Ph.D. Silvin Willemsen

for hosting me at the Multisensory Experience Lab of Aalborg University

in Copenhagen. The time I spent in Denmark was both fruitful in terms of

scientiﬁc results as well as entertaining and enjoyable. I also express my

greatest appreciation to Luis Costa for his invaluable advice that helped

me navigate through the meandering path of the English language and

undoubtedly improved each of my publications.

Conducting research in a place like Aalto Acoustics Lab was an unforget-

table journey ﬁlled with great people and challenging discussions. Working

with an inspiring and supportive crowd of brilliant researchers was noth-

ing short of a delight. Therefore, I would like to thank all my colleagues at

the Acoustics Lab, with a special mention to the past and present members

of the Audio Signal Processing group.

Thanks to all my friends in Poland, Finland, and all the other locations

Preface

where they currently ﬁnd themselves, for helping me ﬁnd a healthy balance

between work and life outside of the university. I am also grateful to my

family, who supported me since my ﬁrst day on the job. Lastly but most

importantly, I express my deepest gratitude to Daniel, who went with me

through all the ups and downs of the last four years.

Helsinki, September 12, 2022,

Karolina Prawda

Contents

Preface 1

Contents 3

List of Publications 5

Author’s Contributions 7

List of Abbreviations 9

List of Symbols 11

1. Introduction 13

2. Sound Decay in Enclosed Spaces 17

2.1 Sound Propagation in Rooms . . . . . . . . . . . . . . . . . 17

2.2 Reverberation Time . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Reverberation Time Formulas . . . . . . . . . . . 20

2.2.2 Effect of Air Absorption . . . . . . . . . . . . . . . 23

2.2.3 Air Absorption Compensation . . . . . . . . . . . 24

2.3 Reverberation Time Prediction . . . . . . . . . . . . . . . . 24

2.3.1 Evaluation of Reverberation Prediction Formulas 25

2.3.2 Absorption Coefﬁcient Calibration . . . . . . . . 26

3. Impulse Response Measurement 29

3.1 Measurement Techniques . . . . . . . . . . . . . . . . . . . 29

3.1.1 Time-Stretched Pulses . . . . . . . . . . . . . . . 30

3.2 Factors Inﬂuencing Measurements . . . . . . . . . . . . . 31

3.2.1 Stationary Noise . . . . . . . . . . . . . . . . . . . 31

3.2.2 Transfer-Function Variation . . . . . . . . . . . . 32

3.2.3 Non-stationary Noise . . . . . . . . . . . . . . . . 34

3.3 Sweep Measurements in Noisy Environments . . . . . . . 34

3.3.1 Rule of Two . . . . . . . . . . . . . . . . . . . . . . 34

Contents

3.3.2 Median in Non-Stationary-Noise Detection . . . 36

4. Reverberation Synthesis 37

4.1 Delay-Based Reverberation . . . . . . . . . . . . . . . . . . 38

4.1.1 Feedback Delay Networks . . . . . . . . . . . . . 38

4.1.2 Velvet-Noise Reverberators . . . . . . . . . . . . 39

4.2 Decay-Rate Control . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Accurate Reproduction of Decay Rate . . . . . . . 42

4.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Reverberation Perception . . . . . . . . . . . . . . . . . . . 44

4.3.1 Echo Density . . . . . . . . . . . . . . . . . . . . . 44

4.3.2 Smoothness of Decay . . . . . . . . . . . . . . . . 45

4.3.3 Decay-Rate Perception . . . . . . . . . . . . . . . 47

5. Summary of the Main Results 49

6. Conclusions 53

References 55

Errata 71

Publications 73

List of Publications

This thesis consists of an overview and of the following publications, which

are referred to in the text by their Roman numerals.

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Evaluation

of Reverberation Time Models with Variable Acoustics. In Proceedings

of the 17th Sound and Music Computing Conference (SMC 2020), Turin,

Italy, June 2020.

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Calibrating

the Sabine and Eyring Formulas. The Journal of the Acoustical Society

of America, Vol. 152, No. 2, pp. 1158–1169, August 2022.

III

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Robust

Selection of Clean Swept-Sine Measurements in Non-Stationary Noise.

The Journal of the Acoustical Society of America, Vol. 151, pp. 2117–2126,

March 2022.

Vesa Välimäki and Karolina Prawda. Late-Reverberation Synthesis

using Interleaved Velvet-Noise Sequences. IEEE/ACM Transactions on

Audio Speech and Language Processing, Vol. 29, pp. 1149–1160, February

2021.

VKarolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Improved

Reverberation Time Control for Feedback Delay Networks. In Proceed-

ings of the International Conference on Digital Audio Effects (DAFx 2019),

Birmingham, UK, September 2019.

Karolina Prawda, Silvin Willemsen, Stefania Seraﬁn, and Vesa Välimäki.

Flexible Real-Time Reverberation Synthesis with Accurate Parameter

Control. In Proceedings of the International Conference on Digital Audio

List of Publications

Effects (DAFx 2020), Vienna, Austria, September 2020.

VII

Karolina Prawda, Vesa Välimäki, and Stefania Seraﬁn. Evaluation of

Accurate Artiﬁcial Reverberation Algorithm. In Proceedings of the 17th

Sound and Music Computing Conference (SMC 2020), Turin, Italy, June

2020.

Author’s Contributions

Publication I: “Evaluation of Reverberation Time Models with

Variable Acoustics”

The idea for the paper stemmed from discussions between the author and

her advisor. The author conducted the measurements used in the study

and analyzed the results. The author implemented all the reverberation

time formulas, the evaluation method, produced all ﬁgures and tables, and

wrote the manuscript with the help of her advisor and supervisor.

Publication II: “Calibrating the Sabine and Eyring Formulas”

The idea for the study took shape over many discussions between the

author and her advisor. The author gathered the measurement data,

performed the analysis, and carried out the air absorption compensation

procedure. The sound absorption calibration was performed by the author

based on the idea and the ﬁrst implementation of the method made by her

advisor. The author produced all ﬁgures and tables, as well as wrote the

manuscript in collaboration with her advisor and supervisor.

Publication III: “Robust Selection of Clean Swept-Sine

Measurements in Non-Stationary Noise”

The idea for the paper was born from discussions between the author

and her advisor. The author gathered the measurements included in the

validation database, conducted the experiments presented in Sections III

and IV, and produced all the ﬁgures and tables. The author conducted the

validation and comparison procedures and wrote 90% of the manuscript.

Author’s Contributions

Publication IV: “Late-Reverberation Synthesis using Interleaved

Velvet-Noise Sequences”

The original idea for the paper came from the author’s supervisor. The

author designed and conducted the listening experiment, improved the

original implementation of the algorithm by adding smearing, segmenta-

tion, and a graphic equalizer as an attenuation ﬁlter, as well as conducted

the comparison and validation procedures. The author produced Figures

1–3 and 7–18 and both tables. The author wrote approximately half of the

manuscript, including Section II-C, Section III-C–E, and Sections IV–VI.

Publication V: “Improved Reverberation Time Control for Feedback

Delay Networks”

The author extended the original idea of her advisor by improving the

attenuation ﬁlter and collaborated with her advisor on the implementation

of the optimization method. The author produced all the ﬁgures and tables

and wrote approximately 90% of the manuscript.

Publication VI: “Flexible Real-Time Reverberation Synthesis with

Accurate Parameter Control”

The author came up with the original idea of the paper, as well as the

technical basis for the plugin, which are described in Sections 2 and 3. The

author participated in the planning of the plugin implementation. The

author conducted the experiments described in Section 4 and produced

Figures 4 and 5. The author wrote 50% of the manuscript, including

Sections 1–2, Sections 3.3–3.4., Section 4.1, and Section 5.

Publication VII: “Evaluation of Accurate Artiﬁcial Reverberation

Algorithm”

The author provided the idea for the paper, conducted both the objective

and subjective evaluation, and analyzed the results. The author produced

all the ﬁgures and tables and wrote 95% of the manuscript.

List of Abbreviations

AR augmented reality

ESS exponential swept-sine

FDN feedback-delay network

FFT fast Fourier transform

FIR ﬁnite impulse response

GEQ graphic equalizer

IIR inﬁnite impulse response

IR impulse response

IRS inverse-repeated sequence

IVN interleaved velvet noise

JND just noticeable difference

LSS linear swept-sine

MLS maximum-length sequence

PCC Pearson’s correlation coefﬁcient

RIR room impulse response

Ro2 Rule of Two

RT reverberation time

SFIR sparse FIR

SNR signal-to-noise ratio

VN velvet noise

VNS velvet-noise sequence

VR virtual reality

List of Symbols

Afeedback matrix

Ai,j element of the feedback matrix

AdB magnitude response of the attenuation ﬁlter

biinput gain of the ith delay line

cioutput gain of the ith delay line

cspeed of sound

ddirect-path gain

Eenergy

ffrequency

fssampling rate

himpulse response

Isound intensity

Ksize of the feedback matrix

lmean free path

Ldelay-line length

msound-absorption coefﬁcient in air

Mnumber of VN sequences in an IVN reverberator

nstationary noise

Nlength of a signal in samples

Ppower

List of Symbols

sioutput of the ith delay line

Ssurface area

t(discrete) time

T60 reverberation time evaluated over 60 dB of decay

T20 reverberation time evaluated over 20 dB of decay

T30 reverberation time evaluated over 30 dB of decay

Vvolume

xinput (excitation) signal in time domain

youtput (microphone) signal in time domain

αˆsound-absorption coefﬁcient

αaverage absorptivity

γdB target gain-per-sample

ρy1,y2Pearson’s correlation coefﬁcient of two signals

ρy1,y2expected Pearson’s correlation coefﬁcient

τtransfer-function variation factor

1. Introduction

Reverberation is an inherent property of sound in physical enclosed spaces,

describing the frequency-dependent decay of sound energy over time. It is

affected by several properties of the space, such as its size and geometry,

the absorbing and scattering properties of its materials, as well as whether

the space is empty or ﬁlled with objects. The acoustic conditions created

by the reverberation make it an important aspect of the design process of

specialized facilities for speech and music, such as concert halls, classrooms,

auditoria, recording studios, and more.

Characterization of the reverberation of a room has been a much re-

searched topic for over a century, ever since the pioneering work by Wallace

C. Sabine was published [

]. Over the decades, multiple techniques for

predicting the sound decay have emerged, starting from straightforward

formulas that simplify the calculations [

], and ending with

complicated procedures that consider the geometry and material properties

of the space in great detail [

]. Although relatively

quick and accurate room acoustic simulations are feasible due to the the

modern computational capacity, reverberation prediction formulas are still

popular among acousticians and researchers. They are used as a point

of comparison for the simulations results, as well as are used in sound

absorption coefﬁcient estimation and measurement. This dissertation dis-

cusses the applicability and accuracy of reverberation prediction formulas,

especially the classical equations by Sabine and Eyring.

The most direct way to learn about the acoustic properties of a space is

to perform a suitable measurement. Techniques of capturing the room’s

sound-energy decay over time, commonly referred to as the room impulse

response, have developed along with reverberation research since the emer-

gence of this branch of acoustics [

]. The methods always include emitting

sound energy into the system under test, which in the case of architectural

acoustics is a room. The energy of such excitation signals must be high

enough to obtain sufﬁcient dynamic range in a possibly broad frequency

range [

] to minimize the negative effect of background noise on the

measurement. At the same time, factors such as time variance and non-

Introduction

Figure 1.1.

Variable acoustics laboratory Arni and the equipment used in room impulse

response measurements.

stationary noise often contribute to measurement uncertainty [

]. A

part of this dissertation is dedicated to estimating the inﬂuence of the

aforementioned aspects of acoustics measurements with the use of the

exponentially swept-sine as an excitation signal. A big database of such

signals was collected by the author in the variable acoustics laboratory

Arni. A system of adjustable acoustic panels in Arni allows for numerous

combinations of absorptivity values and distribution. A view of the labo-

ratory with several panels and measurement equipment is presented in

Fig. 1.1.

Artiﬁcial reverberation aims to recreate the frequency-dependent sound

decay found in physical spaces. It is used as an effect in games, movies, and

music production. The need to add reverberation to recordings predates the

digital signal processing era, with the use of, among others, echo chambers,

plates, and strings [

]. The advent of digital reverberation algorithms in

the 1960s allowed for more ﬂexibility in designing the characteristics of

the decay [

]. At the same time, some problems related to physical

artiﬁcial reverberation, such as high cost, were eliminated or reduced

[

]. The majority of present-day reverberators contain several feedback

loops, which include a delay line and an attenuation ﬁlter. A part of this

dissertation presents solutions that aim at accurate and computationally

efﬁcient reproduction of the characteristics of reverberation. The issues

regarding the perceptual qualities of synthesized impulse responses are

Introduction

discussed as well.

This work is comprised of the research collected in seven peer-reviewed

articles, three of which were published in international journals, whereas

the remaining four were presented at international conferences. These

publications can be topically organized into three groups. Publications I

and II focus on reverberation-time estimations with the use of prediction

formulas, studying their accuracy in numerous conditions regarding the

amount and distribution of sound absorption.

Publication III is centered around impulse-response measurements with

exponential swept-sines, analyzing the inﬂuence of a few external factors

on the measurement. It also introduces a method to reliably determine

whether the captured sweeps are contaminated or devoid of non-stationary

disturbances.

The third group comprises four publications focusing on artiﬁcial rever-

beration algorithms, with Publication IV introducing a method to syn-

thesize impulse responses using interleaved velvet noise. Publications

V–VII are centered around aspects of feedback delay networks, including

attenuation ﬁlters controlling the decay rate, accuracy of reverberation

reproduction, and issues concerning the echo density and reverberation

perception.

The introductory part of this thesis is organized as follows. Chapter

2 describes the decay of sound energy in enclosed spaces and discusses

reverberation-time prediction as well as its accuracy. Chapter 3 provides

an overview of impulse-response measurement techniques and describes

detection of non-stationary noise in sweep measurements. In Chapter 4,

artiﬁcial reverberation algorithms are presented, and the issues of decay-

rate control and echo density are discussed. Chapter 5 summarizes the

scientiﬁc contributions of publications included in the thesis. Chapter 6

offers concluding remarks.

2. Sound Decay in Enclosed Spaces

When the sound produced by a source travels within an enclosed space,

it interacts with both the propagation medium and the surfaces limiting

the space before reaching the receiver (with the exception of the direct

sound that interacts with medium only). While the medium generally

contributes to attenuating the sound energy, the surfaces may either absorb

the sound wave or reﬂect it—either specularly or diffusely [

]. The

reﬂections decay over time and form an impulse response (IR), in the

context of architectural acoustics more speciﬁcally referred to as room

impulse response (RIR).

This chapter explains the propagation of sound through air in enclosed

spaces. It describes the components of an RIR, and presents an intensity-

based model of the sound-energy decay as well as parameters related to

it, with a focus on reverberation time. The inﬂuence of medium-related

absorption on reverberation is discussed. The chapter lists the most pop-

ular reverberation-time prediction formulas and links them with speciﬁc

assumptions regarding the sound absorption conditions in a room. This

chapter presents the evaluation of reverberation predictions using the

most common models, as presented in Publication I and Publication II.

It discusses a few sources of possible uncertainties as well. The chapter

also presents the reduction of such errors via air absorption compensa-

tion and sound absorption coefﬁcient calibration, based on the ﬁndings in

Publication II.

2.1 Sound Propagation in Rooms

Figure 2.1 presents all the basic stages of sound decay in rooms, starting

from the ﬁrst wave that arrives at the receiver after a signal is emitted

from a sound source. This part of an RIR, called the direct sound, most

commonly propagates through the space without obstruction. Thus, the

attenuation along the direct path results from radiation loss, obeying the

inverse-square law, and the viscosity of the air. The delay between the

Sound Decay in Enclosed Spaces

Figure 2.1.

A depiction of an RIR, divided into the direct sound, early reﬂections, and late

reverberation parts.

emission of the direct sound and its reception is determined by the distance

separating the source and the receiver, as well as the medium-speciﬁc and

climate-speciﬁc speed of sound. Hence, the direct sound carries information

about sound source location relative to the receiver [14, 23].

Some time after the direct sound, multiple sound waves arrive at the

receiver. They are already reﬂected from the surfaces within the enclosure,

which include both the space’s limiting surfaces, such as walls, ﬂoor, and

ceiling, as well as objects, e.g., furniture [

]. At ﬁrst, such reﬂections,

called early reﬂections, are typically sparse. Their time of arrival at the

receiver is affected by the geometry of the space, whereas their energy

is inﬂuenced by the absorbing and reﬂecting properties of the surface

materials [14].

With time, the number of reﬂections within the space increases. As the

overlapping sound waves lose their directional cues, the sound energy gets

distributed more evenly throughout the room [

]. When the ability to

discriminate separate reﬂections decreases, the sound ﬁeld approaches

diffuseness and displays statistical properties [

]. This part of the

RIR is referred to as the late reverberation. It is commonly characterized

by its decay rate, which is affected by both medium- and surface-speciﬁc

absorption.

The general model for the sound decay in enclosed spaces can be ex-

pressed in terms of sound intensity [8, 28, 29]:

I(t, f ) = I(0, f) exp −αˆ (f)

lc texp (−m c t),(2.1)

where I(t, f )and I(0, f)are the sound intensities at times tand 0, respec-

Sound Decay in Enclosed Spaces

Sound source switched off

60 dB decay

t= T60

E0 - 60

Time (s)

Energy (dB)

Figure 2.2.

Illustration of the deﬁnition of the RT according to Sabine as the time that

passes from the termination of excitation signal to the sound energy decreasing

to one millionth of its initial value.

tively, and frequency

αˆ

represents the general absorption coefﬁcient of

the room’s surfaces,

is the mean free path, i.e., the mean value of the

distance that a sound particle (ray) travels before it encounters an obstacle

[

], which is determined by the geometry of the enclosure [

the speed of sound propagation in air, and

is the absorption coefﬁcient

of air.

2.2 Reverberation Time

The parameter that is most commonly associated with RIRs and sound

decay is reverberation time (RT). It was introduced in the late 19th century

by Wallace C. Sabine as the time that passes from the termination of the

excitation signal to the moment in which the decaying sound reaches the

threshold of inaudibility [

]. Sabine suggested the threshold value to be a

millionth part of the initial sound energy, which translates to a 60-dB decay,

as shown in Fig. 2.2. Therefore, the common symbol for denoting RT is

T60

where the number 60 refers to the established inaudibility threshold.

It is important to note, however, that achieving 60-dB decay in mea-

sured RIRs is generally difﬁcult due to the presence of noise, especially

given additional requirements speciﬁed in standards (i.e., the the decay

estimation should start 5 dB below the peak level and end 10 dB above

Sound Decay in Enclosed Spaces

the noise-ﬂoor level) [

]. Thus, a number of parameters are used to

quantify RT in real-life conditions, e.g.,

T20, T30

, where the number given

in the subscript speciﬁes the dB-range over which the decay was evaluated

(i.e., 20-dB decay for

T20

, 30-dB decay for

T30

) [

]. Since the theoretical

RT predictions are noise-free, in the present dissertation the symbol

T60

used to underline the compatibility with Sabine’s deﬁnition.

Applying Sabine’s RT deﬁnition to Eq. (2.1), we see that

I(T60(f , m)) = 10−6I(0)

for

t=T60

. Solving this equation for

T60

, we

see that

T60(f , m) = ln(106)l

c(αˆ(f) + l m),(2.2)

which is the most general model for calculating RT values based on the

room geometry and sound absorption due to both surface absorptivity as

well as the propagation medium. In the remainder of this dissertation,

T60(f , m)

denotes RT values when the air absorption is considered, whereas

T60(f , 0) = T60(f)stands for RT without air absorption.

2.2.1 Reverberation Time Formulas

Sabine’s initial experiments led to establishing the relation between the

RT, absorption in the room, and room’s dimensions, which was further

transformed into the ﬁrst formula to calculate the RT values, nowadays

written as

T60(f) = 0.164 V

S α(f),(2.3)

where

is the volume of the space in

is the room surface in

, and

is the frequency-dependent average absorptivity in the room, deﬁned

α=iSiαi/S

, where

are the surface areas in

and

αi

are the

corresponding absorption coefﬁcients of each wall.

The constant 0.164 was derived heuristically by Sabine, but often does

not appear as such in the literature, assuming a range of values between

0.16 and 0.164 [

]. We can see that Eq. (2.2) results from the

term

ln(106)l/c

. Hence, this value depends on the shape of the room and

the considered atmospheric conditions that affect the propagation speed.

For example, for a shoebox room with

l= 4 V /S

[

] and the standard

conditions of temperature, humidity, and atmospheric pressure (leading to

c= 343 m/s), we obtain the value 0.161.

Sabine’s formula assumes that the sound in the enclosure decays con-

tinuously [

]. This implies that the energy of sound at a given point

in time needs to be equal everywhere within the room. This means that

the sound ﬁeld needs to be perfectly diffuse, meeting the requirements

of homogeneity and isotropy, which are unattainable in the majority of

rooms [

]. On the contrary, the late part of an RIR often expresses

directional properties even in big halls, which are more likely to achieve a

good level of diffusion than small rooms [38].

Sound Decay in Enclosed Spaces

The earliest modiﬁcation to the Sabine equation was made by Carl F.

Eyring in 1930 [

]. Instead of being continuous, the sound decay was

assumed to be discrete, with energy losses after each reﬂection from the

room’s surfaces [2, 36]. Thus, the Eyring formula emerged:

T60(f) = 0.164 V

−Sln(1 −α(f)) .(2.4)

Due to the different approach to applying the sound-absorption coefﬁ-

cient in the Eyring equation, the predicted RT is always lower than that

obtained with Sabine’s model for the same values of absorptivity and

α > 0

Additionally, in the case where the room’s surfaces are perfectly absorptive

(

α= 1

), Eq. (2.3) returns an incorrect non-zero result, whilst the denomi-

nator of Eq. (2.4) results in

ln(0)

, which is commonly interpreted as a RT of

0 s [

]. Despite the common understanding that the Eyring formula gives

accurate predictions for high values of

, literature suggests otherwise

[

], claiming that Eyring’s predictions are correct for small values

of average absorptivity (α < 0.5[22, 40]).

Over the next decades, more researchers felt compelled to study the

reverberation theory and created their own prediction formulas based on

modiﬁcations of Sabine’s and Eyring’s equations. The main focus of those

changes was on the distribution of absorption at the room’s surfaces. Both

Sabine’s and Eyring’s formulas assume that the absorption is distributed

evenly within the enclosure, which is rarely the case. Thus, the devel-

opment of the reverberation theory aimed at increasing the predictions

accuracy when the wall absorption coefﬁcients were signiﬁcantly different

from each other.

Figure 2.3 illustrates the use cases for the most commonly used formulas

for

T60

prediction, which are also detailed in Publication I. Apart from the

Sabine’s and Eyring’s equation (Fig. 2.3(a) and (b), respectively) the list

includes the following:

•

Millington-Sette’s formula, which assumes that each wall has a slightly

different average absorption [3, 4], shown in Fig. 2.3(c),

•

Kuttruff’s formula utilizes a statistical approach and introduces the

concept of a relative variance of mean path length [

], and assumes that

each wall has a different absorption coefﬁcient, as shown in Fig. 2.3(c),

•

Fitzroy’s model, the ﬁrst one to consider separate decays along three

axes, equivalent to the three basic axes in a rectangular room (X, Y, Z)

[5], as presented in Fig. 2.3(d),

•

Arau’s formula, which is parallel to Fitzroy’s idea, assumes that the total

reverberation in a room is a geometric average of three axial (X, Y, Z)

decays [6], presented in Fig. 2.3(d), and

Sound Decay in Enclosed Spaces

𝛼 < 0.2 𝛼 < 0.5

𝛼≠𝛼≠𝛼≠𝛼

𝑦

𝑥

𝑧

𝛼≠𝛼≠𝛼𝛼≠𝛼

Sabine Eyring

Millington-Sette

Kuttruff

Fitzroy

Arau Neubauer

(a) (b)

(c)

(d) (e)

Figure 2.3.

Representation of use cases for the RT prediction formulas. (a) Sabine’s

formula is commonly considered accurate when the absorption is small and

distributed evenly on the room surfaces. (b) Eyring’s model is meant for rooms

with moderate, but evenly distributed absorption. (c) Millington-Sette’s and

Kuttruff’s formulas are aimed at spaces where each wall is characterized by

a different absorption coefﬁcient. (d) Fitzroy’s and Arau’s equations assume

decays along the three major axes of the room (X, Y, Z). (e) Neubauer’s model

separates the decay in the XY-plane from the decay along the Z-axis. In all

cases, the respective surfaces and absorptivity symbols are marked with the

same color.

Sound Decay in Enclosed Spaces

•

Neubauer’s equation, an extension to Fitzroy’s approach, in which the

total decay is a weighted average of two components, that in the XY-plane

and that along the Z-axis [7, 34, 35, 41], as depicted in Fig. 2.3(e).

2.2.2 Effect of Air Absorption

Early reverberation theories assumed the attenuation of sound was caused

only by the surface absorption, neglecting entirely or to some extent the

attenuating effect of air. The absorption coefﬁcient of air, as shown in

Eqs. (2.1) and (2.2), was introduced by Knudsen [

] and commonly appears

in the denominator as 4m V , in the example of Sabine’s formula as

T60(f , m) = 0.164 V

Sα(f)+4m V .(2.5)

Knudsen also determined the values of

under different conditions of

temperature and humidity [

], and conducted an in-depth research

on the attenuating properties of different media, mainly various types

of gases [

]. The effect of humidity was also analyzed,

among others, by Harris [

], who described changes in

the speed of sound in various atmospheric conditions. Pielenmeier, who

was also interested in similar issues [

], focused additionally on

ultrasonic propagation [

]. Bass and his numerous collaborators

developed a family of curves that describe the air absorption of sound in

still atmosphere [

], which were later used in standardized

methods to estimate sound attenuation in air [

]. The research on

this topic, however, is mostly applied to the outdoor propagation of sound

[67, 68, 69, 70, 71, 72, 73].

Although the effect of air on the RT values may be signiﬁcant, especially

at high frequencies, the coefﬁcient

is often excluded from RT calculations

[

], or it is applied to the formulas selectively [

]. The sound absorp-

tion in air is commonly omitted for volumes under 200 m

[

], when

measurements or simulations are performed within the audible frequency

range. Neglecting the effect of air absorption may, unfortunately, lead to

considerable errors, and is sometimes recognized as a source of uncertainty

in RT calculations [75].

Publication II shows the effect of bias introduced by air absorption on

the example of RT values. The measurement results performed on days

with signiﬁcantly different atmospheric conditions are compared, and the

difference in obtained RTs is shown. Even though the volume of the mea-

surement space was below 200 m

, the inﬂuence of the air attenuation of

sound was considerable, in the most extreme cases lowering the RT values

by 50% compared to air-absorption-compensated numbers. Therefore, the

effect of air absorption must be taken into account. This is crucial when

the RT measurements from different days are analyzed or compared to

Sound Decay in Enclosed Spaces

results of predictions or computer simulations.

2.2.3 Air Absorption Compensation

The effect of air absorption on the decay of sound, albeit often unaccounted

for in RT calculations, is signiﬁcant enough that compensation is needed.

Numerous techniques to compensate for air absorption propose to extend

RIRs by applying frequency- and time-dependent ﬁltering to cancel the

medium-speciﬁc attenuation [

]. Such procedures are mostly

tailored to outdoor propagation and scale-model measurements [

]. They are, however, not free from problems, mostly related to

an excessive increase in background-noise level, which is ampliﬁed in the

same way as the useful part of the RIR [80, 81, 82, 83].

In many applications, extending the RIR is necessary to compensate

for the air absorption. However, when results of

T60

are considered, e.g.,

in sound absorption measurements [

], certain arttiﬁcial reverberation

algorithms [

] or in the case of measurements in Publication II, such a

procedure can be omitted. The effect of the air absorption can be removed

by simply subtracting the part related to coefﬁcient

from the obtained

values:

T60(f) = 1

T60(f , m)−c m

ln(106)−1

.(2.6)

Although the formulas to estimate the pure-tone attenuation of sound in

air are well developed and standardized [

], there is little agreement

as to what value to choose when full-octave frequency bands are considered

[

]. Thus, Publication II examines the approaches to determine

in octave bands. The comparison is between the pure-tone absorption

coefﬁcient, the coefﬁcients for the center frequency of octave bands, and a

procedure proposed in Publication II. In the last method, the full-octave

band

is obtained by averaging over an arbitrary number of pure-tone co-

efﬁcients from that band. Publication II shows that the averaging removes

the bias related to air absorption from the results of RT measurements.

This approach also performs well in comparison to other methods, giving

robust and reliable results. Thus, the averaging of pure-tone air absorption

coefﬁcients is shown to be applicable in air-absorption compensation.

2.3 Reverberation Time Prediction

The numerical prediction of a room’s reverberation is an important part of

acoustical simulations and of the design of specialized spaces for speech

and music, such as concert halls, auditoria, recording studios, and others.

Additionally, the RT is directly related to other room parameters, such

as clarity and deﬁnition [

], as well as speech intelligibility [

Sound Decay in Enclosed Spaces

] and listening effort [

], making it a critical parameter in the

acoustic adaptation of schools and classrooms.

The attempts to overcome the limitations inherent to Sabine’s model

comprised expanding his initial idea and developing new equations (see

Sec. 2.2.1). Nowadays, the number of available models fuels a need to

evaluate their applicability to speciﬁc spaces and absorption conﬁgurations.

Considering the signiﬁcance of RT as an acoustic descriptor, multiple

efforts to assess the accuracy of predictions were undertaken.

2.3.1 Evaluation of Reverberation Prediction Formulas

Among many attempts to evaluate the accuracy and applicability of rever-

beration prediction formulas, there is no agreement in the results. Numer-

ous studies declare different formulas to be in best agreement with either

measurements or RT values obtained in simulations using specialized

software. The invention of every new equation was accompanied by a claim

of increased reliability compared to the earlier methods [

Bistafa and Bradley [

] as well as Rossell and Arnet [

] show that Arau’s

formula causes the smallest average error, Neubauer points to his own

formula [

] as being the most reliable, and Nowo´

swiat and

Olechowska also prove the accuracy of their own approach [

To address this issue, the author conducted her own examination of

accuracy of the seven reverberation models presented in Sec. 2.2.1. In most

cases, the literature analyzes the predictions either in one measurement

space with few absorption variations [

] or in several different spaces

[

]. The research presented in Publication I investigates the reliability of

the aforementioned equations to predict the RT values for a diverse set of

absorption conditions. The measurements were performed in the variable

acoustics laboratory Arni, a facility within the Acoustic Laboratory of

Aalto University in Espoo, Finland. Arni is equipped with 55 adjustable

panels, whose state can be switched from absorptive to reﬂective, allowing

for a variety of RT values, ranging from 0.2 s to 1.5 s, depending on the

frequency band. The layout of the room and the placement of the panels

are presented in Fig. 2.4.

In Publication I, the measurement of each condition—from 0 to 55 absorp-

tive panels—was presented. Increasing the absorptivity in the room led to

a proportional decrease of the RT values. These numbers are compared

with the seven reverberation formulas presented in Sec. 2.2.1. The analy-

sis showed that, in principle, none of the models predicted the measured

RT values accurately. Fitzroy’s equation came closest to the reference,

although the estimations were on average burdened by a relative error of

more than 10% of the measured RT. The relative errors of the remaining

formulas were greater than 20%.

Sound Decay in Enclosed Spaces

Figure 2.4.

Layout of the variable acoustics laboratory Arni in which the measurements

were performed.

2.3.2 Absorption Coefﬁcient Calibration

The poor performance of the RT prediction formulas demonstrated in

Publication I led to the presumption that the problem is due to multiple

sources of uncertainty. One of them is the air absorption of sound, covered

in Sec. 2.2.2. The remaining ones include: the assumptions regarding the

sound ﬁeld in the room and the sound absorption coefﬁcient values. Here,

the focus is on the latter.

The estimation of the sound absorption coefﬁcient values for RT predic-

tion is troublesome and burdened with uncertainty. Unlike other parame-

ters in the RT formulas, such as the room dimensions and air absorption,

sound absorption coefﬁcient is difﬁcult to estimate reliably. The issue of

the robustness and reliability of the sound-absorption coefﬁcient estima-

tion via measurements is a subject of scientiﬁc debate since the 1930s,

which eventually came to be known as the “absorption coefﬁcient prob-

lem” [

100

101

]. Despite improvements to absorption coefﬁcient

Sound Decay in Enclosed Spaces

measurement methods [

102

103

], the dispute persists [

104

105

106

107

Similarly, the sound absorption coefﬁcients are many times recognized as

a signiﬁcant source of discrepancies between results of acoustic measure-

ments and simulations. As such, they often play an important role in the

calibration of RT prediction models. In the literature, the minimization of

the difference between measured and predicted values, e.g., of RT or clarity

parameters, often involves adjusting the sound absorption coefﬁcients of

room’s materials [

108

109

110

]. The focus of these studies, however,

is rarely on the RT prediction formulas [

108

111

], and more commonly on

geometrical acoustic models [75, 109, 110, 112].

The goal of the research presented in Publication II is to prove that the

simplest reverberation prediction models—namely Sabine’s and Eyring’s

formulas—achieve good accuracy in calculating RT values. This incorpo-

rates removing, or at least decreasing, the uncertainty related to the sound

absorption coefﬁcient.

To this end, a big dataset of RIRs was collected in Arni. It comprised of

over 5 000 combinations of absorptive and reﬂective panel combinations,

allowing for a considerable diversity of absorption distribution conditions.

After removing the bias introduced in the results by the air absorption

as done in Eq. (2.6), the absorptivity for each combination was estimated

from measured RT values based on the modiﬁed Eq. (2.2):

αˆ(f) = ln(106)l

c T60(f).(2.7)

The absolute difference between the measured and assumed (based on

material data and information from the literature) absorptivity serves

as a measure of error for the calibration process of Sabine’s and Eyring’s

formulas.

The results of Publication II show that after adjusting the absorptivity,

the majority of the measured RT values ﬁt within

10% of the predicted

ones, showing that these classical formulas can estimate the reverberation

of a room well regardless of the absorption distribution. Both equations

also show good accuracy for the linear change in absorptivity conditions

presented in Publication II, proving good scalability of the RT predictions.

3. Impulse Response Measurement

The RT of an enclosure can be estimated using prediction formulas or with

virtual acoustic simulations, as already mentioned in the previous chapter.

However, the most direct way to gain knowledge about the acoustics of

a room is through measurements. Nowadays, high-class equipment and

specialized software allow for quick and accurate measurements [

113

114

115

]. The abundance of available excitation signals offers good applicability

for various measurement scenarios [17, 116, 117].

This chapter presents a few of the most common techniques to measure

RIRs, with an emphasis on the exponential-sine-sweep method. The ad-

vantages and drawbacks of each technique are mentioned. The chapter

also presents the phenomena appearing during RIR measurements, such

as stationary noise, time variance and non-stationary noise, and their

negative inﬂuence on the aforementioned methods. Lastly, a method to

detect non-stationary disturbances in sweep measurements is described.

3.1 Measurement Techniques

The techniques to capture RIRs have been evolving since the early days

of acoustic measurements. Starting from methods that produce transient

sounds through sophisticated albeit impractical analog devices, such as

Sabine’s pipe organ [

], and various types of digitally generated impulse-

based excitation signals, all of the techniques aim to produce sound that

can be used to extract information about the acoustical properties of a

space.

One of the most intuitive methods to measure an RIR is to generate an

impulse signal and record it together with the decay. The impulse signal

can be generated digitally as a short burst of broadband noise [8], as well

as by using analog sources. Such techniques include the use of signals

originating from hand claps [

118

119

120

121

], balloon pops [

122

123

gunshots [124, 125], and others [119, 126].

The advantage of analog impulse sources is that they are generally

Impulse Response Measurement

straightforward to use and do not always require powerful and costly

equipment, such as dodecahedron loudspeakers [

127

128

]. On the other

hand, the spectra of impulses generated with analog sources usually is

not ﬂat and may vary between measurements [

120

]. The energy of an

impulse may prove too low to achieve sufﬁcient dynamic range to estimate

room acoustic parameters, especially in the low-frequency range [

120

Additionally, analog sound sources may not prove powerful enough to

ensure an even distribution of sound energy within the entire enclosure.

One way to overcome some of the drawbacks of impulse signals in acous-

tic measurements is to spread the energy of sound over time. This can be

accomplished by emitting a sound with a ﬂat spectrum, e.g. white noise,

into the measured space. However, methods utilizing pseudo-random noise,

such as Maximum-Length Sequence (MLS) [

129

130

131

], Inverse Re-

peated Sequence (IRS) [

132

], and similar [

133

] gained more popularity

than using purely random noise. The advantage of excitation signals based

on pseudo-random noise is their immunity to transient noise [

129

130

However, the presence of “distortion peaks” originating from nonlinearities

in both the measurement equipment and the system under test reduces

their applicability [17, 130].

3.1.1 Time-Stretched Pulses

Another family of measurement methods is based on the expansion and

compression of an impulse [

]. First, the impulse signal is stretched over

time, so that its frequencies are not all excited at once, but consecutively.

This results in a time-domain signal resembling a sine with varying in-

stantaneous frequency. When the measurement is ﬁnished, the obtained

signal is temporally compressed so that the frequencies appear again at

the same time instance followed by the measured sound decay [134, 135].

When the change in the excited frequency over time is linear, the ex-

citation signal is referred to as the linear swept-sine (LSS). The LSS is

depicted in the top part of Fig. 3.1. Since all the frequencies are emitted for

the same amount of time, the spectrum of the LSS is ﬂat in the frequency

region of interest [134, 135].

In an exponential swept-sine (ESS), the time allocated to excite speciﬁc

frequencies changes exponentially, with the low-frequency part of the

signal being emitted for a longer time than the high-frequency part [

as shown in the bottom part of Fig. 3.1. This results in the ESS having

a pink spectrum—the magnitude response decreases as the frequency

increases. This property is advantageous in most of the measurement

scenarios, because the background noise is most prominent in the low

frequencies, often having a detrimental effect on the dynamic range of

the measured signals [

136

137

]. The ESS variation in which the

excitation time of speciﬁc frequencies is controlled through an iterative

Impulse Response Measurement

Figure 3.1.

(Left) Time and (right) time-frequency representation of swept-sine measure-

ment signals. Top to bottom: LSS and ESS.

process to achieve the target dynamic range is also studied in the literature

[138, 139, 140, 141, 142, 143, 144].

The length of the sweep is its most important parameter. An appro-

priately chosen signal duration is crucial not only to obtain sufﬁcient

dynamic range, but also to minimize the measurement uncertainty and

to push the distortion artifacts into non-causal part of the deconvolved

RIR [

136

137

145

146

147

148

]. At the same time, the vulnerability

of the LSS and ESS to non-stationary noise and excessive time variance

increases proportionally to the signal duration [19].

3.2 Factors Inﬂuencing Measurements

In room acoustic measurements, any type of acoustic event different from

the observed signal is considered to be noise which may have a destructive

effect on the measurement, affecting the captured RIR and introducing

error to the estimated parameters [

149

]. The selection of the type of

excitation signal might be based on the characteristics of background

noise in the system under test and the possibility of the occurrence of

non-stationary noise, and the presence of factors amplifying time variance.

3.2.1 Stationary Noise

At this point in the dissertation, the division of ambient noise into station-

ary and non-stationary noise needs to be made. In the present work, any

noise event that exhibits constant properties, such as frequency content

Impulse Response Measurement

and energy, through the entire measurement is considered stationary. An

example is noise caused by the measurement equipment. If these proper-

ties change, such an event is considered to be non-stationary. An example

is randomly occurring transient noise. Stationary noise is, in principle,

unavoidable, whereas non-stationary noise may or may not occur during

the measurement procedure. In the remainder of this work, the terms

“background noise” and “ambient noise” are, therefore, used to refer to the

stationary noise only.

Stationary noise is mainly characterized by two properties: its frequency

response and energy (or power). Usually, such noise has the biggest portion

of its energy in the low-frequency region, but occasionally high-energy mid-

and high-frequency components occur as well. Noise can be described with

numerous parameters, but in this dissertation the signal-to-noise ratio

(SNR) is of interest, since the noise is not analyzed on its own, but in

relation to the measurement signal.

The SNR is an important parameter from the standpoint of acoustic

measurements since it quantiﬁes the dynamic range of the RIR, i.e., the

distance between the loudest element of the RIR (usually the direct sound)

and the noise ﬂoor, under which the decay is masked by the ambient noise.

Depending on the parameter to be extracted from the RIR, the requirement

posed on the dynamic range varies, as already discussed in Chapter 2.2.

The SNR is usually expressed with the ratio of the power of the signal of

interest (excluding noise) to the power of noise itself. Since the power may

be expressed in terms of the signal energy

P[x] = 1



t=1

|x(t)|2=1

NE[x],(3.1)

where

is the length of the signal

in samples, the SNR can be calculated

as the ratio of powers or energies:

SNR =P[x]

P[n]=P[y]−P[n]

P[n]=E[y]−E[n]

E[n]=E[x]

E[n],(3.2)

where

is the measured signal comprising of the signal of interest

and

noise component

. In the two formulas above, the notation with braces

imply that the parameter is calculated for the entire discrete signal, such

P[x]

is the power of the signal

. The brackets indicate operation on a

single sample (or a portion of samples) of a signal, i.e.,

|x(t)|2

is the squared

absolute value of the signal xat a discrete time t.

3.2.2 Transfer-Function Variation

Another type of contamination in acoustic measurements that is, in prin-

ciple, unavoidable, is time variance or transfer-function variation, which

results from changes in the measurement environment [

150

151

136

Impulse Response Measurement

149

152

]. For the majority of the measurement techniques to be fully

accurate, the system under test needs to be time-invariant, a requirement

that is inevitably violated in practically any room acoustic measurement

scenario. MLS is particularly vulnerable to time variances, but other tech-

niques, such as the ESS, whilst demonstrating more robustness to such dis-

turbances, may still produce erroneous results [

136

151

]. Sound-pressure

level, reverberation time, and clarity index are among the parameters that

are commonly affected by time variances [18, 149].

There are two categories of time variances: interperiodic and intrape-

riodic. The tempo of the former kind is slow, causing differences to be

noticeable when two separate measurements are compared [

151

]. In-

terperiodic time variances are caused by ﬂuctuations in the atmospheric

conditions between the measurements, such as changes in temperature

and humidity that affect the sound propagation speed [

149

151

]. As a

result, a phase shift is created between two measurements along with a

change in the time delay between the emission and capture of the sound.

This causes errors in the averaging process when methods such as MLS

and swept-sine with synchronous averaging (when multiple measured

signals are averaged) are used [

151

153

]. This effect is more

pronounced at high frequencies [18, 19, 149, 154].

Intraperiodic time variances are quick enough to induce signiﬁcant

changes to a single measurement [

151

]. They stem from sudden

changes in temperature, air movements due to wind, ventilation, hu-

man interference, or displacement of the measurement equipment due to

vibration [18, 149, 151, 152].

Quantifying the effects of time variance on the measurements is a dif-

ﬁcult task plagued by considerable uncertainty. This is due to the un-

predictable nature of the time variances and difﬁcult, if not impossible,

monitoring of relevant phenomena. In the literature, quantiﬁcation based

on measurements is often done by comparing several recorded signals or

the parameters obtained from them, e.g. RT values [

136

149

]. Usually,

the assumption of such procedures is that one of the signals acts as a

“reference” against which the others are compared. This implies that this

“reference” should be devoid of both inter- and intraperiodic time variances,

which is hardly ever the case, thus rendering such methods unreliable.

Predicting the inﬂuence of time variances is, on the other hand, done

separately for inter- and intraperiodic changes. In the case of the former

type, the most established method is time stretching [

136

149

151

In the latter kind, delay modulation might be used to model the properties

of rapid air ﬂuctuations [

]. The results of both approaches are difﬁcult

to relate to real-life measurements. Additionally, when using the delay

modulation technique, an assumption of periodic changes is made, which

is rarely, if ever, true [18].

Impulse Response Measurement

3.2.3 Non-stationary Noise

Non-stationary noise is a type of disturbance that lasts shorter than the

duration of the excitation signal with sound decay. Transient sounds, short

bursts of non-impulsive noise, and sound dropouts caused by errors in the

measurement equipment (when a part of the signal is not emitted or not

recorded) belong to this category.

As discussed before, MLS and IRS techniques are particularly immune

to impulsive noise, whilst the effect of other types of non-stationary dis-

turbances were not researched in their context. On the other hand, the

LSS and ESS signals are vulnerable to all the aforementioned kinds of

non-stationary noise, which may introduce considerable error in the RT

estimation [

149

]. This is true as long as the frequency spectrum of the

disturbance and the instantaneous frequency of the signal intersect. If, on

the other hand, these frequencies are different, after deconvolution the

disturbance may be pushed to the non-causal part of the RIR and can be

discarded.

3.3 Sweep Measurements in Noisy Environments

Since the sweep’s vulnerability to non-stationary noise poses a problem

in the measurements, a method that performs a reliable detection of such

disturbances in captured sweep signals is needed. Although such a method

was proposed by Guski [

149

], it is burdened with a few sources of uncer-

tainty, such as separating the measured sweep with the sound decay from

the rest of the recorded sound.

3.3.1 Rule of Two

Publication III introduces a different approach, called Rule of Two (Ro2) in

which the selection of a pair of clean sweeps from a series of measurements

is based on the similarity between measured ESS signals. This similarity

is expressed by the means of Pearson’s correlation coefﬁcient (PCC).

The basic principle of the Ro2 procedure is that in a repeated measure-

ment, ESS signals devoid of non-stationary noise display high similarity

to each other and low similarity to sweeps that are contaminated with

non-stationary disturbances. This translates to a high and low value of

PCC, respectively. To categorize the measured signals into one of these

classes, a detection threshold is required. Therefore, the Ro2 is formulated

if ρy1,y2>ρy1,y2then y1and y2are a clean pair,(3.3)

where

and

are two ESS (microphone) signals,

ρy1,y2

is their measured

correlation, and ρy1,y2is the detection threshold.

Impulse Response Measurement

The deﬁnition of PCC requires a minimum of one pair of measurements

for the comparison. However, if the non-stationary noise occurs in at

least one of the two signals,

ρy1,y2

cannot directly indicate which one is

contaminated. Therefore, if the correlation is lower then a threshold value,

another ESS needs to be captured, and its correlation to the previous

signals needs to be compared. The measurement can end when at least

two sufﬁciently clean sweeps are captured, so that the PCC values point

out the contaminated ESSs.

The PCC threshold

ρy1,y2

is estimated with respect to the correlation re-

duction resulting from the expected contamination—stationary noise and

time variance. The expected PCC is ﬁrst determined based on the measure-

ment’s SNR. However, this initial threshold value is calculated with the

assumption that the stationary noise terms of all compared measurements

are uncorrelated, which might be, in fact, incorrect.

In Publication III, the noise terms in the sweep measurements are treated

as correlated—two extreme cases of fully correlated and anticorrelated

noise are considered. However, two perfectly correlated noise terms are

virtually indistinguishable from the parts of the measurement that include

the ESS signals. Thus, in Publication III, the threshold

ρy1,y2

is calculated

with the assumption of anticorrelated noise. Such an assumption accounts

for the possibility of the background noise containing harmonic content

(such as noise from electric devices). It also poses a less strict detection

threshold than the assumption of fully uncorrelated Gaussian noise.

Since the expectation of the time-invariant system is not true in most

cases of the RIR measurements, the transfer-function variation acts as

an expected contamination that affects the similarity between the sweeps.

Therefore, another adjustment in the detection threshold that depends

on the values of transfer-function variation factor

, determined from

the time-variance energy, is introduced in Publication III. Not having a

“reference” RIR captured in a perfect time-invariant environment prevents

us from reliably quantifying the amount of time variance in each of the

measured signals.

Publication III proposes to infer the transfer-function variation energy

from the difference of energy between the two measured ESSs, and chooses

that each sweep contributes equally to this difference. Thus, the transfer-

function variation factor is obtained

τ=E[y1−y2]−E[n1−n2]

E[x∗h],(3.4)

where

and

are noise terms associated with the measured signals

and

, respectively, and input (excitation) signal

is convolved with the

IR of the system under test, h.

Impulse Response Measurement

3.3.2 Median in Non-Stationary-Noise Detection

A necessary component for selecting clean ESSs in the Ro2 is a robust esti-

mation of noise power (energy) used in the detection-threshold calculations.

Both power and energy are subject to contamination by non-stationary

noise, such as impulses and short noise bursts. Although power is essen-

tially a mean value of squared noise samples (and energy is a scaled mean

value), any non-stationary disturbance skews it in the positive direction,

since it contains high-valued samples acting as outliers. This might lead

to an incorrectly calculated detection threshold and, as a result, falsely

classiﬁed sweeps.

When the stationary noise has a Gaussian distribution, its squared

samples follow the chi-squared distribution with one degree of freedom.

Although for chi-squared distribution mean and median do not have the

same value, as it happens for the Gaussian distibution, the relation be-

tween the mean and median for chi-squared distribution is known [

155

Thus, in Publication III we propose using the median as a more robust

estimator. With a breakdown point of 0.5 (compared to zero for the mean),

such an estimator is not affected if less than half of the signal’s samples are

contaminated [

155

156

157

]. We show that the change of estimator suc-

cessfully increases the resilience of noise-power estimation in the presence

of non-stationary noise.

Similarly, the transfer-function variation factor

is vulnerable to contam-

ination by non-stationary noise. Since

is estimated from a pair of mea-

sured sweeps, its value is higher if one of the ESSs contains non-stationary

noise. In Publication III,

is ﬁxed for the entire set of measurements, and

its distribution contains many high-valued outliers. Again, the median is

used as a base for the estimation of τ.

4. Reverberation Synthesis

Artiﬁcial reverberation is an audio effect that augments audio and music

with an impression of space by imitating or enhancing decay characteristics

of sound. This effect is used in numerous ways and in a broad range of

applications, most notably in audio production of music, ﬁlms, and games

[

158

159

160

]. Artiﬁcial reverberation can be added to dry recordings to

enhance the ambiance. In such cases, the produced reverberation can be

adjusted to an extent to ﬁt the aesthetic climate of the entire production.

However, a more precise and realistic decay synthesis is needed in virtual

and augmented reality (VR and AR, respectively), and when simulating

architectural acoustics in the design process of spaces for speech and

music [

161

162

]. Another application for artiﬁcial reverberation are

reverberation-enhancement systems, aimed at modifying the acoustics of

architectural objects digitally, without changing the properties of the space

itself [163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173].

The history and development of sound-decay synthesis dates back many

decades. Early techniques of artiﬁcial reverberation focused primarily on

appending or extending reverberation for the purpose of ﬁlm and music

production. Physical reverberation production techniques involved echo

chambers [

174

175

], plates [

176

], and springs [

177

], which offered rich,

diffuse sound, at the same time being relatively inﬂexible and high in cost.

Analog methods, such as tape delays [

176

178

] and oil-can delays [

179

]

allowed for more freedom in sound design, creating abstract, psychedelic

effects. Many of the aforementioned methods are still used to this day

to achieve speciﬁc artistic goals. Additionally, the efforts to model them

digitally make them more applicable in modern audio production [

180

181

182, 183, 184, 185, 186, 187, 188, 189].

The digital artiﬁcial-reverberation architectures available nowadays

can be roughly divided into three groups: convolution algorithms, delay

networks, and physical room models [

]. Due to their high computational

complexity, models based on room geometry are usually used for off-line

rendering and acoustic simulations [

190

191

192

193

194

]. They can

also be utilized in hybrid approaches, where they serve as early reﬂections

Reverberation Synthesis

generators, whilst the later part of an RIR is produced with delay-based

methods [195, 196].

The convolution of an audio signal with a measured or simulated RIR

produces rich and high-ﬁdelity reverberation. However, since convolving

an RIR is equivalent to using a ﬁnite impulse response (FIR) ﬁlter of the

same length, the computational cost of such an operation is high, especially

for long decays [

197

198

]. An improvement to the computational load

of convolutional reverbs is made with the use of the fast Fourier transform

(FFT) [

199

200

] and block processing of the input signal and the RIR [

Of the three groups, the focus of this dissertation is on the delay-based

artiﬁcial reverberation. In this chapter, the digital artiﬁcial-reverberation

methods are presented, with an overview of both early and recent tech-

niques. The use of attenuation ﬁlters is analyzed as well, with an eye

on an accurate approximation of the target RT values and avoiding ﬁlter

instabilities. The ability of certain techniques to produce perceptually

dense RIRs is also discussed, followed by investigations of the subjective

reception of synthesized reverberation.

4.1 Delay-Based Reverberation

The ﬁrst digital artiﬁcial-reverberation technique was invented 60 years

ago, when Manfred Schroeder and Ben Logan introduced a delay system

in which two comb ﬁlters—feedback and feedforward—were connected to

create an allpass ﬁlter [

]. Such an intervention allowed for a structure

that produced a series of decaying echoes with a ﬂat magnitude response.

Schroeder’s system created sound devoid of strong coloration speciﬁc to

using only a feedback or a feedforward comb ﬁlter. The structure that

utilized both feedback and allpass ﬁlters allowed for control over the decay

rate and ensured an increase in echo density over time [14].

4.1.1 Feedback Delay Networks

The early work by Schroeder was the foundation for an algorithm intro-

duced by Gerzon [

201

], who arranged comb-ﬁlter sets into a recirculating

network interconnected with an orthogonal matrix, ensuring no loss of

energy outside of the operation of attenuation ﬁlters. The architecture

was adapted by Stautner for multi-channel reproduction [

202

], and later

revisited by Jot, who redeﬁned the design [

203

]. Jot’s formulation of

the feedback delay network (FDN) resulted in the algorithm’s popularity,

making it the state-of-the-art artiﬁcial-reverberation architecture to the

present day.

The block diagram of an example FDN is shown in Fig. 4.1. In the time

Reverberation Synthesis

z-L

s (t)

y(t)x(t)

s (t)

Figure 4.1. Block diagram of an FDN consisting of three delay lines.

domain, the FDN can be expressed as [30, 204]

y(t) =



cisi(t) + d x(t),(4.1a)

si(t+Li) =



Ai,j sj(t) + bix(t),(4.1b)

where

x(t)

and

y(t)

are the input and output signals, respectively, at

discrete time

si(t)

is the output of the

th delay line, and

Ai,j

is the

element of a

K×, K

feedback (scattering) matrix

that interconnects all

the delay lines.

and

are input and output gains, respectively, and

the direct-path gain.

The versatility of the FDN architecture allows for a wide variety of

applications and extensions. It is used to produce binaural [

195

196

] and

multichannel reverberation [

205

206

], as well as to synthesize multiple-

slope decay for coupled spaces [

207

208

]. Other artiﬁcial reverberation

techniques, such as scattering delay networks [

209

210

], digital waveguide

networks [

204

], digital waveguide mesh [

211

], and ﬁnite difference time

domain methods [

191

192

193

194

] have a close relation to the FDN

structure.

4.1.2 Velvet-Noise Reverberators

As the number of echoes in an RIR grows over time, after a certain point

the decay resembles random noise with an exponentially decaying enve-

lope [

212

213

214

215

216

]. Therefore, some artiﬁcial-reverberation

algorithms utilize this property to model the late part of the RIR with ran-

dom or pseudo-random signals [

214

215

216

]. In practice, since Gaussian

noise has a ﬂat spectrum, it cannot be used in reverberation production

unprocessed. Filtering is necessary to imitate both the air and surface

Reverberation Synthesis

0 50 100 150 200 250 300 350 400

Sample index

-1

Signal value

0 50 100 150 200 250 300 350 400

Sample index

-1

Signal value

Figure 4.2.

(Top) VN with one non-zero pulse occurring once in every 20 samples. (Bottom)

Example of an IVN comprising four sequences. Each sequence is marked with

a different line style. In both panes, the blocks of 20 samples are marked with

vertical dotted lines. Although the IVN is made with very sparse sequences,

each of them can be appropriately delayed, so that the pulse density of the

resulting signal ensures perceptual smoothness.

absorption, introducing frequency-dependent attenuation [213, 214, 215].

Computational complexity is an important issue in artiﬁcial reverbera-

tion, and methods to reduce it are important for the progress of the ﬁeld.

This problem is especially signiﬁcant in noise-based reverberation, since

Gaussian noise requires that every sample is processed.

To this end, Karjalainen and Järveläinen introduced a special kind of

quasi-random noise that constitutes of sparsely distributed pulses that

assume values of

−

1, 0, and 1 only [

216

]. The perceptual smoothness of

such kind of signal despite its sparseness earned a name for itself—velvet

noise (VN). An example of a VN signal with non-zero values appearing

once every 20 samples is shown in the top pane of Fig. 4.2.

The advantage of VN is that a non-zero pulse appears only once in a block

of samples. Given a density of 2205 pulses per second (i.e., one non-zero

sample in a block of 20 for

= 44.1 kHz, as in Fig. 4.2), which ensures

perceptual smoothness [

217

], the computational cost of convolution with

VN is up to 95% lower compared to convolution with Gaussian white noise.

Additionally, the use of

1’s for pulse amplitudes requires only additions

and no multiplications, thus possibly generating further savings [

218

Hence, the low complexity of the VN makes it beneﬁcial for reverberation

synthesis.

The ﬁrst reverberation algorithm utilizing VN was based on a simple

recursive structure [

216

]. The early part of the RIR was synthesized using

an FIR ﬁlter, whilst the VN was inserted in the delay line as a sparse FIR

(SFIR) ﬁlter. This approach was later improved by using short decaying

Reverberation Synthesis

noise sequences that were cross-faded and overlapping in time [

219

]. This

approach, named switched-convolution reverb, avoided repetitions present

in the original implementation by switching the VN sequence with each

pass through the feedback loop [

220

]. The technique was later extended for

the purposes of direction-dependent decay [

221

]. The idea of reverberation

modelling with VN was continued by applying ﬁltered sparse sequences

to approximate segments of the late part of an RIR, at the same time

closely following the target decay of each of such fragments [

222

223

Recently, VN signals were inserted in an FDN architecture to enhance the

diffuseness of produced sounds [224, 225, 226].

The latest developments in the area involve a reverberator with a few

branches, each of which include an SFIR ﬁlter made with a very sparse

VN sequence (VNS), having

1/M

th the density of a regular VN, where

is the number of parallel branches in the reverberator. The outputs

of the branches are delayed and summed, making the VNSs interleave,

thus achieving the pulse density that ensures perceptual smoothness. The

algorithm, called interleaved velvet-noise (IVN) reverberator, is introduced

in Publication IV. An example of an output of the IVN reverberator com-

prising four branches is presented in the bottom pane of Fig. 4.2.

4.2 Decay-Rate Control

An important property of an artiﬁcial-reverberation algorithm is to produce

IRs that decay at a target rate, which translates to the desired reverber-

ation time. In all of the aforementioned reverberators, this decaying

property is realized by the attenuation ﬁlters.

The ﬁrst digital reverberator by Schroeder produced reverberation that

had the same decay rate across the whole frequency range, which does

not accurately imitate the natural decay of sound in physical spaces. To

improve the perceived naturalness of sound, Moorer inserted a one-pole

ﬁlter in Schroeder’s reverberator loop gain [

212

] that allowed for high

frequencies to diminish faster than the low ones, imitating the sound

absorption in air. Moorer’s seemingly simple idea led to the development of

more complicated ﬁlter designs, such as Jot’s biquad ﬁlters used to control

the

T60

of an FDN in three independent bands [

227

] that can be changed

or extended to control decay rates over an arbitrary frequency resolution.

With a growing interest in the reproduction of accurate reverberation

and the increase in available computational power, the attenuation ﬁlters

grew in size. In [

196

], a 13th order ﬁlter, composed of single shelving

ﬁlters (low-, high-, and band-shelving) [

228

], was used with an addition

of a bandpass ﬁlter to further control the magnitude response of the edge

frequencies. After that, graphical equalizers (GEQs) began to appear in

the context of decay-rate control in several frequency bands [229, 230].

Reverberation Synthesis

4.2.1 Accurate Reproduction of Decay Rate

Regardless of the ﬁlters used in a speciﬁc reverberator implementation,

the attenuation required to obtain the target decay in the frequency region

of interest is based on the reference RT values [203]:

γdB(f) = −60

fsT60(f),(4.2)

where

is the sampling rate. Here,

γdB

is a parameter called gain-per-

sample and is expressed in dB. For all the delay lines to reach the

−

60-dB

attenuation at the same time, the magnitude response of the loop ﬁlter

AdB(f)must be adjusted by the length of its delay line Lin samples:

AdB(f) = LγdB (f).(4.3)

Regardless of the approach used to estimate the target ﬁlter gain, it

is crucial that such a ﬁlter accurately achieve the desired attenuation,

possible in many frequency bands. To achieve this goal, Publication V

proposes incorporating a cascaded GEQ, introduced by Välimäki and Liski

[

231

], that ensures command-gain errors no bigger than

±1

dB within a

±12 dB range.

The GEQ controls the attenuation in ten full-octave bands by using peak-

notch ﬁlters [

232

], one for each band, and a two-step process to minimize

the error in gain estimation [

231

]. The drawback of this approach is that

the magnitude response of each of these ﬁlters approaches 0 dB outside

of its designated frequency band. This poses a problem in the lowest and

highest bands of interest, where the sudden increase in gain results in a

surge of the respective RT values and may affect the decay in the adjacent

bands as well. As a remedy, Publication V proposes smoothing the magni-

tude response of the ﬁlter by using a broadband median gain, as suggested

in [

233

]. Should any further unwanted increase in the magnitude response

of the attenuation ﬁlter occur, it can be reduced by using low-order ﬁl-

ters, such as shelving ﬁlters [

234

]. A comparison of an attenuation-ﬁlter

magnitude before and after smoothing is depicted in Fig. 4.3.

4.2.2 Stability

The issue of stability is discussed in the context of accurate synthesis of

sound-energy decay in multiple frequency bands. An attenuation ﬁlter

is, in the case of this dissertation, realized as a time-invariant inﬁnite-

impulse-response (IIR) ﬁlter, which requires all of its poles to lie within

the unit circle in order to maintain stability [

235

236

237

238

]. If this

condition is met in the overall magnitude of the attenuation ﬁlter (which

can include several low-order ﬁlters), the IR of the reverberator will decay

instead of being sustained or ampliﬁed. In most cases, it is sufﬁcient that

Reverberation Synthesis

0.1 1 10

Frequency (kHz)

-1.5

-1.2

-0.9

-0.6

Magnitude (dB)

Figure 4.3.

Frequency response of a GEQ smoothed by using broadband gain (black)

compared to the response without smoothing (lilac). The RT values were set

to be equal in every frequency band.

all the linear gains of the loss ﬁlter are less than one, or equivalently, its

magnitude response stays below 0 dB in the entire frequency range.

The requirement for stability, however, is sometimes violated, when

the decay rate is controlled in multiple frequencies with large differences

between adjacent RT values. Such a situation can occur when reproducing

the reverberation of unusual spaces, e.g. with one dimension signiﬁcantly

larger than the other two [

221

239

], or when using artiﬁcial reverberation

to create an artistic effect. If the attenuation ﬁlter cannot approximate the

target magnitude response with sufﬁcient precision, reaching or exceeding

the unit-gain limit is possible [230].

An example of a target magnitude response that is too demanding to be

followed accurately by an attenuation ﬁlter is depicted in Fig. 4.4, together

with the resulting RT values. The parameters of the reverberator—target

RT values and a delay line of 20 ms—were speciﬁcally chosen to cause

instability of the system.

The problem of RT-error minimization is of a nonlinear nature, with the

same error in magnitude being more critical to the system stability close to

the 0-dB limit and less relevant when the attenuation is high (see Fig. 4.4).

Thus, a nonlinear error minimization between the target and obtained

RT values is an efﬁcient way to increase the reproduction accuracy and

lower the risk of instabilities. In Publication V, the nonlinear approach

is further extended with a weighting matrix emphasizing inaccuracies

close to the 0-dB limit. It is shown that this method successfully reduces

the discrepancies in the most crucial parts of the loss ﬁlter’s magnitude

response.

Reverberation Synthesis

0.1 1 10

Frequency (kHz)

-40

-20

Magnitude (dB)

0.1 1 10

Frequency (kHz)

Reverberation time (s)

Figure 4.4.

(Left) An example of instability caused by the attenuation-ﬁlter frequency

response leading to (right) an inﬁnite RT. The black solid line shows the target

values, the solid dark violet line represents the obtained values, whereas the

lilac areas indicate the parts of the response where instabilities occur along

with the respective increase in the RT values. It should be noted that the

large increase in RT corresponding to attenuation approaching, but not yet

reaching, the 0 dB limit is not considered an instability.

4.3 Reverberation Perception

Artiﬁcial-reverberation algorithms are mainly assessed using objective

measures, such as the ability to reproduce certain decay characteristics

or the required computational complexity. Such parameters describe well

the applicability of a speciﬁc algorithm to produce reverberation in many

situations. Objective evaluation alone is, however, insufﬁcient to qualify

the algorithm in terms of accurate reproduction of existing RIRs. This is

often essential, e.g., when the comparison between natural and synthesized

sound occurs continuously (for instance, in AR applications).

The scope of reverberation perception is wide and consists of many as-

pects, many of which are not easily reproducible digitally [

240

241

242

243

244

]. The subjective quality “reverberance” is most closely associated

to RT value, although not thoroughly described by it. Another quality that

can be somewhat assessed objectively is “diffuseness", which is connected

to the RIR echo density [245].

4.3.1 Echo Density

Echo, or reﬂection, density is a property of sound that describes the amount

of reﬂections reaching the receiver within a time unit. Although these

reﬂections may not be consciously perceived as separate sound events [

they are responsible for the perception of the texture of sound, its timbre,

and the size and shape of the enclosure [

246

247

]. With this in mind,

achieving sufﬁcient echo density at a rate suitable for the reproduced space

is an important issue when designing artiﬁcial reverberation structures.

In the case of FDNs, the main elements that contribute to the increase

Reverberation Synthesis

of echo density are the delay lines and the feedback matrix. Even though

FDNs have been the state-of-the-art parametric artiﬁcial-reverberation

algorithm for decades, there is no clear and reliable set of rules on the

choice of the number of delay lines and their lengths [

204

248

]. A common

design guideline is to choose mutually prime delays. Such a criterion is,

however, insufﬁcient to obtain an adequately rapid increase in the echo

density, especially when other unwanted effects, such as clustering and

low-order dependencies, are not avoided [248, 249].

The primary criterion for choosing the feedback matrix for an FDN is

ensuring losslessness of the structure, i.e., the energy of the system should

not decay when the attenuation ﬁlters are not in use [

202

203

250

]. Apart

from this, the matrices are also used to enhance speciﬁc properties of the

FDN, such as the increase in the echo density, computational efﬁciency,

and spectral ﬂatness [26, 225, 226, 251, 252, 253].

To this end, Publication VI presents a real-time implementation of an

FDN algorithm in which the user is able to manipulate elements of the

architecture, including the delay-line lengths and distribution. The study

also discusses the effect of good and ﬂawed choices on the perceived smooth-

ness of the produced reverberation. It is shown that delay-line lengths

should be suited to the duration of the target decay. Relatively too short or

too long decays do not contribute well to the perceived smoothness of the

sound, while still employing computational resources.

Similarly, the implementation presented in Publication VI allows the

user to choose the FDN order and type of the feedback matrix. It is shown

that fulﬁlling the losslessness criterion is not equivalent to producing high-

quality reverberation. The analysis presented in the study shows that,

for the considered scalar matrices, the minimal order of the FDN that is

capable of achieving sufﬁcient echo density in a short enough time is 16.

4.3.2 Smoothness of Decay

In noise-based reverberators, the produced IR is densely populated with

reﬂections from the start. This has the desired effect on the perception of

smoothness of sound, but also contributes to a high computation cost since

each sample has to be processed, regardless of whether it actually can be

registered by the listener or not. The problem is greatly reduced by using

pseudo-random noise, such as velvet noise. However, its parameters must

be chosen carefully to achieve good perceptual density whilst maintaining

low computational complexity of the signal.

It was established in literature that velvet noise containing about 2000

samples per second is sufﬁcient to sound as smooth or smoother than

Gaussian noise [

214

215

216

217

254

]. In Publication IV, the parame-

ters of an IVN structure are chosen with this fact in mind. However, the

perceptual smoothness ensured by the reﬂection density of IVN may be

Reverberation Synthesis

0 0.15 0.3

-20

-10

Magnitude (dB)

0 0.15 0.3

-20

-10

Magnitude (dB)

0 0.15 0.3

Time (s)

-20

-10

Magnitude (dB)

Figure 4.5.

IRs (dark violet) and their envelopes (black) produced with IVN reverberator.

The effect in the IR when (top) all the sequences begin at the same time with-

out smearing or segmentation, (center) the beginning times are smeared, and

(bottom) both the beginning times are smeared and the decay is segmented.

All the envelopes are computed as root-mean-square values of the signals,

determined over a sliding window of 2000 samples.

compromized by using the same sequences multiple times. Too few or too

short recurring VNSs create a repetitive, ﬂuttery sound [

220

221

]. The

results of the subjective evaluation presented in Publication IV show that

an IVN reverberator utilizing four sequences (see Fig. 4.2) with mutu-

ally prime lengths ensures sufﬁcient smoothness without unreasonable

extension of the structure’s complexity.

Attenuating repeated VNSs creates a series of decaying steps that may

result in audible drops in sound energy. To remedy this, Publication IV

proposes to offset the starting times of all except the ﬁrst VNSs, creating a

smearing effect. Further improvements can be made by segmenting the

decay, i.e., dividing each step of decay into smaller parts and gradually

decrease their level [

255

]. In Publication IV, three segments within each

step are deemed sufﬁcient, causing only a marginal increase in the com-

putational complexity (two multiplications). The effect of smearing the

sequence onsets and segmentation on the early parts of the IVN reverbera-

tion is presented in Fig. 4.5.

Reverberation Synthesis

4.3.3 Decay-Rate Perception

In the context of auditory perception of humans, the just-noticeable dif-

ference (JND) describes the minimal perceptible deviation of a certain

characteristic of sound. In the ﬁeld of artiﬁcial reverberation, the JND is

often used to evaluate the ability of an algorithm to accurately reproduce

the target RT values [

230

]. However, the use of the JND is not straightfor-

ward. When assessing sound-decay reproduction algorithms, the JND may

depend on the type of sound [

256

257

], ranging from 3% for reverberated

speech [

254

] to over 20% for band-limited noise and music [

257

258

259

Also, a decay with more than one slope, occurring in coupled spaces, affects

the JND [260].

Publication V and Publication VII show that when using a sufﬁciently

detailed attenuation ﬁlter, achieving RT values that differ by less than

5% from the reference numbers is possible for “regular” RIRs, i.e., concert

halls, auditoria, etc. According to the international standard [

], such

accuracy should ﬁt within one JND for RT values. The perceptual evalua-

tion conducted in Publication VII proves that although many of the human

listeners could not distinguish between the measured and synthesized

reverberation, this was not the case for the majority of the listening test

participants. This result is consistent with the ﬁndings of similar studies

analysing the perceptual qualities of artiﬁcial reverberation [

196

261

262

Therefore, the need for a more comprehensive understanding of precise

reproduction of measured RIRs remains.

5. Summary of the Main Results

This section presents the main results of the featured publications that

are related to the author’s work.

Publication I - "Evaluation of Reverberation Time Models with

Variable Acoustics"

In Publication I, popular RT prediction formulas are tested on a large

dataset of measured reverberation-time values from a variable acoustics

laboratory. In all, the work evaluates seven RT models, including the

classical formulas by Sabine and Eyring as well as formulas that are meant

for rooms with non-uniformly distributed absorption, such as Fitzroy’s and

Kuttruff’s equations. In two scenarios comprising measurements with

both uniformly and non-uniformly distributed absorption and a wide range

of RT values, Fitzroy’s formula proved to give the smallest error, with

Sabine’s and Kuttruff’s models following.

Publication II - "Calibrating the Sabine and Eyring Formulas"

Publication II discusses the problem of the accuracy of RT predictions

obtained with Sabine’s and Eyring’s formulas. The study presents results

of measurements in the variable acoustics laboratory Arni, showing the

distribution of RT values for different placement of absorptive and re-

ﬂective elements in the room. The study shows that, for measurements

conducted in different atmospheric conditions, the uncertainty related to

the air absorption of sound is high, and thus highlighting the need for com-

pensation of air absorption. The compensation procedure is presented as

well. The study also introduces a method to calibrate the sound-absorption

coefﬁcient to reduce the error of RT estimations. The calibration considers

the gradual change in absorption occurring in the measurement space and

the non-normal distribution of the RT values. It is shown that with the cor-

Summary of the Main Results

rectly estimated absorptivity, both Sabine’s and Eyring’s formula predict

the reverberation of a measured space with high accuracy—the majority of

the measured RT values lie within ±10% of the predicted results.

Publication III - "Robust Selection of Clean Swept-Sine

Measurements in Non-Stationary Noise"

Publication III introduces a method called the Rule of Two, which identiﬁes

a pair of clean swept-sine signals from a series of measurements. The

selection process is based on the similarity between two measured ESS

signals, which is expressed by the Pearson correlation coefﬁcient. The

PCC values above a set threshold classify two sweeps as clean, whilst a

low correlation indicates that at least one signal from the pair contains

contamination that may render a deconvolved RIR unsuitable for further

use. The study investigates the effects of expected contamination, such as

background noise and transfer-function variation, on the PCC values and

threshold estimation. It also identiﬁes different types of contamination

by non-stationary disturbances, such as short noise bursts, impulsive

events, and sound dropouts, and presents the impact they have on the

correlation between two signals. The use of the median is established in the

process of non-stationary noise detection and transfer-function variation

quantiﬁcation. Ro2 is tested on a large dataset of ESS measurements. The

results show that Ro2 is a reliable method that increases the robustness of

acoustic and audio measurements.

Publication IV - "Late-Reverberation Synthesis using Interleaved

Velvet-Noise Sequences"

Publication IV introduces an algorithm to synthesize the late part of an

RIR. Based on the fact that late reverberation resembles random noise,

the technique approximates the target decay with sparse velvet-noise

sequences. These sequences are implemented so that the non-zero samples

appear only within a limited sample range in a signal. When the sequences

are combined in parallel, the non-zero samples do not overlap, creating a

sparse but smooth-sounding signal. The listening test conducted in this

work proved that four combined sequences are sufﬁcient to achieve a non-

repetitive sound. Additional operations, such as altering the start times

of the sequences and segmentation, creating an intermediate step in the

decay, further enhanced the perceptual smoothness of the reverberation.

Each of the sequences is then ﬁltered by its own attenuation ﬁlter to

achieve the target RT values. The resulting algorithm is proven to be a

computationally affordable and accurate technique for synthesizing late

Summary of the Main Results

reverberation.

Publication V - "Improved Reverberation Time Control For Feedback

Delay Networks"

In Publication V, an accurate GEQ is inserted as an attenuation ﬁlter

into the FDN reverberator structure to provide accurate approximation

of the target reverberation time in the resulting impulse response. The

GEQ presented in the work by Välimäki and Liski [

231

] is used for this

purpose. Additional operations of shifting and scaling, as well as using

a high-shelf ﬁlter for attenuating the frequencies above 16 kHz further

ﬂatten the magnitude response of the ﬁlter, decreasing the approximation

error. Furthermore, a weighted gain optimization is introduced in the

ﬁlter-design process to improve the accuracy for cases of extreme changes

in reverberation-time values between neighboring frequency bands. The

algorithm is tested on two cases: an RIR of a concert hall and an artiﬁcial

extreme case. In the former scenario, the error between the target and the

obtained RT values does not exceed a JND of 5%. In the latter case, how-

ever, the differences are larger and exceed the 5% JND. The optimization

successfully lowers the approximation error in critical areas that are most

likely to result in ﬁlter instability.

Publication VI - "Flexible Real-Time Reverberation Synthesis with

Accurate Parameter Control"

Publication VI presents an efﬁcient real-time implementation of an FDN

reverberator. The plugin allows for the accurate control of the RT values

in octave frequency bands, detects instabilities in the attenuation ﬁlter,

and indicates them to the user. At the same time, the user can alter other

relevant parts of the FDN architecture: the number and lengths of de-

lay lines and the properties of the feedback matrix. The implementation

provides real-time feedback on the inﬂuence of the user’s choices on the

computational load of the plugin and the quality of the produced reverber-

ation. The study concludes that an FDN of order at least 16 can produce

high-quality sound, provided that a sufﬁcient level of mixing is ensured

by the feedback matrix. The choice of delay-line lengths is discussed as

well, listing good and bad design practices. Finally, the study discusses the

direct effect that the choice of FDN parameters has on the computational

cost of the implementation, with the CPU usage growing exponentially as

the FDN order is increased.

Summary of the Main Results

Publication VII - "Evaluation of Accurate Artiﬁcial Reverberation

Algorithm"

Publication VII conducts an extensive evaluation of the performance of

an FDN reverberator with accurate control over RT values in ten octave

frequency bands. The assessment was comprised of two parts: an objective

and a subjective evaluation. In the former, the ability of the algorithm to

reproduce three different RIRs—of an ofﬁce, a lecture hall, and a concert

hall—was tested in terms of the accuracy of decay reproduction. It is shown

that in the majority of the cases, the error between the target and obtained

RT values lies within 10%, and in the best cases does not exceed 5%,

which is the JND of RT speciﬁed in the ISO standard [

]. The subjective

evaluation showed that the IR that was the most accurately reproduced

by the FDN was the hardest to distinguish from the measured one in a

listening test. However, it was possible to spot differences between the

two, showing that the complexity and unique features of a real sound are

difﬁcult to imitate with an artiﬁcial-reverberation algorithm.

6. Conclusions

In the introductory part of this thesis, various aspects of reverberation

were discussed and analyzed. The overview was divided into three parts:

RT prediction, IR measurement techniques, and digital synthesis of rever-

beration.

The ﬁrst part began by describing the propagation of sound in enclosed

spaces. The components of an RIR—direct sound, early reﬂections and

the reverberation tail—were described, with a special focus on the late

part. The ﬁrst chapter reviewed Sabine’s deﬁnition of RT and listed the

most popular RT-prediction formulas together with the assumptions and

use cases associated with them, based on Publication I. Publication II dis-

cussed the inﬂuence of the air-absorption coefﬁcient on RT estimations and

proposed a method to calculate it for full-octave frequency bands. The need

to compensate for the attenuation of sound in air in RT measurements and

simulations was highlighted as well. The ﬁrst part also elaborated on the

accuracy of RT predictions in relation to results obtained from RIR mea-

surements in variable acoustics space for multiple absorptivity conditions,

following the studies conducted in Publication I and Publication II. This

part of the thesis was concluded by showing that the classical RT prediction

formulas by Sabine and Eyring achieved high prediction accuracy. Good

results were obtained regardless of the absorption distribution in a room,

when the uncertainty due to the sound-absorption coefﬁcient values used

in calculations was reduced.

In the second part of the overview, various methods to capture IRs were

outlined, with an emphasis on the ESS. ESS is nowadays the most widely

used excitation signal in acoustical measurements due to its excellent

SNR, which is particularly important at low frequencies. The second

part addressed the ESS’s high sensitivity to non-stationary noise and

elaborated on types of harmful non-stationary disturbances that Publica-

tion III discussed. It also described two types of unavoidable, expected

contamination—stationary noise and time variance. The procedure to es-

tablish whether a pair of measured ESS signals was free of non-stationary

contamination, called Ro2, was presented based on Publication III. Chap-

Conclusions

ter 3 elaborated on the role of robust estimators in contamination detection

as well.

The ﬁnal part of the overview discussed the techniques used to synthe-

size sound-energy decay. It described artiﬁcial-reverberation algorithms

based on feedback ﬁlters, with the most attention paid to FDNs and IVN

reverberator introduced in Publication IV. The issue of decay-rate control

in artiﬁcial reverberation was discussed, based on the ﬁndings presented

in Publication V. The real-time implementation of the proposed solution,

which utilized an FDN with a GEQ, was presented in Publication VI. In

Chapter 4, the perceptual aspects regarding artiﬁcial reverberation were

highlighted. Smoothness of reverberation was discussed, following the

ﬁndings in Publication IV and Publication VI. In the case of the FDN, the

focus was on the inﬂuence of delay-line lengths and that of the feedback

matrix on the echo density. In the IVN reverberator, the effect of the

number of velvet-noise sequences was considered, showing that four sufﬁ-

ciently long VNSs were enough for perceptually smooth reverberation. The

objective and subjective evaluation of an FDN with a GEQ showed that

digitally created sound could many times trick human listeners to perceive

it as being the same or very similar to a real RIR. The need for further

improvements in the ﬁeld of artiﬁcial reverberation was also highlighted,

as discussed in Publication VII.

The continuation of the research presented in this dissertation can take

several directions. The database of IR measurements collected in the

variable acoustics laboratory Arni allows for a number of diverse research

projects. The inﬂuence of atmospheric conditions on the measurement

accuracy and acoustic parameters estimation may be examined. Also,

such a database is potentially useful for devising a method to remove

non-stationary disturbances from ESS measurements.

A natural continuation of the research on artiﬁcial reverberation is to

achieve higher ﬁdelity in reproducing real-world RIRs. Since the computa-

tional capacity of processors is increasing rapidly, using more complicated

systems becomes feasible. Employing tools such as machine learning and

deep learning could potentially aid in achieving good accuracy of objective

parameters as well as in elevating perceptual plausibility.

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Errata

Publication I

The denominators in Equations (8) and (9) should be squared, resulting in

[ρww iSwi]2

and

[ρcf (Sc+Sf)]2

, respectively. In Equation (12), the index of

the second sum operator should be k, reading K

k=1.

Publication I

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Evaluation

of Reverberation Time Models with Variable Acoustics. In Proceedings of

the 17th Sound and Music Computing Conference (SMC 2020), Turin, Italy,

June 2020.

Reprinted with permission.

Evaluation of Reverberation Time Models with Variable Acoustics

Karolina Prawda, Sebastian J. Schlecht and Vesa V¨

alim¨

aki

Aalto University, Acoustics Lab, Dept. of Signal Processing and Acoustics, FI-02150 Espoo, Finland

{karolina.prawda, sebastian.schlecht, vesa.valimaki}@aalto.fi

ABSTRACT

Reverberation time of a room is the most prominent pa-

rameter considered when designing the acoustics of phys-

ical spaces. Techniques for predicting reverberation of en-

closed spaces started emerging over one hundred years ago.

Since then, several formulas to estimate the reverberation

time in different room types were proposed. Although

validations of those models were conducted in the past,

they lack testing in a space with a high granularity of con-

trollable absorptive and reﬂective conditions. The present

study discusses the reverberation time estimation

techniques by comparing various formulas. Moreover, the

reverberation time measurements in a variable acoustic lab-

oratory for different combinations of reﬂective and absorp-

tive panels are shown. The values calculated with the pre-

sented models are compared with the ones obtained via

measurements. The results show that all formulas pre-

dict reverberation time values inaccurately, with an aver-

age error of 16% or larger. Among the analyzed models,

Fitzroy’s formula gives the smallest error.

1. INTRODUCTION

Reverberation is considered as one of the most important

qualities of sound within the physical space [1–3] and there-

fore central in designing acoustics of halls and rooms. The

ﬁrst attempt to invent a theory to predict the reverbera-

tion time value of a given space was made by Sabine [4],

who introduced a formula based on experimental results.

Over the decades, many improvements were made to his

model to allow more accurate predictions for spaces with

both uniformly and unevenly distributed absorption [3, 5,

6]. However, studies show that in many cases those formu-

las do not give results close enough to measured reverber-

ation time values to be reliable [3,5, 7–10].

As the variable acoustic solutions are gaining popularity

in the ﬁeld of acoustic treatment of spaces, there are few

works that study the change in reverberation time values in

a room with varying absorption [9, 11, 12]. In most cases,

however, only a few different combinations were studied.

The present paper presents measurements of reverbera-

tion time in a variable acoustics space with a high level

of absorption granularity. It further compares the obtained

2020 Karolina Prawda, Sebastian J. Schlecht and Vesa

V¨

alim¨

aki et al. This is an open-access article distributed under the terms of the

Creative Commons Attribution 3.0 Unported License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original author

and source are credited.

values with the predictions calculated by several reverber-

ation time models.

The paper is organized as follows. Section 2 presents re-

verberation time formulas. Section 3 describes measure-

ments in the variable acoustics laboratory. Section 4

presents the results of measurements and reverberation time

predictions using the formulas discussed in Section 2. It

also discusses the differences between measured and pre-

dicted values and reveals, which of the models provides

the best results. Section 5 summarizes the work presented

in the paper, concludes the ﬁndings, and presents ideas for

further research.

2. REVERBERATION TIME FORMULAS

Sabine deﬁned reverberation time as the time needed for

the sound energy to decrease by 60 dB from its original

level after the termination of the excitation signal [4].

Sabine’s prediction is given by

T60 =0.161V

Sα + 4mV ,(1)

where Vis the volume of a space, Sis the room sur-

face, 0.161 is an experimentally determined coefﬁcient,

αis the average absorptivity in a room. αis deﬁned as

α=PiSiαi/S, where Siare the areas and αiis the cor-

responding absorption of each wall, and mis the attenua-

tion coefﬁcient of the air, the value of which depends on

the frequency of sound and the air humidity.

For the Sabine formula to predict the reverberation time

of the room accurately, a number of requirements must be

met: the energy of sound must be equally diffused through-

out the space, which means that the walls are not paral-

lel, there are no big differences between the basic dimen-

sions (length, width, and height), and the absorption is

small (α < 0.2[5]) and uniformly distributed on all walls

[3, 5, 6,13]. In practice, all of those conditions are almost

never met, making the Sabine formula applicable only in a

small percentage of rooms [3].

Since the Sabine formula proved useful only in consid-

erably live spaces, Eyring introduced a new reverberation

theory based on the mean free path between sound reﬂec-

tions [13]. The mean free path in an enclosed space char-

acterized by a diffuse ﬁeld is expressed by l= 4V/S

[14–16]. This leads to the following formula:

T60 =0.161V

−Sln(1 −α).(2)

The Eyring formula is designed for rooms with consid-

erable absorption [17]. Both Equations (1) and (2) as-

sume that all surfaces have the same average absorption,

although in reality, the absorption coefﬁcients of the walls,

the ﬂoor, and the ceiling can vary greatly. This was ad-

dressed by Millington [18] and Sette [19], who introduced

the following formula:

T60 =0.161V

−PiSiln(1 −αi).(3)

Another improvement to reverberation time prediction

was made by Kuttruff [16]. Similarly to Eyring, he based

his model on the mean free path approach. He suggested,

however, statistical distribution of sound, introducing a rel-

ative variance of the path length γ2= (l2−l2)/l2. Kut-

truff’s model took into account the shape of the room and

distribution of the absorption, as well as corrected the av-

eraging of the sound absorption coefﬁcient, yielding the

following equation:

T60 =0.161V

−Sln(1 −α)1 + γ2

2ln(1 −α).(4)

Kuttruff’s formula repeatedly gives good reverberation

time predictions for rooms, where all walls but one have

similar absorption, but not when the absorption is

distributed asymmetrically [3].

Although the above formulas present the progress in re-

verberation time estimation over the years, all of them still

assume that the absorption coefﬁcients of the room’s sur-

faces are approximately equal. The ﬁrst model that in-

cluded geometrical aspects of the sound ﬁeld with unevenly

distributed absorption was presented by Dariel Fitzroy [20].

His empirically derived equation assumes a relation within

three possible decay rates along the three basic axes in a

rectangular room and is expressed by

T60 =0.161V

S2X

−Sj

ln(1 −αj),(5)

where j=x, y, z denotes the current axis, Sjis the total

area of the opposite parallel walls along the axis, and αj

is the average absorption coefﬁcients for each pair of op-

posite walls. Fitzroy’s model is reported to work best for

relatively large spaces, such as concert halls [17], but only

when they are of rectangular shape [3].

A similar approach was adopted by Arau-Puchades [21],

who described the reverberation time of a room to be a geo-

metric weighted average of the reverberation times in three

orthogonal directions. The absorption coefﬁcients are de-

termined for each pair of the parallel walls, yielding the

following formula:

T60 =Y

j"0.161V

−Sln(1 −αj)+4mV #Sj

.(6)

A further modiﬁcation to Fitzroy’s formula was proposed

by Neubauer [3, 22, 23], who used the fact that both

Fitzroy’s and Kuttruff’s models were based on the concept

by Eyring. He introduced a similar correction to Fitzroy’s

equation as was earlier done by Kuttruff to Eyring’s for-

mula. Therefore, Kuttruff’s correction was split into two

parts – one for the ceiling and ﬂoor and another for the

remaining walls. Neubauer’s formula is expressed by

T60 =0.32V

S2h(l+w)

α∗

+l·w

α∗

cf ,(7)

where h,w, and lare the room dimensions height, width,

length in meters, and ¯α∗

ww and ¯α∗

cf are the average effec-

tive absorption exponents of the walls and the ceiling and

the ﬂoor, respectively:

α∗

ww =β+"Piρwi(ρw i −ρww )S2

ρww PiSwi #,(8)

α∗

cf =β+"ρc(ρc−ρcf )S2

c+ρf(ρf−ρcf )S2

ρcf (Sc+Sf)#,(9)

where ρ= 1 −αis the reﬂection coefﬁcient and β=

−ln(1/ρ).

3. ACOUSTIC MEASUREMENTS

This section discusses the measurements conducted and

equipment used during this study in the variable acoustics

laboratory Arni at the Acoustics Lab of Aalto University,

Espoo, Finland. Examples of measured impulse responses

are available online 1.

3.1 Variable acoustic space

The Arni room is of rectangular shape, with dimensions

8.9 m ×6.3 m ×3.6 m (length, width, and height). The

walls and the ceiling of the room are covered with variable

acoustics panels made from painted metal and ﬁlled with

absorptive material. On the front of the panels, rectangular

slots are cut out from the surface. The slots can be opened,

letting the sound reach the absorptive material inside, or

closed, making the surface reﬂective. The dimensions of

a single panel are 0.6 m ×0.4 m ×2.4 m (length, width,

and height). There is a total of 55 panels in the variable

acoustics laboratory including 8 on three of the walls, 11

on the fourth wall, and 20 on the ceiling.

3.2 Measurement setup

During the measurements, two Genelec 8030A loudspeak-

ers were used as sound sources. Five G.R.A.S. 1/2-inch

free-ﬁeld microphones of type 46AF served as receivers.

The positions of sound sources and receivers are marked

in Fig. 1. Moreover, G.R.A.S. power model of type 12AG

was used as an ampliﬁer. All equipment was connected

to an HP ZBook laptop via MOTU UltraLite mk3 Audio

Interface.

The measurement signal was a 3-second long exponential

sine sweep. It was played three times for each panel con-

ﬁguration through each sound source, resulting in 6 record-

ings for each microphone, making a total of 30 test signals

recorded for each panel conﬁguration. All in all, 56 panel

1http://research.spa.aalto.fi/publications/

papers/smc20-RTmodels/

Figure 1: Layout of the variable acoustics laboratory Arni showing the panels and the sound sources and receiver locations.

The arrows show the order of panels closing on the walls and the ceiling.

Material 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz

Panel open

[26]

0.86 0.77 0.66 0.45 0.38 0.42

Panel

closed

[26]

0.09 0.05 0.05 0.04 0.02 0.03

Wall [27] 0.02 0.03 0.03 0.04 0.05 0.05

Floor [27] 0.02 0.03 0.03 0.03 0.02 0.03

Curtain

[28]

0.45 0.95 0.99 0.99 0.99 0.99

Table 1: Sound absorption coefﬁcients of materials used as

the basis for determining the αin T60 calculations.

conﬁgurations were measured, the ﬁrst one having all pan-

els open (conf. no. 1). In the following conﬁgurations the

panels were being closed one by one (conf. no. 2 = 1 panel

closed, conf. no. 3 = 2 panels closed, and so on). Addi-

tional 20 conﬁgurations were measured by closing only the

panels on the ceiling, while the ones on the four remaining

walls were open. After the acoustic measurements, the re-

verberation time was estimated for each conﬁguration ac-

cording to [24], using the functions included in the IoSR

Matlab Toolbox [25].

3.3 Measurement accuracy

The T60 were averaged for each conﬁguration according to

T60,n(k) = 1

m=1

T60,m,n(k),(10)

where M= 30 is the number of values obtained for one

panel conﬁguration (30 = 5 positions ×2 sources ×3

sweeps), kis the frequency index, and nis the conﬁgura-

tion number. The standard deviations were obtained using

the equation:

σn(k) = sPM

m=1(T60,m,n (k)−T60,n(k))2

M−1.(11)

4. COMPARING MEASURED AND MODELED T60

The measured values of the reverberation time were com-

pared with the results of calculations of T60 using the for-

mulas presented in Sec. 2. Two scenarios were tested: in

the ﬁrst one, all panels were closing following the direction

showed by the arrow in Fig. 1. In the second one, only the

panels on the ceiling were closing, whilst the panels on the

remaining four walls stayed open. The absorption coefﬁ-

cients of materials used for the calculations are presented

in Table 1.

4.1 All panels open to all closed

The measured and modeled values for six octave frequency

bands for the case of all panels closing are shown in Fig. 2.

Figure 2a shows that the modeled values ﬁt the measured

ones well for 250 Hz. For 500Hz–1 kHz frequency bands,

depicted in Fig. 2b–2d, the predictions underestimate the

measured reverberation times. For 4 kHz, presented in

Fig. 2e, the predicted values are lower than the measured

ones for all formulas except for Fitzroy’s, which provides

accurate results for the last two combinations. For 8 kHz

depicted in Fig. 2f, all formulas give too low RT values

when most of the panels are open. When the number of

0 10 20 30 40 50

Number of closed panels

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Reverberation time [s]

Measured

Sabine

Eyring

Millington-Sette

Fitzroy

Arau

Kutruff

Neubauer

(a) 250 Hz

0 10 20 30 40 50

Number of closed panels

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Reverberation time [s]

(b) 500 Hz

0 10 20 30 40 50

Number of closed panels

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Reverberation time [s]

0 10 20 30 40 50

Number of closed panels

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Reverberation time [s]

(d) 2 kHz

0 10 20 30 40 50

Number of closed panels

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Reverberation time [s]

Measured

Sabine

Eyring

Millington-Sette

Fitzroy

Arau

Kutruff

Neubauer

(e) 4 kHz

0 10 20 30 40 50

Number of closed panels

0.2

0.4

0.6

0.8

1.2

1.4

1.6

Reverberation time [s]

(f) 8 kHz

Figure 2: Results of reverberation time measurements and predictions at different octave bands for the case of all panels

closing one by one. The shaded area above and below the measured values represents one standard deviation from the

mean. The dotted vertical lines mark the end of each wall.

Conﬁguration Sabine Eyring Millington-Sette Fitzroy Arau Kuttruff Neubauer

All panels open

[%]

µ23.55 27.35 34.41 17.94 26.78 22.94 25.13

σ7.32 15.67 13.88 8.58 16.13 12.93 15.39

All panels closed

[%]

µ31.22 30.31 29.06 18.81 28.51 26.38 27.74

σ13.04 16.74 17.09 10.38 12.49 16.54 16.63

All combinations

[%]

µ27.33 28.47 32.46 24.34 33.58 25.90 26.94

σ9.64 12.89 13.95 11.55 12.90 11.48 11.74

Table 2: Average difference µand standard deviation σfrom the measured T60 values for all panels open, all panels closed,

and from all panel combinations. The smallest result on each row is highlighted.

0 10 20 30 40 50

Number of closed panels

Mean difference between measured

and modeled T60 values[%]

Sabine

Eyring

Millington-Sette

Fitzroy

Arau

Kuttruff

Neubauer

Figure 3: The mean difference between the measured and

modeled reverberation time values for the scenario of all

panels in the room closing. The dotted vertical lines mark

the end of each wall.

closed panels grows, however, the accuracy of Eyring’s,

Millington-Sette’s, Kuttruff, and Neubauer’s models

increases, whilst Fitzroy’s quickly goes from producing

too low to too high T60 values.

The difference between the measured and modeled rever-

beration time values was averaged over all frequency bands

and presented in Fig. 3. None of the used formulas predict

the T60 of the room with less than 17% error. The smallest

differences were obtained by using Fitzroy’s formula.

Additionally, the averaged difference for maximum ab-

sorption (all panels open), minimum absorption (all panels

closed) and the average difference over all combinations

were calculated according to:

µ=1

n=1

∆e

T60,n(k),(12)

where ∆e

T60,n(k) = |e

T /T 60,n(k)−1| · 100%,Tis the RT

predicted with a particular model, and Nis the number of

conﬁgurations over which the difference is averaged (N=

1for cases of all panels open and all panels closed, whilst

for all combinations N= 56).

The standard deviation was obtained using the formula:

σ=sPN

n=1 PK

k=1(∆ e

T60,n(k)−µ)2

NK −1.(13)

The averaged differences and standard deviations are

shown in Table 2. The results conﬁrm that Fitzroy’s for-

mula provides the best predictions, by giving the smallest

difference between measured and modeled values. Sabine’s

formula has the least variation in the predicted RT.

4.2 Panels on the ceiling closing

Figure 4 presents the comparison between the measured

and modeled RT values for the scenario when only the pan-

els on the ceiling are closing, whilst the ones on the four

remaining walls stay in the open conﬁguration.

The models predict the RT values similar to the measured

ones only for the ﬁrst ten panels closed for 250 Hz pre-

sented in Fig. 4a. For all the remaining frequency bands,

depicted in Fig. 4b–4d, the T60 is underestimated by all

the formulas. None of the models mimic the increase in

the measured RT values that starts around 10th panel and

is the most prominent for 250 Hz in Fig. 4a and visible for

500 Hz and 1kHz in Fig. 4b and Fig. 4c, respectively.

The differences between the measured and predicted RT

values for the case of the ceiling closing is shown in Fig. 5.

The predicted values are different from the measured ones

by at least 16%. Similarly as in the scenario when all pan-

els in the room were closing, Fitzroy’s model provides the

smallest error. However, when the absorption decreases, as

more panels are closed, Sabine’s formula gives very simi-

lar results to Fitzroy’s.

The averaged differences between measured and calcu-

lated RT values were calculated using Eq. (12) (N= 1

for all panels open and all panels closed, whilst N= 20

for all combinations) and are presented together with their

standard deviations obtained with Eq. (13) in Table 3 for

the cases of all the panels on the ceiling open, all the pan-

els on the ceiling closed, and all combinations. Fitzroy’s

formula gives the smallest error when the absorption in

the room is high, but Sabine’s equation performs similarly

when the absorption is low. Moreover, for the case of all

panels on the ceiling closed, Sabine’s, Fitzroy’s Kuttruff’s,

and Neubauer’s models perform very similarly, which is

depicted in Fig. 5 when the number of closed panels ap-

proaches 20 and in Table 3, where the means of the above

5 10 15 20

Number of closed panels

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reverberation time [s]

Measured

Sabine

Eyring

Millington-Sette

Fitzroy

Arau

Kutruff

Neubauer

(a) 250 Hz

5 10 15 20

Number of closed panels

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reverberation time [s]

(b) 500 Hz

5 10 15 20

Number of closed panels

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reverberation time [s]

5 10 15 20

Number of closed panels

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reverberation time [s]

(d) 2 kHz

5 10 15 20

Number of closed panels

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reverberation time [s]

(e) 4 kHz

5 10 15 20

Number of closed panels

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reverberation time [s]

(f) 8 kHz

Figure 4: Results of reverberation time measurements and predictions for the case of panels on the ceiling closing one

by one, whilst the rest remaining open. The shaded area above and below the measured values represents one standard

deviation from the mean.

Conﬁguration Sabine Eyring Millington-Sette Fitzroy Arau Kuttruff Neubauer

All panels open

[%]

µ22.49 26.49 33.02 16.74 26.50 21.83 24.52

σ6.13 14.35 14.23 9.09 13.72 12.20 13.17

All panels closed

[%]

µ27.96 32.52 39.12 28.57 34.86 28.50 29.01

σ11.07 16.69 15.64 17.03 15.49 16.32 14.79

All combinations

[%]

µ24.52 29.25 35.05 22.58 30.33 25.47 26.45

σ7.08 12.23 14.45 10.76 11.73 10.09 10.00

Table 3: Average difference µand standard deviation σfrom the measured T60 values for all panels open, all panels closed,

and all panel combinations for the case of only the panels on the ceiling closing. The best result on each row is highlighted.

5 10 15 20

Number of closed panels

Mean difference between measured

and modeled T60 values[%]

Sabine

Eyring

Millington-Sette

Fitzroy

Arau

Kuttruff

Neubauer

Figure 5: The mean difference between the measured and

modeled reverberation time values for the scenario when

only the panels on the ceiling are closing.

formulas for the “All panels closed” case differ by 1.05

percentage point or less.

4.3 Error Propagation of Absorption Coefﬁcients

Incorrectly speciﬁed absorption coefﬁcients are a common

source of error in RT predictions. To check whether the

results presented in the study are reliable, we added noise

to the absorption coefﬁcient values of the ﬂoor, walls and

curtains and calculated the RT for all the conﬁgurations

and models. The simulation was repeated 1000 times and

the mean and standard deviations of all the trials were cal-

culated. The experiment showed that even considerable

changes in the absorption coefﬁcients (up to 50% of the

initial values) did not change the fact that all the formulas

underestimated the RT and that the predictions made with

the Fitzroy’s model were the most accurate.

5. CONCLUSIONS

The study compares seven reverberation time models, which

were introduced over the past decades, and discusses their

applications. It compares the RT estimations obtained with

the models with the results of measurements conducted

in the variable acoustics laboratory. The comparison was

conducted for two scenarios. In the ﬁrst one, the absorp-

tion in the room was decreased by changing the conﬁgura-

tion of acoustic panels from open to closed on four walls

and the ceiling. In the second one, only the panels on the

ceiling were being closed, whilst the rest remained open.

The results show that all the formulas produce inaccurate

T60 predictions, with the difference between measured and

modeled values above 16% in every conﬁguration.

Fitzroy’s model performed best in both scenarios, which is

a reasonable result due to the fact that it was developed for

rectangular rooms with non-uniformly distributed absorp-

tion. It also attempts at producing bumps in the estimated

RT at the beginning and the end of each wall, which in

measured values are especially visible in low frequencies.

Neubauer’s and Arau’s models, however, were also in-

troduced with assumptions similar to Fitzroy’s, but the es-

timated RT values are far from the measured ones, espe-

cially when the absorption in the room decreases. This

is especially surprising since the intuition is that both of

those models should perform well in situations when the

reverberation time along one axis is different from the re-

maining two (e.g. in the second scenario).

Kuttruff’s formula is the second best one for the ﬁrst sce-

nario (all panels closing). Its average error is large, but

stable across all combinations. However, in most cases,

it does not follow the increase of the reverberation time,

which is visible in the ﬁrst scenario after 20 panels are

closed, and in the second scenario after 10 panels are closed.

The poor performance of that formula may be due to the

fact that in most of the combinations, the absorption in

the room is asymmetric. However, in the cases when it

is symmetric or close to symmetric, Kuttruff’s model still

produces a considerable error.

The fact that Sabine’s formula performs so well is surpris-

ing, especially taking into account that the smallest error is

produced when the absorption is high, and it grows con-

siderably with the decrease of the absorption. Another un-

expected result is Eyring’s formula returning higher errors

than Sabine’s model in most cases since it should estimate

the T60 better in rooms with considerable absorption. The

large difference between the predictions made with both of

those models and measured RT values may be due to the

fact that the formulas require the sound ﬁeld in the room

to be thoroughly diffuse, which is not achieved with any of

the panel combinations.

All in all, the error obtained with the discussed models

shows that there is a strong need for a more accurate and

ﬂexible way to predict the reverberation time. This should

be a focus of future research in room acoustics. Addition-

ally, more measurements with the same amount of absorp-

tion distributed differently in a room need to be conducted,

since the situation in which the absorption changes linearly

along the surfaces of the room is unlikely in real life.

Acknowledgments

This work was supported by the “Nordic Sound and Music

Computing Network—NordicSMC”, NordForsk project

number 86892.

6. REFERENCES

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Publication II

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Calibrating

the Sabine and Eyring Formulas. The Journal of the Acoustical Society of

America, Vol. 152, No. 2, pp. 1158–1169, August 2022.

Reprinted with permission.

Calibrating the Sabine and Eyring formulas

Karolina Prawda,

Sebastian J. Schlecht,

and Vesa V€

alim€

aki

Acoustics Lab, Department of Signal Processing and Acoustics, Aalto University, FI-02150 Espoo, Finland

ABSTRACT:

Of the many available reverberation time prediction formulas, Sabine’s and Eyring’s equations are still widely used.

The assumptions of homogeneity and isotropy of sound energy during the decay associated with those models are

usually recognized as a reason for lack of agreement between predictions and measurements. At the same time, the

inaccuracy in the estimation of the sound-absorption coefﬁcient adds to the uncertainty of calculations. This paper

shows that the error of incorrectly assumed sound absorption is more detrimental to the prediction precision than the

inherent error in the formulas themselves. The proposed absorption calibration procedure reduces the differences

between the measured and predicted reverberation time values, showing that an accuracy within 610% from the tar-

get reverberation time values can be achieved regardless of the absorption distribution in a room. The paper also dis-

cusses the oft neglected air absorption of sound, which may introduce considerable bias to the measurement results.

The need for an air-absorption compensation procedure is highlighted, and a method for the estimation of its parame-

ters in octave bands is proposed and compared with other approaches. The results of this study provide justiﬁcation

for the use of the Sabine and Eyring formulas for reverberation time predictions. V

C2022 Author(s). All article

content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://

creativecommons.org/licenses/by/4.0/). https://doi.org/10.1121/10.0013575

(Received 14 April 2022; revised 28 July 2022; accepted 1 August 2022; published online 24 August 2022)

[Editor: Ning Xiang] Pages: 1158–1169

I. INTRODUCTION

One of the most important properties of sound in physical

spaces is reverberation, which is affected by the size, the

geometry, and even the materials of the surfaces of the enclo-

sure. Reverberation is also a signiﬁcant factor affecting the

perception of sound, having an inﬂuence on, e.g., clarity of

music and intelligibility of speech. A parameter commonly

used to describe the sound decay in a room is called reverbera-

tion time (RT), already established in the literature over

100 years ago.

Since then, several methods to estimate a

room’s RT have been developed, involving both simple for-

mulas and more sophisticated models that take into account an

enclosure’s geometry and sound-absorption distribution.

2–11

However, the basic models introduced by Sabine

and Eyring

are still the best known and the most frequently used.

The accuracy of RT prediction is crucial when design-

ing spaces for speech and music, such as concert halls and

auditoriums. Over the years, many studies have been pub-

lished attempting to evaluate the correctness of RT estima-

tion formulas, both in spaces with big volumes, such as

concert halls,

and in small rectangular rooms,

12–14

mostly

classrooms.

15–19

The results generally reveal that the consid-

ered formulas are burdened with a certain error, often prov-

ing them to be too unreliable to use in acoustic design. This

study identiﬁes the sources of the uncertainty associated

with two classical reverberation formulas and introduces

ways to reduce the errors to achieve sufﬁcient accuracy for

practical designs.

One of the sources of error in reverberation calculations

is the sound-absorption coefﬁcient. Its insufﬁcient measure-

ment accuracy and reproducibility has been known since the

1930s

20,21

and has earned a name for itself: “the absorption

coefﬁcient problem.”

22,23

Even though the measurement

procedure has been standardized and is occasionally

updated,

24,25

the issue still remains and is a subject of scien-

tiﬁc debate.

26–31

Another source of uncertainty is the neglect of the inﬂu-

ence that the absorption of sound in the air has on a room’s

RT, causing discrepancies between predicted and measured

RT values. The early works by Sabine and Eyring consid-

ered the attenuation of sound in a room to be a result of sur-

face absorption only.

1,2

The neglected attenuating effect of

the medium was introduced in RT formulas by Knudsen.

However, some studies omit the air absorption in RT calcu-

lations

10,33

or use it with selected formulas only.

The present work investigates the ability of Sabine’s

and Eyring’s formulas to accurately predict the RT value for

different sound-absorption conditions in a small rectangular

room with variable acoustics. The RT values estimated from

captured room impulse responses (RIRs) and from the afore-

mentioned models are compared for different amounts and

distributions of the absorbing and reﬂecting elements. The

study presents the absorption calibration procedure that is

adapted to the measured RT values.

Additionally, the study shows the effect of air absorp-

tion on the RT measurements and highlights the need for its

Electronic mail: karolina.prawda@aalto.ﬁ

Also at: Media Lab, Department of Art and Media, Aalto University,

FI-02150 Espoo, Finland.

1158 J. Acoust. Soc. Am. 152 (2), August 2022 V

CAuthor(s) 2022.0001-4966/2022/152(2)/1158/12

ARTICLE

...................................

compensation. A method to estimate the air-absorption coef-

ﬁcient of sound for full-octave bands, based on the standard-

ized calculations for pure-tone absorption, is proposed.

The remainder of this paper is organized as follows:

Sec. II presents the Sabine and Eyring RT prediction formu-

las and discusses the sound ﬁeld conditions associated with

each of them. The issues related to sound-absorption-coefﬁ-

cient calibration are elaborated upon in Sec. III. Section IV

recapitulates the methods to calculate the air absorption for

pure tones, proposes a new procedure that translates it to a

coefﬁcient for full-octave bands, and compares the new

method with the other approaches. Section Vdescribes

comprehensive RT measurements conducted in a variable

acoustic laboratory. Section VI presents the results of the

measurements, compares them with the predictions

obtained with Sabine’s and Eyring’s formulas, and shows

the calibration of sound-absorption coefﬁcients. Section VII

discusses the results of the study, and Sec. VIII concludes

the paper.

II. RT ESTIMATION

This section reviews popular RT estimation formulas

and models. It also discusses their development and assump-

tions regarding the properties of the sound ﬁeld.

A. Sound-decay model

Considering the loss of sound energy resulting from the

absorption at the boundary of the enclosure, the rate of

sound decay is described in terms of sound intensity,

2,32,34

Iðt;fÞ¼Ið0;fÞexp 

aðfÞ



lct



;(1)

where I(t)andI(0) are the sound intensities at times tand 0,

respectively, and frequency f;cis the speed of sound propaga-

tion in air; and 

lis the mean free path. Here, ^

arepresents the

general absorption coefﬁcient of the room’s surfaces regard-

less of the approach adopted to calculate it. The RT value at

time t¼T60ðfÞis obtained when Iðt;fÞ=Ið0;fÞ¼106(i.e., a

60-dB decay), resulting in the following formula:

T60ðfÞ¼ ln ð106Þ

aðfÞc:(2)

Here, the propagation in an ideal gas is assumed with no air

absorption.

B. Reverberation time formulas

The earliest work on the RT estimation comes from

Wallace Sabine, who also deﬁned this parameter as the time

needed for sound to become inaudible. As a result of his

experimental work, Sabine introduced the following formula

to predict the RT of concert halls:

T60ðfÞ¼0:164 V

S

aðfÞ;(3)

where Vis the volume of a space in m3,Sis the room sur-

face in m2, 0.164 is an experimentally determined coefﬁ-

cient (although different values between 0.16 and 0.164 are

commonly used

6,10,33,35,36

), and 

ais the average absorptivity

in the room, deﬁned as 

a¼PiSiai=S, where S

are the sur-

face areas in m2and a

are the corresponding absorption

coefﬁcients of each surface. Equation (3) is equivalent to

Eq. (2) when the standard atmospheric conditions and a

shoebox room are considered. To generalize Sabine’s for-

mula and make it applicable in other scenarios as well, the

present study uses Eq. (3) in the form that is closer to Eq.

(2), when

aSðfÞ¼

aðfÞ:(4)

Although Sabine’s formula is commonly used to predict

the RT of different types of rooms, only very speciﬁc rooms

meet the requirements that make these estimations accu-

rate.

10,37

Since the formula assumes continuous decay of

sound,

the key condition is that, at any given moment,

sound energy is diffused equally throughout the space, i.e.,

it is homogeneous and isotropic. In practical terms, achiev-

ing the diffuse sound ﬁeld is commonly translated to a num-

ber of additional guidelines for the enclosure: e.g., walls are

not parallel, basic dimensions (height, width, and length)

have no big differences between them, and the small absorp-

tion

[

a<0:2 (Refs. 17 and 39)] is uniformly distributed

on all surfaces.

2,10,12,17

To improve the RT predictions for rooms with consider-

able absorption, Eyring

proposed another reverberation the-

ory, which was based on the mean free path of sound

particles, which for rectangular rooms is 

l¼4V=S,

2,6,40,41

and the stepwise exponential energy decay, changing by

1

aafter a specular reﬂection. The Eyring formula uses

aEðfÞ¼ln ð1

aðfÞÞ:(5)

The Eyring formula is designed to give more accurate

RT estimates in “dead,” i.e., highly absorptive, rooms

(although Eyring does not specify a particular threshold for

a room to be “dead” or “live”

) than Sabine’s formula and

consistently gives lower RT values for the same absorption

coefﬁcient.

If the room’s surfaces are perfectly absorptive,

i.e., 

a¼1, Sabine’s formula gives a non-zero number,

whereas Eyring’s results in ln ð0Þ, which in literature is

interpreted as T60ðfÞ¼0s.

However, this common view of Eyring’s formula accu-

racy for high sound absorption is contested.

38,40

Eyring’s

theory assumes that the sound decay in a room is a discrete

process, which could be described by a probability of the

interaction between the sound particle and the surface.

For such assumptions to be justiﬁed, the sound ﬁeld is

required to be homogeneous and isotropic.

It is discussed

in literature that the combination of such assumptions and

prerequisites does not result in more accurate RT estima-

tions when 

ais high,

37,38,40

but rather returns correct predic-

tions for small values of average absorptivity [

a<0:5

(Refs. 39 and 42)].

J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al. 1159

https://doi.org/10.1121/10.0013575

III. SOUND-ABSORPTION-COEFFICIENT

CALIBRATION

In this study, we show that the classical equations by

Sabine and Eyring can achieve sufﬁcient accuracy of RT

predictions, even when the assumptions of homogeneous

and isotropic sound ﬁeld are not met. This is accomplished

by reducing the inaccuracy in the estimation of the average

absorptivity 

Several studies exist that aim at calibrating room acous-

tics simulations to obtain parameter values matching the

measurement results.

43–46

However, applying the calibration

to Sabine’s or Eyring’s models is rare,

as most of the

research focuses on simulations in software, such as

ODEON

47–49

or CATT-Acoustic.

50–52

Also, in the majority

of the studies, there is no record of collecting the atmo-

spheric data during measurements.

In the present study, the measurements are performed in

an environment that allows changing the amount and distri-

bution of sound absorption in the room (described in more

detail in Sec. V). Hence, the 

acalibration is conducted with

a similarly adjustable case in mind.

Here, we assume that the change of the total absorption

within the space 

ais directly proportional and linearly

related to the amount of absorbing material. The model for

the calibrated 

ais



ac;XðnÞ¼bXþnDX;(6)

where nis the number of ﬁxed portions of absorbing mate-

rial added to the space (in our case, one variable acoustics

panel changing its state from reﬂective to absorptive, cf.

Sec. V). The adopted approach consists of a base absorption

coefﬁcient bXrepresenting the most reﬂective conﬁguration

of the measured space, i.e., with the smallest possible 

a. The

change of absorptivity resulting from the addition of absorb-

ing material is symbolized by a step value DX.

Here, the subscript X 2fS;E;Tgdenotes the particular

formula to be calibrated. Subscript S represents Sabine’s

model and E Eyring’s method. Additionally, subscript T

denotes absorption values that are based on laboratory mea-

surements or the literature. Since neither Sabine’s nor

Eyring’s model considers absorption distribution, the

change in 

ais also considered as distribution-independent.

To verify that the predictions by Sabine’s and Eyring’s

models can come close to measurement results, the average

absorptivity 

ais ﬁrst ﬁtted to match ^

abased on the obtained

values of RT. Stemming directly from the relation between

the sound-absorption coefﬁcient and the RT values

(T60 1=

a), the problem of optimizing 

ac;Xis of nonlinear

nature. Here, error minimization is performed on the abso-

lute difference between the measured ^

afor combination k

with certain absorption conditions (the minuend) and an 

estimated from either of the equations for the same combi-

nation (the subtrahend). The total number of analyzed

absorption combinations is denoted as K.

In the case of Sabine’s equation, the problem is formu-

lated as

bS;DS¼arg min

bS;DSX

k¼1

ak

ac;SðnkÞj

¼arg min

bS;DSX

k¼1

akðbSþnkDSÞj;(7)

whereas for Eyring’s formula, it is deﬁned as

bE;DE¼arg min

bE;DEX

k¼1

akðln ð1

ac;EðnkÞÞÞj

¼arg min

bE;DEX

k¼1

akþln ð1ðbEþnkDEÞÞj:(8)

Assuming that the distribution of the measured RTs is not

normal, the absolute values are not squared. This avoids

amplifying the effect of outliers on the calibration, making

the process more robust.

IV. AIR-ABSORPTION COMPENSATION

This section presents the formulas to theoretically deter-

mine the air-absorption coefﬁcient m. The applicability of a

pure-tone mto determine the absorption in full-octave bands

and its further use in the air-absorption compensation proce-

dure is discussed as well.

A. Attenuation of sound in air

To account for both the absorbing properties of the

enclosure’s surfaces and the decay caused by sound propa-

gating through air, Eq. (1) is extended with a second

exponential,

32,34

Iðt;fÞ¼Ið0;fÞexp 

aðfÞ



lct



expðmctÞ:(9)

This results in the following changes in Eq. (2):

T60ðf;mÞ¼ ln ð106Þ

ð^

aðfÞþ

lmÞc:(10)

Equivalent corrections are also made to Sabine’s and

Eyring’s formulas. When the air absorption is omitted, i.e.,

m¼0, we write T60ðf;0Þ¼T60 ðfÞ.

The value of the air-absorption coefﬁcient mdepends

on the frequency of sound and the atmospheric conditions. It

is derived from the attenuation of sound in the air, aa, which

is expressed in dB/m,

mðfÞ¼ aaðfÞ

10 log ðeÞ;(11)

where e denotes Euler’s constant (e ¼2:71828).

The attenuation of sound in the air is a function of the

relaxation frequencies of oxygen frO and nitrogen frN , which

are calculated, respectively, from

53,54

1160 J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al.

https://doi.org/10.1121/10.0013575

frO ¼pa

24 þ4:04 104hv

0:02 þhv

0:391 þhv



(12)

and

frN ¼pa



1=29þ280hv

exp 4:170 Ta



1=3

1

"# !!;(13)

where pais the ambient atmospheric pressure in kPa,

pr¼101:325 kPa is the reference ambient atmospheric

pressure, Tais the ambient atmospheric temperature in K,

Tr¼293:15 K is the reference ambient atmospheric temper-

ature, and hvis the molar concentration of water vapour,

presented as a percentage and dependent on the relative

humidity (RH). Based on these quantities, the pure-tone

sound-attenuation coefﬁcient for atmospheric absorption for

a speciﬁc frequency fis expressed as

aaðfÞ¼8:686f21:84 1011 pr

Ta



1=2

þTa



5=2

0:01275 exp 2239:1



frO

rO þf2



þ0:1068 exp 3352:0



frN

rN þf2



#!:(14)

B. Air-absorption coefficient in octave frequency

bands

The air-absorption coefﬁcient obtained with Eqs.

(11)–(14) is applicable for estimating the effect of the atmo-

spheric conditions on the decay of pure tones only.

However, the measured RT values as well as those esti-

mated with the prediction formulas are usually given for

full- or third-octave bands. Therefore, a suitable representa-

tion of the air-absorption coefﬁcient mfor the whole band

needs to be determined.

A few approaches estimate the effect that air absorption

has on sound decay in the frequency bands. One, introduced

by Sisler and Bass,

integrates the power spectral density of

the signal over third-octave bands. They showed that when

the atmospheric absorption is comparable with the boundary

absorption, the differences between pure-tone and band RT

values are signiﬁcant.

An approach following a similar reasoning was adapted

by Wenmaekers et al.

to calculate the effective air absorp-

tion in full-octave bands. A special requirement of this

method is the RT value without the air absorption, which is

unavailable in many scenarios utilizing air-absorption com-

pensation, e.g., when estimating the room acoustic parame-

ters from scale model measurements. The RT would have to

be obtained either through measurement or simulation, and

this is an additional source of uncertainty in further

calculations.

Considering that there is no agreement in the literature

on how to use the air-absorption coefﬁcient m,

34,56,57

asim-

ple experiment was conducted to compare the air-absorption

coefﬁcient values obtained with available methods. First, the

mvalues for all the pure tones within the frequency range of

interest (125 Hz–8 kHz) were calcu lated for Ta¼298:15 K,

RH of 50%, and pa¼101:325 kPa, the standard pressure.

The obtained values are marked in Fig. 1with a solid

black line and serve as the reference for further calcula-

tions, for which the atmospheric condition parameters were

ﬁxed at the aforementioned values. The outcomes of the

rest of the calculations are assumed to fall close to the ref-

erence. The values of the center frequencies of each octave

band are highlighted, since often they are used as nominal

values of m.

The air-absorption coefﬁcients of center frequencies

were compared with the values averaged over a number of

pure-tone ms from the whole considered range. The amount

of pure tones used for averaging changed with each band,

spanning from 88 for the 125-Hz band to over 5000 for the

8-kHz band. The results of this experiment are presented in

Fig. 1. The differences between the air-absorption coefﬁ-

cients of the center frequencies and obtained with averaging

are small, with the averaged mbeing between 8.2% and

8.4% higher than for the center frequencies.

V. RT MEASUREMENTS

This section presents the measurements conducted and

equipment used in this study in the variable acoustics labo-

ratory Arni at the Acoustics Lab of Aalto University, Espoo,

Finland.

A. Measurement space and setup

The Arni room is rectangular in shape with dimensions

8.9 m 6.3 m 3.6 m (length, width, and height, respectively).

FIG. 1. Air-absorption coefﬁcient values obtained with different methods

for Ta¼298:15 K, RH of 50%, and pa¼101:325 kPa. The vertical dashed

lines mark the octave bands. w.r.t., with respect to.

J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al. 1161

https://doi.org/10.1121/10.0013575

The walls and the ceiling of the room are covered with variable

acoustics panels made from painted metal sheets and ﬁlled with

absorptive material. On the front of the panels, rectangular slots

are cut out from the surface. The slots can be opened, letting

the sound reach the absorptive material inside, or closed, mak-

ing the surface reﬂective. The dimensions of a single panel are

0.6 m 0.4 m 2.4 m (length, width, and height). The absorp-

tive material is 25 cm thick, allowing the closing mechanism to

move behind the front surface of the panel. There are in all 55

panels in the variable acoustics laboratory, including eight on

three of the walls, 11 on the fourth wall, and 20 on the ceiling.

The panels on the three walls are placed directly on the ﬂoor,

whereas those on the fourth wall are hanging 63 cm above the

ﬂoor due to the heating installations situated on that wall. The

view of Arni and the equipment used in the measurements are

presented in Fig. 2.

During the measurements, a 01 dB LS01 omnidirec-

tional loudspeaker served as the sound source. There were

a few types of receivers used in the procedure: two

G.R.A.S. (Holte, Denmark) 1/2-in. diffuse-ﬁeld micro-

phones of type 40AG, two G.R.A.S. 1/2-in. free-ﬁeld

microphones of type 46AF, and one Br€

uel & Kjær

(Nærum, Denmark) 1/2-in. diffuse-ﬁeld microphone of

type 4192. A G.R.A.S. power module of type 12AG was

used as an ampliﬁer. All the equipment was connected to a

measurement laptop via a MOTU UltraLite mk3 Audio

Interface. The atmospheric data were gathered using a

Testo 174H Mini data logger. The positions of the sound

source, the receivers, and the atmospheric data logger are

marked in Fig. 3.

The measurement signal was a 3-s-long exponential

sine sweep

59,60

with frequency response spanning from

20 Hz to 20 kHz. The sweep was played ﬁve times for each

panel conﬁguration with 2 s of silence in between to allow

the sound to fully decay, achieving signal-to-noise ratios

between 40 and 50 dB. The total number of recorded

measurement signals for each panel conﬁguration was 25

(5 sweeps 5 receivers). The atmospheric data were col-

lected once for each measurement of ﬁve sweeps.

In total, K¼5312 panel combinations were measured.

This included the scenarios in which all panels are absorp-

tive, all panels are reﬂective (one combination each); one

panel is absorptive, one panel is reﬂective (55 combinations

each); and 2–54 panels are absorptive (100 combinations

each). The panel state transition implies a gradual shift of

the total absorption in the room by a value proportional to

the number of absorptive and reﬂective panels. The database

of the measured RIRs is available online.

B. Measurement accuracy

Since the measurement procedure was automated—

changing of the panel states, outputting the excitation signal,

recording the measurement, and writing to a ﬁle were exe-

cuted using a PYTHON script—the process was not continu-

ously monitored. Therefore, non-stationary noise events

disturbing the measurement were not caught during the pro-

cess and had to be detected in a post-processing stage. To

identify which of the measured sweeps are free from non-

stationary disturbances, the procedure called rule of two

(Ro2) was employed.

From the signals free of contamina-

tion, i.e., those classiﬁed by Ro2 as having high values of

Pearson’s correlation coefﬁcient, one sweep per receiver

position in a combination was used later in the result

analysis.

When presenting results, the median of the T

values

across all combinations with nabsorptive panels for all

receiver positions was used as an estimator of the “main”

value of a combination’s RT. The results also show the

trends in the RT change with the decrease in the room’s

absorption. The median RT was estimated as

FIG. 2. Variable acoustics laboratory Arni and the equipment used in the

measurements.

FIG. 3. Layout of the variable acoustics room Arni showing the positions of

the panels, sound source, and receivers.

1162 J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al.

https://doi.org/10.1121/10.0013575

T60;n¼medianðT60;kjnk¼nÞ;(15)

where n¼0;1;2;…;55 is the number of absorptive panels

in a combination, and n

is the number of absorptive panels

for combination k.

VI. RESULTS

This section presents the results of RT measurements

and compares them with the outcome of estimations

obtained with the Sabine and Eyring formulas. The distribu-

tion of the obtained RT values is discussed, and the air-

absorption compensation is performed. Furthermore, the

section analyzes the correction that sound-absorption cali-

bration introduces to the predicted RT.

A. Air-absorption compensation

The RT values were calculated from the measured RIRs

using the DecayFitNet software

due to the achieved

robustness of the RT predictions in the presence of the back-

ground noise. The correctness of decay estimations was

assessed using the mean squared error (MSE) between the

predictions and measured RIRs. The median MSE for all

combinations and six frequency bands was 3.34 dB with a

median absolute deviation of 2.01 dB, showing good agree-

ment between the recorded and estimated decays.

DecayFitNet estimates the decay parameters of the ana-

lyzed RIRs, such as the initial energy level and the decay

rate, together with the background noise level. Thus, it is

possible to resynthesize the energy-decay curve extended

beyond the noise ﬂoor.

Such a procedure was used in the

present research; hence, the RT values presented here were

calculated as T

times, over the 60-dB decay range,

between –5 and –65 dB. The results for the octave bands

250 Hz–8 kHz are shown in Figs. 4(a)–4(f).

Due to the enormous number of measurements, they

were conducted over a long period of time (approximately 2

weeks), during which the atmospheric conditions varied

considerably. Figure 5shows the change in RH, ambient

temperature, and ambient atmospheric pressure during the

entire measurement period. The inﬂuence of these changes

on the predicted RT values was signiﬁcant, especially in the

4- and 8-kHz bands, as shown in Figs. 4(e) and 4(f), with

the values of median RT with air absorption.

The effect of air absorption adds considerable bias to

the measurement results, such as introducing drops to the

RT values, when increase is expected. For instance, for con-

ditions with zero and one absorptive panels, the RT values

in the 4- and 8-kHz bands, denoted with white circles in

Figs. 4(e) and 4(f), are lower than when two or three panels

were absorptive. During the time these conditions were mea-

sured, the temperature was relatively stable, staying at

between 19.5 Cand19.9

C. Bigger differences were regis-

tered in the atmospheric pressure, which dropped from

101.9 kPa for zero and one absorptive panels to between 100.5

and 100.7 kPa for conditions with two and three absorptive

panels. The most prominent discrepancies, however, were

observed in RH, which for the ﬁrst two sets of measurements

oscillated around 28.5%, whereas the latter two sets of mea-

surements were performed in RH changing from 40% to 50%.

To increase the robustness of the result analysis, the air

absorption is compensated.

In the present work, the effect of the air absorption is

subtracted from the measurement results according to the

following formula:

T60ðfÞ¼ 1

T60ðf;mÞcm

ln ð106Þ



1

:(16)

To choose the mused in this formula to be either the mof

the center frequency of the octave band or the result of aver-

aging of pure-tone coefﬁcients as discussed in Sec. IV B,

both of those approaches were used on the measurement

results. The RT values compensated with the use of averag-

ing formed a smoother curve with more shallow drops than

those compensated with the center frequency m. Hence,

averaging was chosen as the air-absorption-compensation

method in the remainder of this paper.

The results compensated for air absorption are dis-

played in Figs. 4(a)–4(f) and consistently used in the

remainder of this paper. The biggest change in RT values

after the compensation is observed at high frequencies, such

as in the 4- and 8-kHz bands. However, the same effect,

although much less signiﬁcant, is observed already for RTs

over 1 s in the 250-Hz and over 0.6s in 500-Hz band in Figs.

4(a) and 4(b).

The above changes oppose the common approach of

considering air absorption only for spaces with volumes

over 200 m

(Refs. 6and 17) (the nominal volume of Arni is

201.9 m

, but when the adjustable panels are in their closed

state, the volume is considered to decrease by 0.58 m

per

panel). Air-absorption compensation is, therefore, advised

when analyzing the results of acoustic measurements. It is

especially needed when comparing results of measurements

done in different atmospheric conditions, e.g., on different

days, or comparing measured and simulated RT data.

B. Distribution of RT values

The measured values show that the change in the place-

ment of absorptive and reﬂective panels produces a rela-

tively broad distribution of RT values around the median.

Because of Arni’s dimensions, the Schroeder frequency

reaches as high as 200 Hz when the total absorption within

the room is low. This results in a low modal density below

that frequency, impeding the measurement of the RT values

in the 250-Hz band and possibly causing wider spreads

around the median as well as outliers. Similarly, the distri-

bution of the absorptive and reﬂective panels seems to affect

the distribution of the measured RT values. Such an effect

appears to be more prominent in 250-Hz to 1-kHz bands, as

seen in Figs. 4(a)–4(c). For high frequencies, the distribution

is narrower, even though some outliers are spotted in the 8-

kHz band in Fig. 4(f).

J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al. 1163

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In the 1–8 kHz bands, most of the results in Fig. 4lie

within 610% from the median, whereas in the low frequen-

cies, the majority of RT values are inside 625% from the

median. The exact numbers specifying the spread of values

for each band are presented in Table I, conﬁrming wider dis-

tributions in low frequencies.

For the majority of combinations, the distribution of RT

values cannot be classiﬁed as normal. This justiﬁes the

choice of median as an estimator (since the mean value is

more susceptible to contamination by outliers, it would

prove less robust). An example of a probability distribution

plot showing the measured values relative to the median for

the 1-kHz frequency band and all panel conditions is pre-

sented in Fig. 6.

C. Predicted RT values

In this work, the sound-absorption coefﬁcients used to

obtain the average absorptivity 

ain Sabine’s and Eyring’s

formulas were taken either from the data provided by the

material’s manufacturer (in the case of variable acoustic

panels and curtains) or from the literature (in the case of

wall and ﬂoor materials). Their values are presented in

Table II.

The results of calculations using table-based 

aand

m¼0 in the Sabine and the Eyring formulas are compared

with the measured RT values and are displayed in Fig. 7

with blue and red dots, respectively. They show that both

formulas predict the reverberation of the room with a con-

siderable error. The best estimations are obtained in the

250-Hz band, where both models follow the median RT

FIG. 4. RT values with air absorption and air-absorption compensation for octave frequency bands 250 Hz–8kHz. The median is shown for values with air

absorption and air-absorption compensation, whilst the measurement points (each denoted with a gray dot) are presented only for the air-absorption-compen-

sated case. Removing the air absorption from the results mostly affects high frequencies, such as 4–8 kHz, but is visible in all bands when the number of

absorptive panels is low.

FIG. 5. Change of values of (top) RH, (middle) ambient temperature, and

(bottom) ambient air pressure for each of k¼1;…;Kcombinations during

the entire measurement period. The temperature was the most stable,

whereas humidity varied the most.

1164 J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al.

https://doi.org/10.1121/10.0013575

closely (Sabine with a slight offset) below the value of

approximately 0.8 s. At 500 Hz and 1 kHz, the low T

combinations are estimated correctly by the formulas, but

the error increases rapidly when the T

is high. The pre-

dicted RT values for frequencies 2–8 kHz are too low

across all combinations in Fig. 7. The average error per-

centage across all combinations for both formulas in each

octave band is shown in Table III, further conﬁrming the

prediction inaccuracy.

An important observation is that whilst the results of

the measurements are distributed around the median value,

the estimations are not. The inability of Sabine’s and

Eyring’s formulas to adapt to a changing absorption distri-

bution is an error inherent to those equations, which was

already discussed in literature

40,67

and is not further exam-

ined in the present study.

D. Calibrated sound-absorption coefficient

The calibration of sound-absorption coefﬁcients was

performed according to Eqs. (7) and (8) for Sabine’s and

Eyring’s formulas, respectively. A median absorption for

combination kand all ﬁve receivers was used as ^

akwith

K¼5312. The base values of absorptivity in Arni,bSand

bE, were equivalent to the case when all the panels are in

their reﬂective state, whereas the difference in absorption,

DSand DE, indicated the change of panel state from reﬂec-

tive to absorptive. Similar calculations were conducted

for bTand DT, following the example given below for the

500-Hz band,

bT;500 ¼1

Sð0:05 55 Spanel þ0:03 Swall

þ0:03 Sfloor þ0:95 ScurtainÞ¼0:1531;

DT;500 ¼1

Sð0:05 54 Spanel þ0:77 Spanel þ0:03 Swall

þ0:03 Sfloor þ0:95 ScurtainÞbT;500 ¼0:0059:

(17)

The comparison between the values of 

abased on

Table II, as well as 

ac;Sand 

ac;E, is presented in Fig. 8. The

differences between bT;bS, and bEare small for the 250-Hz

band, but the table-based numbers are signiﬁcantly higher

than the calibrated ones in all the remaining bands. As DSis

consistently higher than DE, no such relation is observed

between bSand bE.

The results of the calibration for all measurements are

presented in Fig. 9. Ideally, both measured and predicted RT

values would be equal, forming a diagonal line. However,

this is impossible since the measured values are inﬂuenced

by absorption distribution, which is not accounted for either

in Sabine’s or Eyring’s formulas. Therefore, in Fig. 9, the

estimated absorption coefﬁcients create vertical lines that

assign several values of measured T

for each number pre-

dicted with the Sabine or Eyring formulas. However, the

stepwise nature of calibrated values does not contribute to

an excessively wide distribution of the results, mostly ﬁtting

within 610% of the target diagonal.

The exact proportions of measured RT values not

exceeding the 10% and 25% limits are presented in Table

IV, which shows that the spread is almost the same for both

analyzed formulas, with slight advantage in Eyring’s model

in the 250-Hz and 1-kHz bands. The results of Table IV also

TABLE I. Proportion of measured RT values ﬁtting within 10% and 25%

from the median for each frequency band in Fig. 4.

Frequency 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz

610% of ~

T60;n65% 79% 94% 99% 99% 99%

625% of ~

T60;n97% 98% 99% 100% 100% 100%

FIG. 6. Distribution of RT values for the 1-kHz band and each panel condi-

tion [cf. Fig. 4(c)]. The relative probabilities for each absorption condition

are offset to ensure better readability.

TABLE II. Sound-absorption coefﬁcients of individual materials used for

obtaining 

afor T

calculations as well as bTand DTin Fig. 8.

Material 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz

Panel absorptive (Ref. 64) 0.86 0.77 0.66 0.45 0.38 0.42

Panel reﬂective (Ref. 64) 0.09 0.05 0.05 0.04 0.02 0.03

Wall (Ref. 65) 0.02 0.03 0.03 0.04 0.05 0.05

Floor (Ref. 65) 0.02 0.03 0.03 0.03 0.02 0.03

Curtain (Ref. 66) 0.45 0.95 0.99 0.99 0.99 0.99

J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al. 1165

https://doi.org/10.1121/10.0013575

align well with the distribution of measured RT values pre-

sented in Table I. The percentage of average error of cali-

brated predictions relative to the measured RT values shows

great improvement compared with the error from before the

calibration, cf. Table III.

It is crucial to note that some amount of error in the pre-

dictions is expected, as both formulas are claimed to be

accurate only in a diffuse sound ﬁeld (cf. Sec. II B). The

sound ﬁeld within Arni, however, is assumed to not be fully

diffused. Factors such as unevenly distributed absorption

(even when all the panels are in the same state, other

elements of the interior prevent the uniformity of absorptiv-

ity distribution) and the shoebox shape of Arni

both indi-

cate lack of isotropy and homogeneity.

Considering that the just noticeable difference (JND)

for reverberation perception varies depending on the type of

sound

70–74

and the character of decay,

ranging from

around 3% for speech signals

to over 20% for band limited

noise and music,

71,74,76

the calibrated predictions ﬁt within

those constraints. This precision is crucial, considering that

design decisions for concert halls and auditoriums are made

based on the RT values estimated with Sabine’s or Eyring’s

formulas.

VII. DISCUSSION

There are many possible sources of uncertainty in RT

predictions. A few of them are mentioned in this paper,

namely, the incorrectly assumed sound-absorption coefﬁ-

cient, the error inherent to the analyzed formulas, and

neglect of the effect of the air. The results may be biased

also by the uncertainty coming from the measurement

equipment, the method used to calculate the RT values from

obtained RIRs, and the destructive effect of stationary and

non-stationary noise and time variance.

77,78

To account, at least partially, for the diffuse reﬂections

in the decay, the modiﬁed sound-absorption coefﬁcient

based on the diffusion model

was also examined in the

present study. Due to its not showing a signiﬁcant improve-

ment in predictions with Eyring’s formula, it was not

included in Sec. VI for the sake of clarity of presentation.

The outcome of this method is well in agreement with the

results presented in the literature for the sound-absorption

coefﬁcient values under 0.6.

The present study shows that when the sound-

absorption coefﬁcient is chosen carefully, the traditional

Sabine and Eyring RT prediction formulas achieve good

accuracy. However, the estimation of the sound-absorption

coefﬁcient remains challenging. The current method to mea-

sure the absorption of materials is based on the Sabine for-

mula,

which may not agree well with multiple-slope

FIG. 7. Comparison between measured (cf. Fig. 4) and predicted RT values

for 

a. In the majority of cases, the estimations are not consistent with the

measurement results.

TABLE III. Average absolute error of RTs predicted with Sabine’s and

Eyring’s formulas in each octave band using 

a, relative to the measured RT

values, cf. Fig. 7.

Frequency 250Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz

Average error Sabine 7% 12% 14% 22% 23% 34%

Average error Eyring 6% 19% 21% 29% 29% 40%

FIG. 8. Comparison of sound-absorption coefﬁcients based on data from

Table II and the results of calibration of Sabine’s and Eyring’s models.

1166 J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al.

https://doi.org/10.1121/10.0013575

decays observed in reverberation chambers.

This approach

creates an error buildup when an inaccurately estimated 

ais

used to calculate the RT in a room not fulﬁlling the condi-

tions required by the Eyring and Sabine formulas.

Apart from that, standardized measurements require the

use of a speciﬁc amount, edge-to-area ratio, and arrangement

of absorbing material to achieve signiﬁcant effect on the

acoustics of a reverberation chamber, as well as to eliminate

the edge effect that skews the results.

Such placement of

acoustic materials is, however, rarely used in acoustic adap-

tations of concert halls, auditoriums, and other specialized

facilities. Thus, the ideal 

ain the measured space, under-

stood as a value that will return the measured RT when used

in one of the prediction formulas, might be, in fact, signiﬁ-

cantly different from the one resulting from the values

obtained in the laboratory. Therefore, the debate over the

accuracy of RT formulas will probably be resolved only

with an improved procedure to estimate sound absorption in

practical settings.

VIII. CONCLUSIONS

This study analyzes two of the most popular RT estima-

tion formulas—Sabine’s and Eyring’s equations—and veri-

ﬁes their accuracy in predicting RT values on a big dataset

of measured impulse responses. The results show that, even

in a scenario where the absorption distribution in the room

varies considerably between measurements, both of the

aforementioned models predict the T60ðfÞto within approxi-

mately 610% precision after the proposed calibration.

Both formulas assume that the sound ﬁeld in a consid-

ered room is homogeneous and isotropic, an unachievable

requirement for the vast majority of rooms and a source of

error in the calculations. The uncertainty in results may also

come from using an absorption coefﬁcient that is not equiva-

lent to the actual absorption in the room. The present work

shows that the error resulting from incorrectly assumed

absorptivity preponderates the inaccuracy coming from the

non-diffuse sound ﬁeld. We show that by using a calibrated

sound-absorption coefﬁcient, a sufﬁciently accurate RT esti-

mation, within the limits of JND for reverberation percep-

tion, is achieved.

The results show that Sabine’s and Eyring’s formulas

display good scalability of predictions with the change of

total absorptivity in the room. This means that having cor-

rect estimations of the base and step value of the sound-

absorption coefﬁcient, the equations return accurate results

regardless of the amount and distribution of absorption

added or subtracted from the initial acoustic conditions.

This can be valuable in case of acoustic adaptation, where

results of a few measurements or simulations (base absorp-

tion and absorption change) offer a possibility to precisely

tune the designs to desired RT values without need for extra

computations or additional measurements.

Another important issue, many times neglected in the

RT estimation, is the absorption of the air. The results pre-

sented in this paper show that the air absorption may intro-

duce a signiﬁcant bias to the measurement results.

Therefore, we emphasize that it is necessary to compensate

for the air absorption in RT measurements and simulations.

The study also proposes a method to establish the air-

absorption coefﬁcient for full frequency bands based on the

pure-tone calculations used in standards. Compared with

FIG. 9. Comparison between measured and calibrated (with bS;DSand

bE;DE, respectively) RT. The calibrated values are in good agreement with

the measured ones. Similar to the RT shown in Fig. 7, the distribution is

wider in low frequency bands.

TABLE IV. Proportion of predicted calibrated RTs ﬁtting within 10% and

25% from the measured target values in each octave band in Fig. 9and the

average absolute error of calibrated predictions using bS;DSand bE;DE,

respectively, relative to the measured RT values.

Frequency 250Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz

Sabine 610% 55% 75% 89% 98% 99% 98%

Sabine 625% 94% 99% 99% 99% 99% 99%

Average error Sabine 4% 3% 2% 2% 1% 2%

Eyring 610% 56% 75% 92% 98% 99% 98%

Eyring 625% 95% 99% 99% 99% 99% 99%

Average error Eyring 3% 3% 2% 2% 1% 2%

J. Acoust. Soc. Am. 152 (2), August 2022 Prawda et al. 1167

https://doi.org/10.1121/10.0013575

other approaches for air-absorption compensation, the pro-

posed method is shown to not introduce an error dependent

on the RT values. The procedures proposed in this paper

improve the accuracy of RT predictions in room acoustics.

ACKNOWLEDGMENTS

This work was supported by the Nordic Sound and

Music Computing Network—NordicSMC, NordForsk

Project No. 86892.

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https://doi.org/10.1121/10.0013575

Publication III

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Robust Se-

lection of Clean Swept-Sine Measurements in Non-Stationary Noise. The

Journal of the Acoustical Society of America, Vol. 151, pp. 2117–2126,

March 2022.

Reprinted with permission.

Robust selection of clean swept-sine measurements

in non-stationary noise

Karolina Prawda,

Sebastian J. Schlecht,

and Vesa V€

alim€

aki

Acoustics Lab, Department of Signal Processing and Acoustics, Aalto University, FI-02150 Espoo, Finland

ABSTRACT:

The exponential sine sweep is a commonly used excitation signal in acoustic measurements, which, however, is

susceptible to non-stationary noise. This paper shows how to detect contaminated sweep signals and select clean

ones based on a procedure called the rule of two, which analyzes repeated sweep measurements. A high correlation

between a pair of signals indicates that they are devoid of non-stationary noise. The detection threshold for the

correlation is determined based on the energy of background noise and time variance. Not being disturbed by non-

stationary events, a median-based method is suggested for reliable background noise energy estimation. The pro-

posed method is shown to detect reliably 95% of impulsive noises and 75% of dropouts in the synthesized sweeps.

Tested on a large set of measurements and compared with a previous method, the proposed method is shown to be

more robust in detecting various non-stationary disturbances, improving the detection rate by 30 percentage points.

The rule-of-two procedure increases the robustness of practical acoustic and audio measurements. V

C2022 Author(s).

All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY)

license (http://creativecommons.org/licenses/by/4.0/). https://doiorg/10.1121/10.0009915

(Received 15 December 2021; revised 24 February 2022; accepted 9 March 2022; published online 24 March 2022)

[Editor: Ning Xiang] Pages: 2117–2126

I. INTRODUCTION

An impulse response (IR) measurement is one of the

most common procedures to assess the acoustic qualities of

various systems, including physical spaces, such as concert

halls and rooms,

1–5

electronic devices,

audio software,

and

more. IR measurements can be conducted using a variety of

excitation signals, including impulses, which are produced

with sources such as pistols and balloon pops;

8,9

noise-based

methods, e.g., maximum-length sequence (MLS)

10–12

and

inverse repeated sequence (IRS);

and linear and exponen-

tial swept-sine signals. Each of these methods has its strong

sides and shortcomings.

The exponential swept-sine (ESS) as an excitation sig-

nal for measuring IRs was introduced in the form used now-

adays by Farina over 20 years ago.

Currently, it is widely

used as it provides the best consistency and highest robust-

ness of measurements.

14,16

The ESS technique also rejects

most of the harmonic distortion,

a bane of noise-based

methods, such as MLS and IRS.

14,17–19

However, ESS is

sensitive to non-stationary noise, which causes artifacts in

IRs obtained in the deconvolution process, and may lead to

errors in the estimation of acoustic parameters.

14,20–23

The

present study investigates the ESS technique and discusses

the disturbances that may occur during measurements and

negatively affect the resulting IRs. A novel method to dis-

criminate between clean and corrupted sweeps is proposed.

The ESS technique is known for its excellent signal-to-

noise ratio (SNR)

that results from a long excitation of

low frequencies, which are usually more susceptible to con-

tamination by background noise than high frequencies. This

feature of swept-sine signals can be employed to achieve

target SNR values for different frequencies by adjusting the

time over which speciﬁc frequencies are excited.

24–30

The

vulnerability of the ESS method to non-stationary noise,

however, grows proportionally with the length of the sweep

signal. This may force a compromise between lengthening

the ESS signal to increase the SNR and shortening it to min-

imize the risk of the occurrence of non-stationary noise.

19,20

In this light, Stan et al.

recommend using the swept-sine

technique only for measurements in empty, quiet spaces.

Currently, there is no established method to identify

non-stationary noise in sweep measurements. Manual detec-

tion works only for singular measurements, but in the case

of numerous unsupervised measurements, automatic detec-

tion is necessary. Guski

31,32

presented an algorithm address-

ing the problem of automatic classiﬁcation of contaminated

sweeps. Relying, however, on the separation of IR and back-

ground noise, this method is prone to errors when estimating

decay and noise ﬂoor. Therefore, the need for a simpler and

more reliable procedure remains.

This paper proposes to identify clean and contaminated

ESS measurements based on their similarity to each other,

expressed by the Pearson correlation coefﬁcient (PCC). Used

in applications such as pattern recognition

and as a criterion

for ﬁlter optimization,

PCC proved to be more advanta-

geous than the mean square error criterion. Similarly, cross

correlation was used as a measure to estimate IRs sensitivity

to small changes in sound-source position

as well as for

robust IR measurement against nonlinearities.

36–38

This sug-

gests that parameters related to similarity are good indicators

Electronic mail: karolina.prawda@aalto.ﬁ

Also at: Art and Media Lab, Dept. of Media, Aalto University, FI-02150

Espoo, Finland.

J. Acoust. Soc. Am. 151 (3), March 2022 V

CAuthor(s) 2022. 21170001-4966/2022/151(3)/2117/11

ARTICLE

...................................

of changes in audio signals, even when the environment is

not free from noise.

The present work studies the problem of ESS measure-

ments corrupted by non-stationary noise and introduces a

procedure called the rule of two (Ro2). Ro2 is a method to

identify a pair of clean sweeps, those not contaminated by

non-stationary noise, from a series of measurements in a

noisy environment. The method is based on the correlation

between measured ESS signals. Various factors impacting

the correlation are examined. The threshold separating clean

sweeps from corrupted ones is determined. The Ro2 proce-

dure is tested on a big dataset of ESS measurements and is

compared to another method aimed at detecting impulsive

noise in sweep measurements.

The remainder of this paper is organized as follows.

Section II discusses the correlation between acoustic signals

and describes the proposed method. In Sec. III, the expected

contamination, such as stationary noise and time variance,

are presented. Section IV elaborates on the types of non-

stationary contamination and their effect on the correlation.

Section Vdescribes the validation procedure for the pro-

posed method, discusses the experimental results, and com-

pares the proposed method with another technique. Section

VI concludes the paper.

II. METHOD

This section tackles the detection of non-stationary

events in an ESS signal and proposes a novel method called

Ro2. The correlation of acoustic signals is also discussed.

A. Problem formulation

Assessing whether the signal obtained during acoustic

measurement is free of non-stationary noise or other arti-

facts is often a difﬁcult task. Therefore, a good practice is to

record a few test signals so as to be able to choose the best

one, should unexpected acoustic events occur. In this case,

recordings of the same conditions of the system under test

can be compared to one another.

Given two acoustic measurements y

and y

,wewant

to determine whether they are clean or not. Assuming that

contamination is a random occurrence, we measure the

similarity of y

and y

as an indicator of contamination:

if the similarity is low, then the contamination is indi-

cated (in either one or in both of the signals), whereas a

great similarity denotes an uncontaminated pair. We pro-

pose PCC as a robust measure of similarity. PCC is

deﬁned as

qy1;y2¼covðy1;y2Þ

ry1ry2

N1X

k¼1

ðy1ðkÞly1Þðy2ðkÞly2Þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

N1X

k¼1

ðy1ðkÞly1Þ21

N1X

k¼1

ðy2ðkÞly2Þ2

(1)

where covðy1;y2Þis the covariance of signals y

and y

,ry1

and ry2are their standard deviations, ly1and ly2are the

mean values, and Nis the number of samples in the signals.

In acoustic measurements, the mean of the measured signals

is removed,

transforming Eq. (1) to

qy1;y2¼X

k¼1

y1ðkÞy2ðkÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k¼1

y1ðkÞ2X

k¼1

y2ðkÞ2

s:(2)

Assuming that the system under test is free of noise,

and neither the system nor the measurement equipment

changes between or during the recording of both signals,

i.e., y

¼y

, then qy1;y2¼1. Consecutive measurements,

however, are never strictly the same, and the PCC of two

clean measurements is impacted by two classes of factors:

(1) expected disturbances including stationary background

noise and time-variances of the measured system and (2)

non-stationary occurrences such as impulsive noise or sound

dropouts.

31,40

Note that in the present study the term “clean signal”

refers to a measured signal that contains stationary back-

ground noise and effects of time variance only, whereas the

term “contaminated” is used for the signals containing both

expected and unexpected disturbances.

An example of a set of measured ESS signals is shown

in Fig. 1, where one of ﬁve sweeps is contaminated with

impulsive noise. The corresponding PCC matrix is presented

in Table Itogether with the total energy of each sweep in dB.

The contaminated signal displays lower similarity with the

other sweeps, while also having higher energy than the rest.

B. Proposed method

The proposed method presents a systematic criterion to

distinguish expected disturbances from non-stationary noise

FIG. 1. (Color online) Spectrogram of a measurement consisting of ﬁve

consecutive swept-sine signals. The arrow points to an impulsive noise

event appearing in sweep #3 at about 14 s.

2118 J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al.

https://doi.org/10.1121/10.0009915

to create a meaningful and robust measure for the level of

contamination. The Ro2 method requires a correlation

threshold ^

qy1;y2separating clean signals from contaminated

ones. Thus, the Ro2 is

if qy1;y2>^

qy1;y2then y1and y2are a clean pair:(3)

When non-stationary noise occurs in the measure-

ment, the PCC value does not point directly towards the

contaminated sweep. Thus, when qy1;y2<^

qy1;y2,themea-

surement should be repeated and the correlation of all cap-

tured signals should be estimated. The measurement can

end when at least two signals fulﬁll the requirement in Eq.

(3). Section III shows how to determine the threshold

qy1;y2.

III. EXPECTED CONTAMINATION

In the following, we present the two main sources of

“unavoidable” impacts on correlation that are identiﬁed

here, namely, background noise and time-variance. The fol-

lowing discussion leads to the determination of the expected

PCC ^

qy1;y2, which serves as the detection threshold.

A. Effect of background noise

The term “background noise” in acoustic measurements

refers to any type of unwanted extra sound event. Since this

deﬁnition includes non-stationary noise, the distinction

needs to be made that in this study “background noise” is

used to describe only the stationary noise.

The presence of stationary noise in the sweep signals

affects their correlation. Therefore, in Eq. (2),weneedto

consider two noise signals n

and n

with zero mean.

For this subsection, the background noise is the only dis-

turbance such that the measurement signal is a mixture of

the signal with the background noise, i.e., y1¼xþn1,

where

x¼sh(4)

is the convolution of the ESS sand room impulse response

h, denoted by an asterisk . Similarly, y2¼xþn2. The

resulting correlation is then

qy1;y2¼X

k¼1

ðxðkÞþn1ðkÞÞðxðkÞþn2ðkÞÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k¼1

ðxðkÞþn1ðkÞÞ2X

k¼1

ðxðkÞþn2ðkÞÞ2

s:(5)

If the noise signals are uncorrelated with the ESS signals as

well as with each other, i.e., PN

k¼1n1n2¼0;PN

k¼1xn

1¼0,

and PN

k¼1xn

2¼0, then, Eq. (5) can be simpliﬁed to

qy1;y2¼X

k¼1

xðkÞ2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k¼1

ðxðkÞ2þn1ðkÞ2ÞX

k¼1

ðxðkÞ2þn2ðkÞ2Þ

s:(6)

Thus, the signal energies are related by

qy1;y2¼Ex

½

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðEx

½

þEn

½

ÞðEx

½

þEn

½

p;(7)

where the energy of a signal is computed as

½

¼X

k¼1

jxðkÞj2:(8)

When the noise signal energies are equal, i.e.,

E½n1¼E½n2¼E½n, the PCC can be estimated using the

SNR value,

qy1;y2¼Ex

½

þEn

½

¼SNR

1þSNR ;(9)

where the SNR is expressed in terms of signal energies,

SNR ¼Ex

½

:(10)

In practice, E½xis unknown as it is affected by the room

impulse response h. However, it can be inferred from the

difference E½x¼E½y1E½n1¼E½y2E½n2.

B. Correlated background noise

Equation (9) provides an expected PCC value based on

the assumptions that (1) the sweep responses are identical

and (2) the background noise is uncorrelated and stationary.

In the following, we discuss these assumptions.

The background noise is likely to contain strong har-

monic content caused, for instance, by electric humming.

Depending on the phase relation between measurements,

harmonic background noise can be strongly correlated.

Let us consider the two extreme cases: when the noise

signals n

and n

are fully correlated positively or negatively

(anticorrelated). Thus, n1¼6n2, ergo PN

k¼1n1n2¼6E½n.

This yields the following bounds to Eq. (9):

TABLE I. PCC matrix for a series of ﬁve sweeps, cf. Fig. 1. Sweep #3 is

less similar to the other signals, indicating the presence of non-stationary

noise. The largest energy of sweep #3 also suggests the presence of addi-

tional noise. The smallest PCC values and the largest energy are

highlighted.

Sweep # 1 2 3 4 5 Energy (dB)

1 1.000 0.999 0.995 0.999 0.999 71.76

2 0.999 1.000 0.995 0.999 0.999 71.75

30.995 0.995 1.000 0.994 0.994 71.83

4 0.999 0.999 0.994 1.000 0.999 71.74

5 0.999 0.999 0.994 0.999 1.000 71.73

J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al. 2119

https://doi.org/10.1121/10.0009915

½

En

½

þEn

½

^

qy1;y2;fEx

½

þEn

½

þEn

½

¼1;(11)

where f¼qn1;n2is the correlation between two stationary

noise terms. Note that perfectly correlated background

noise, as part of the measurement signal, is virtually indis-

tinguishable from an ESS.

To this end, an experiment showing the relation

between PCC and SNR values was conducted. A 3-s-long

ESS was synthesized and convolved with a synthetic IR

having reverberation time (RT) of 2 s. This signal was

then added to a set of white and pink noise signals

having various energies so that different values of SNR

could be obtained. The noise signals were either uncorre-

lated (f¼0) or anticorrelated (f¼1). The PCC values of

these combined signals were calculated using Eqs. (9)

and (11).

To simulate background noise with harmonic content,

sawtooth waves were added to the aforementioned noise sig-

nals. The phase shifts between these signals were random-

ized between 0 and p. The results of the experiment, shown

in Fig. 2, indicate that for clean signals, i.e., without non-

stationary noise, PCC calculated as a function of SNR

reaches high values close to unity. The results show that the

spectral characteristics of the stationary noise have a negli-

gible effect on the correlation as long as the noise does not

contain periodic components that may result in sharp peaks

or dips in the spectrum, e.g., sine waves.

The results also illustrate that harmonic content in a

noise signal can heavily inﬂuence the correlation in both

positive and negative directions. In Fig. 2, phase shifts for

different SNR values create lines parallel to the ones result-

ing from the assumptions of uncorrelated and anticorrelated

stationary noise. Small phase shifts close to 0 produce a

highly correlated signal, whereas the increase in phase shift

towards pdecreases the PCC values, placing the signals

with the biggest shift between the uncorrelated and anticor-

related boundaries.

In principle, estimating the correlation of the back-

ground noise between measurements is possible if there is

a sufﬁciently long time interval without any other signals.

However, the time intervals between measurements can be

several seconds such that the stationarity of the back-

ground signals needs to be fulﬁlled precisely to reliably

estimate the correlation. Here, we adopt the worst-case

scenario of anti-correlated noise as a lower bound for the

expected PCC.

C. Transfer-function variation

The measured system itself can undergo change. For

instance, the position of the microphone and loudspeakers

may vary due to vibrations, or the propagation paths can be

altered due to variations in the air caused by temperature

and humidity ﬂuctuations or air movement (e.g., due to ven-

tilation).

17,31,40–43

Unlike background noise, the measure-

ment variations impact the impulse response hdirectly such

that the signal model is

x¼sðhþvÞ;(12)

where his some “ideal” room impulse response and vis the

variation of the impulse response between measurements.

Thus, the energy relation of two measurements are

½

¼Esh

½

þEsv1

½

þEn

½

;

½

¼Esh

½

þEsv2

½

þEn

½

:(13)

The difference between the two measurements is then

y1y2¼sðv1v2Þþn1n2:(14)

The energy relation is

1y2

½

¼Esðv1v2Þ½þEn

1n2

½

:(15)

We choose the variation energy between two measurements

such that E½v1¼E½v2and E½sv1¼E½sv2. Thus,

Esðv1v2Þ½¼Esv1

½

þEsv2

½

¼2Esv1

½(16)

since the variations v

and v

are uncorrelated (as the corre-

lated part belongs to hby deﬁnition). Thus, we deﬁne the

transfer-function variation factor

s¼Esðv1v2Þ½

Esh

½¼2Esv1

½

Esh

½

¼2Esv2

½

Esh

½

;(17)

where E½sðv1v2Þ can be retrieved from the difference

y1y2using Eq. (15), and E½shcan be retrieved from

the measurement as E½sh¼E½y1E½sv1E½n1

using Eq. (13) and Eq. (16).

The PCC of the transfer-function variation is

FIG. 2. Comparison of the PCC values for different SNRs of uncorrelated

and anticorrelated stationary noise, as well as noise with harmonic content.

2120 J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al.

https://doi.org/10.1121/10.0009915

qy1;y2¼X

k¼1

y1ðkÞy2ðkÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

k¼1

y1ðkÞ2X

k¼1

y2ðkÞ2

¼Esh

½

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðEsh

½

þEsv1

½

ÞðEsh

½

þEsv2

½

¼Esh

½

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð1þs=2Þ2ðEsh

½

Þ2

q¼1

1þs=2:(18)

Therefore, the transfer-function variation factor sserves as a

tolerance parameter for the expected PCC from Eq. (11),

qy1;y2;f;s¼

qy1;y2;f

1þs=2:(19)

The effect of time variances on the impulse response

measurements can be modeled with time-stretching

or by

introducing sinusoidal jitter to the signal.

The complexity

and unpredictability of such variations, however, might ren-

der these experiments insufﬁcient to predict svalues cor-

rectly. Therefore, in this study, transfer-function variation is

estimated from the measured signals in Sec. V.

IV. NON-STATIONARY NOISE EVENTS

During an acoustic measurement, various non-

stationary disturbances can occur. Such artifacts are, e.g.,

impulses, low-frequency noises, and sound dropouts, which

originate from door slams, heavy vehicles moving outside

of the measured space, and errors in measurement software.

This section examines how different types of non-

stationary noise impact the PCC values, depending on their

energy, frequency content, and time of occurrence. The

effect of contamination on the correlation threshold estima-

tion is also discussed.

A. Impulsive noise

The relation between the energy added to the sweep and

the drop in PCC values can be concluded from Eq. (5) when

we consider that one of the signals is contaminated with

additional non-stationary noise nns, which is also assumed to

be zero-mean and uncorrelated with both sweeps and sta-

tionary noise signals. Following the same reasoning leading

from Eq. (5) to Eq. (7), we arrive at the following formula:

qy1;y2¼Ex

½

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðEx

½

þEn

½

þEn

½

ÞðEx

½

þEn

½

p;(20)

where E½nnsis the energy of non-stationary noise.

The theoretical values of correlation estimated with the

aforementioned formula are presented in Fig. 3. Equation

(20) predicts the general trend of decrease in the PCC with

the growing energy difference between the clean and

contaminated ESS. The energy difference DEis a quotient

of the energy of the two signals,

DE¼Ey

½

¼Ex

½

þEn

½

þEn

½

þEn

½ :(21)

In the experiment, a synthetic sweep signal containing

stationary noise, as described in Sec. III B, with an SNR of

84 dB was further contaminated with impulsive noise and

low-frequency noise. Broadband, lowpassed, and band-

passed impulses served as impulsive disturbances. Lowpass-

ﬁltered white Gaussian noise lasting 1 s was used to elevate

the noise ﬂoor of the measurement. One hundred signals of

each type of non-stationary noise were used. The disturban-

ces appeared at different times within the sweep, and their

energy varied as well to obtain various changes to the con-

taminated signal’s energy.

All signals used in the experiment had a different fre-

quency content: the broadband impulse spanned across all

frequencies, the lowpassed one had its cutoff frequency at

100 Hz, the bandpassed extended between 500 and 5000 Hz,

whereas the white noise was lowpass ﬁltered at 300 Hz.

The results of this experiment, presented in Fig. 3, show

that increasing the signal’s energy causes the PCC values to

drop in accordance with Eq. (20). They also reveal that the cor-

relation between the sweeps may vary for disturbances that add

the same amount of energy. This phenomenon is especially

prominent for the narrowest-band disturbances, such as low-

passed impulse and low-frequency noise, which might be

because non-stationary disturbance is not completely uncorre-

lated, but displays similarity to either the ESS or stationary

noise. This is especially possible in the low-frequency region,

where the sound is usually less diffuse.

Note that the energy differences represented by DEand

mentioned in Table Iare often small (<0.1 dB) and may

fall below the uncertainty of the measurement equipment

FIG. 3. Comparison of the PCC values for different non-stationary noise

types for synthetic sweep with pink stationary noise and SNR of 84dB. The

areas cover the minimal and maximal values of measured correlation for

the respective type of disturbance and DE.

J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al. 2121

https://doi.org/10.1121/10.0009915

uncertainty. Therefore, the Ro2 should be applied to mea-

surements performed within a short time, using the same

measurement equipment, and the same settings.

B. Median-based background noise energy estimation

Another problem related to the presence of non-

stationary noise in the measurement is the possibility of con-

taminating the background noise used to estimate the SNR

and thus, PCC threshold. A wrongly estimated noise energy

E½n1leads to an underestimated PCC threshold and thus to

the incorrect classiﬁcation of clean and contaminated

sweeps.

When a non-stationary random event contaminates the

background noise, the affected samples carry more energy

than the clean ones. Therefore, the contamination skews the

amplitude distribution of noise in the positive direction. The

nature of the stationary noise does not allow for the use of

the PCC values as a discriminant for ﬁnding the non-

stationary noise, as with sweep signals. However, if the

amplitude distribution of the noise signal is Gaussian, i.e.,

n1ðkÞNð0;r2Þ, a robust estimator can be used.

The energy of Gaussian noise is essentially a scaled

mean value of the squared signal. The drawback of such an

estimator is its high sensitivity to outliers, resulting in false

estimations for contaminated signals. The median, however,

is less inﬂuenced by the outliers than the mean, since its

breakdown point, i.e., the maximum proportion of contami-

nated observations that do not force the estimator to result in

an aberrant value, is higher than that of mean: The break-

down point for the median is 0.5 whereas it is 0 for the

mean.

45–47

This means that if at least 50% of samples are

not contaminated, the median values are not skewed.

For squared samples from the Gaussian distribution, the

mean and median are related by a constant scaling factor

bv¼1:4826.

Thus, the robust noise energy estimate is

½

¼X

k¼1

n1ðkÞ2¼Nb2

vmedianðn1ð1Þ2;…;n1ðNÞ2Þ:(22)

To demonstrate the effectiveness of this method, a 2-s-

long noise signal with Gaussian distribution and different

values of noise power was contaminated at random times

with impulsive noise (as described in Sec. IV A) and 200-

ms-long lowpassed Gaussian noise. The non-stationary dis-

turbances were scaled by random factors to achieve various

effects on the noise energy. Then, the mean and scaled

median values of noise energy were compared to a target

value—the energy of noise without contamination.

The results presented in Fig. 4show that while the

mean energy value can change by tens of decibels in the

presence of non-stationary noise, the median remains essen-

tially unchanged and very close to the target (within 1 dB).

Additionally, using the median instead of the mean does not

add any processing to the detection process, making it the

recommended procedure for calculating the background

noise energy.

C. Sound dropouts

The software-related dropouts do not add energy to the

contaminated signal, but reduce it instead. Additionally,

skipping the samples creates two cropped sweeps, one of

which is shifted with respect to the clean sweep. Therefore,

the energy difference introduced by the dropout is of less

importance to the correlation between two signals than the

time at which the skipping occurrs.

To estimate the effect of the time of the dropout on the

correlation, the synthesized sweeps, as used in Sec. III B,

were contaminated with sound dropouts. The dropouts were

simulated by deleting small portions of the signal, ranging

from one to ten samples, at different times throughout the

ESS and shifting the remaining portion of the sweep by the

respective number of deleted samples. The dropouts were

broadband disturbances, since a discontinuity is an impul-

sive event affecting all frequencies.

The relation between the drop in the PCC values at

the time at which the samples were skipped is depicted in

Fig. 5. The results show that if the dropout happens in the

beginning of the sweep, the correlation between the clean

and corrupted sweeps is very low. However, if the contami-

nation occurs later in the signal, the PCC drop is less promi-

nent. Additionally, the dropouts appearing after the ESS has

ﬁnished playing (in the present case, after 3.0 s) affect the

correlation only marginally.

V. VALIDATION

In this section, the database used for testing the pro-

posed method is presented. The results of using Ro2 on this

dataset are presented, and the transfer-function variation is

determined. The proposed method is also compared with

FIG. 4. Comparison between the mean and median energy of a contami-

nated noise signal. The colored solid lines show non-stationary disturbances

of different energies corrupting background noise signals, with the loudest

impulse at the top and the quietest at the bottom. The median energy

remains closest to the target in all cases.

2122 J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al.

https://doi.org/10.1121/10.0009915

another procedure for coping with non-stationary noise in

acoustic measurements.

A. Validation database

Ro2 was validated on a database of swept-sine measure-

ments collected in the Arni room at the Acoustics Lab of

Aalto University, Espoo, Finland.

5,48

Arni is a rectangular

room, with dimensions 8.9 m 6.3 m 3.6 m (length, width,

and height, respectively). The room’s walls and ceiling are

equipped with acoustic panels that can switch their state

between open and closed, changing the amount of absorp-

tion and thus varying the acoustics within the space. A view

of the space and measurement equipment is shown in Fig. 6.

The equipment used during the measurements included

a 01 dB LS01 omnidirectional loudspeaker (sound source),

two G.R.A.S. 1/2-in. diffuse-ﬁeld microphones of type

40AG, two G.R.A.S. 1/2-in. free-ﬁeld microphones of type

46AF, one Br€

uel & Kjær 1/2-in. diffuse-ﬁeld microphone of

type 4192, a G.R.A.S. power module of type 12AG, a mea-

surement laptop, and a MOTU UltraLite mk3 Audio

Interface. The measurement signal was a 3-s-long ESS

that was played ﬁve times for each panel conﬁguration with

2 s of silence in between to allow the sound to fully decay.

The total number of measurements was 5342, amounting to

26 710 sweep s recorded with each microphone.

Due to the size of the database and the time required for

its collection, the measurements were conducted automati-

cally, without human supervision. Therefore, when an

unwanted acoustic event occurred, no action was taken to

discard the corrupted recording and repeat the measurement.

This approach led to many sweeps being contaminated with

non-stationary noise of unknown origin, type, and energy.

Examples of sounds recorded during the measurements are

available online.

B. Ro2 measurement and selection procedure

The Ro2 detection proceeds as follows: before every

measurement, a short period of silence (background noise)

is captured, and its energy is calculated from Eq. (22). Next,

an ESS is captured, and its energy is calculated as well. In

the event that the noise and sweep signal lengths are differ-

ent, their energies cannot be compared, and thus, the signal

power can be used instead. The procedure is repeated so that

two sweeps are measured. The expected PCC value is then

computed from Eq. (9), and the lower bound for the

expected PCC is calculated from Eq. (11). Next, the toler-

ance resulting from transfer-function variation [obtained

from Eq. (18)] is applied according to Eq. (19). Finally, the

detection threshold is compared to the sweeps’ PCC esti-

mated from Eq. (1).

If the measured PCC is on or above the threshold, both

sweeps are classiﬁed as clean, and the measurement can

end. If, on the other hand, the correlation is below the

threshold, the presence of non-stationary noise is indicated.

The measurement must continue until two sufﬁciently

highly correlated sweeps are obtained. The ESSs which dis-

play low correlation with the clean sweeps are marked as

contaminated and are discarded.

C. Transfer-function variation estimation

The transfer-function variation factor was estimated for

all measurements based on the difference between signals,

as in Eqs. (15)–(17). Since this was done for both clean and

contaminated sweeps, many values are skewed in the posi-

tive direction due to non-stationary disturbances. To elimi-

nate outliers, values of sthat were higher than three median

absolute deviations (MADs) from the median were

discarded.

The distribution of transfer-function variation factors is

displayed in Fig. 7. The ﬁgure shows that abnormal values

of sstart just above the adopted threshold, with a prominent

rise in the number of outliers between 103and 102.

FIG. 5. Effect of sound dropouts on the PCC between sweep signals. The

solid lines show the effect for the number of dropped samples from one

(topmost plot) to ten (bottommost plot), whereas the dashed line indicates

the PCC without dropouts.

FIG. 6. Variable acoustics laboratory Arni and the equipment used in the

measurements.

J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al. 2123

https://doi.org/10.1121/10.0009915

In the present study, s¼0:00019 (indicated in Fig. 7

with a solid line) was used to set the tolerance to the PCC

threshold. Thus, the detection criterion is not as strict as

when using the SNR-based threshold. Note, however, that

the value of sdepends on, among others, the length of the

sweep, the time between consecutive measurements, and the

characteristics of air movement within the measured space.

Therefore, although the spresented here may be used as a

guideline for similar conditions, ideally it should be esti-

mated for each measurement scenario separately.

D. Correlation-based detection

In the present study, a period of silence before the emis-

sion of each sweep was used for background noise energy

estimation. It was certain that there would be no late part of

the decay or long-ringing modes present in that part of the

measurement.

The validation results are presented in Fig. 8, where a

two-dimensional histogram shows the relative probability of

the expected and measured PCC values. The distribution

reveals two clusters: the ﬁrst one is located along the diago-

nal, where both the expected and measured PCC values are

similar. It contains clean signals and includes the largest

number of occurrences (the color bar limits the probabilities

to increase readability). The second cluster consists of

clearly contaminated sweeps. It is located below the diago-

nal in Fig. 8, meaning that the measured PCC is consider-

ably lower than the values expected based on Eq. (9).

The SNR-motivated prediction of PCC values is indi-

cated in Fig. 8with a dashed blue line. Such a threshold is

visibly too strict since the majority of measured signals fall

below it. This proves that the tolerances due to transfer-

function variation and background noise correlation need to

be considered in Ro2. The lower bound for the expected

PCC, ^

qwith f¼1, is indicated in Fig. 8with a dash-

dotted blue line. Although most of the signals are now

classiﬁed as clean, a large number of measurements still lies

below this threshold.

The ﬁnal detection threshold ^

qwith tolerance s

accounting for time variance is marked with a solid blue

line in Fig. 8.^

qwith f¼1, and s¼1:9104identiﬁes

most of the sweeps from the top cluster as clean, whereas

the signals from the bottom cluster are considered contami-

nated. This threshold also considers excessive time variation

as contamination, due to which sweeps that do not otherwise

contain non-stationary noise are discarded. The threshold

represented by ^

qwhen f¼1 and s6¼ 0 is recommended

when using the Ro2 procedure.

E. Comparison with a previous method

The proposed correlation-based detection is compared

with the procedure developed by Guski,

31,32

since it is the

only other method created speciﬁcally for the purpose of

identifying impulsive noise in sweep measurements. In this

approach, the detection is conducted by ﬁrst separating the

sweep with the IR from the background noise using the iter-

ative approach by Lundeby et al.

Then, the logarithmic

ratio between the maximum value and the root mean square

value of the stationary noise is calculated. If the ratio is in

the range of values typical for Gaussian noise, i.e.,

12–14 dB, the measurement is classiﬁed as clean. However,

if the ratio is higher, namely, 20 dB or more, contamination

is indicated. Therefore, in the present study, the value of

20 dB is used as the threshold discriminating between clean

and corrupted sweeps for the Guski method. The implemen-

tation provided in the ITA Toolbox

was employed when

testing this procedure.

First, the detection rate for non-stationary disturbances

from Sec. IV is compared. Since the signals were

FIG. 7. (Color online) Histogram of measured values of transfer-function

variation. The dashed line marks the median, whereas the solid line shows

the threshold of three MADs separating the outliers. FIG. 8. (Color online) Two-dimensional histogram of expected and measured

values of PCC. The dashed line represents the expected values based on t he

SNR, the dash-dotted line shows the lower bound for ^

qwith f¼1, whereas

the solid line represents a threshold including transfer-funct ion variation, ^

with f¼1, and s¼1:9104, which is the proposed threshold.

2124 J. Acoust. Soc. Am. 151 (3), March 2022 Prawda et al.

https://doi.org/10.1121/10.0009915

synthesized, it was known that f¼0 and s¼0. Thus, the

strictest threshold could be used for the Ro2. The results for

each type of non-stationary noise are presented in Table II.

They show that the Ro2 is superior in terms of separating

clean and contaminated sweeps, with detection rates being

higher for each type of disturbance.

The last row in Table II reveals that both methods per-

form worst when sound dropouts occur in the ESS. In the

case of the Ro2, however, only the dropouts occurring after

about 3.5 s are undetected (cf. Fig. 5). Guski’s method, how-

ever, was unable to detect this kind of disturbance alto-

gether. This was an expected result since Guski’s method

was not intended for identifying such a type of non-

stationary noise.

For the remaining types of non-stationary noise, the

Ro2 did not correctly identify the sweeps containing distur-

bances of low energy (cf. Fig. 3). The Guski method, on the

other hand, proved inconsistent in this regard, wrongly clas-

sifying the ESSs corrupted with both low- and high-energy

non-stationary noise.

The comparison was also performed on the dataset of

measured sweeps. The signals that were marked as contami-

nated by the Ro2 were further analyzed by a human annota-

tor. The measurements were checked in terms of audibility

of non-stationary disturbances as well as their visibility in

spectrograms, since often the signal itself may mask the

contamination, rendering it inaudible. The sweeps falling

below the detection threshold due to the excessive transfer

function variation were not incorporated in further experi-

ments, as the Guski method was insensitive to time

variance.

In the annotation process, 283 contaminated signals

were selected to be analyzed with the Guski method. The

results of the comparison are shown in Table III. The total

number of measurements marked as contaminated by the

Ro2 served as a reference, constituting 100% of detected

non-stationary disturbances. Seventy percent of these sig-

nals were also correctly identiﬁed by Guski’s method, while

30% were false positives. The human annotation revealed

that the majority of unidentiﬁed disturbances were short

low-frequency noise bursts. The Guski procedure also over-

looked a small number of ESSs including impulsive noise.

Both experiments show that the Ro2 outperforms

Guski’s method regardless of the type of contamination. Its

efﬁciency and robustness prove that it is the best available

method for separating clean sweeps from those containing

non-stationary noise. The Ro2 method can thus be recom-

mended for acoustic measurements in situations where non-

stationary noise may occur, which include most practical

scenarios.

VI. CONCLUSION

The paper introduces a novel method, called the rule of

two or Ro2, to identify a pair of clean exponential swept

sines in a series of repeated sweep measurements. The clas-

siﬁcation is based on the similarity between the ESS signals,

expressed by means of Pearson’s correlation coefﬁcient. A

detection threshold separates signals containing expected

contamination, such as background noise and time variance,

from those contaminated by non-stationary noise. This study

also shows that using the median to estimate the background

noise energy helps avoid the bias caused by non-stationary

events.

If the resulting PCC value between two measured

sweeps is above the threshold, the measurements are marked

as clean, and both signals can be used in further analysis. If,

on the other hand, the correlation is lower than the thresh-

old, the presence of non-stationary noise is indicated and the

signals must be discarded. Therefore, the measurement

should be repeated until a pair of highly correlated ESSs is

found.

In the large set of thousands of experiments reported in

this study, the Ro2 procedure proved to be reliable and easily

applicable in acoustic measurements. It also performed better

than the previous established procedure for non-stationary

noise detection, proving its robustness and efﬁciency. The

Ro2 procedure increases the reliability of practical acoustic

and audio measurements using sine sweeps.

ACKNOWLEDGMENTS

This work was supported by the “Nordic Sound and

Music Computing Network—NordicSMC,” NordForsk

Project No. 86892.

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https://doi.org/10.1121/10.0009915

Publication IV

Vesa Välimäki and Karolina Prawda. Late-Reverberation Synthesis using

Interleaved Velvet-Noise Sequences. IEEE/ACM Transactions on Audio

Speech and Language Processing, Vol. 29, pp. 1149–1160, February 2021.

Reprinted with permission.

109

IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 29, 2021 1149

Late-Reverberation Synthesis Using Interleaved

Velvet-Noise Sequences

Vesa Välimäki , Fellow, IEEE, and Karolina Prawda

Abstract—This paper proposes a novel algorithm for simulating

the late part of room reverberation. A well-known fact is that

a room impulse response sounds similar to exponentially decay-

ing ﬁltered noise some time after the beginning. The algorithm

proposed here employs several velvet-noise sequences in parallel

and combines them so that their non-zero samples never occur

at the same time. Each velvet-noise sequence is driven by the

same input signal but is ﬁltered with its own feedback ﬁlter which

has the same delay-line length as the velvet-noise sequence. The

resulting response is sparse and consists of ﬁltered noise that decays

approximately exponentially with a given frequency-dependent

reverberation time proﬁle. We show via a formal listening test

that four interleaved branches are sufﬁcient to produce a smooth

high-quality response. The outputs of the branches connected in

different combinations produce decorrelated output signals for

multichannel reproduction. The proposed method is compared

with a state-of-the-art delay-based reverberation method and its

advantages are pointed out. The computational load of the method

is 60% smaller than that of a comparable existing method, the

feedback delay network. The proposed method is well suited to

the synthesis of diffuse late reverberation in audio and music

production.

Index Terms—Acoustics, audio systems, digital signal

processing, ﬁltering algorithms.

I. INTRODUCTION

ARTIFICIAL reverberation algorithms often produce an

impulse response that is reminiscent of exponentially de-

caying ﬁltered noise [1]. Moorer ﬁrst suggested that the tail of a

room impulse response could be described well using pseudo-

random noise [2]. This paper proposes a novel noise-based

reverberation algorithm, which is easy to design and efﬁciently

ﬁlters the input signal with its response.

Room impulse responses are generally known to sound sim-

ilar to exponentially decaying noise after a short time from the

onset. In practice, the noise must be ﬁltered and not white noise,

and different frequency bands must decay at a different rate [1],

[2]. This observation led to the idea that an artiﬁcial reverberation

algorithm could be based on a random-noise generator. In fact,

some reverberation algorithms, such as the well-knownfeedback

Manuscript received May 29, 2020; revised December 18, 2020 and January

29, 2021; accepted February 2, 2021. Date of publication February 22, 2021;

date of current version March 19, 2021. This work was supported by NordForsk

(Aalto UniversityProject 86892, NordicSMC). The associate editor coordinating

the review of this manuscript and approving it for publication was Dr. Alexey

Ozerov. (Corresponding author: Vesa Välimäki.)

The authors are with the Acoustics Lab, Department of Signal Pro-

cessing and Acoustics, Aalto University, FI-02150 Espoo, Finland (e-mail:

vesa.valimaki@aalto.ﬁ; karolina.prawda@aalto.ﬁ).

Digital Object Identiﬁer 10.1109/TASLP.2021.3060165

delay network (FDN) [3]–[5], can be converted into a pseudo-

random noise generator by turning off the decay of sound. In

practice, replacing all ﬁlters and attenuating coefﬁcients in the

algorithm with unity gains (i.e., no signal-processing operation)

leaves only delay line and summation operations. A reverber-

ation algorithm must also be able to efﬁciently convolve an

arbitrary input signal—the signal to be processed—with its noisy

response.

Rubak and Johansen proposed to use a ﬁnite-impulse response

(FIR) ﬁlter with random coefﬁcients as a loop ﬁlter in the feed-

back loop of a reverberation algorithm [6], [7]. This appears to

be a computationally efﬁcient method to generate a decaying re-

sponse with the help of random noise, but, unfortunately, the de-

cay rate of the system is also affected by the randomness, because

the loop ﬁlter has a random magnitude response. Karjalainen and

Järveläinen further elaborated this idea by cascading a random

FIR ﬁlter with the feedback loop structure, allowing accurate

control of the reverberation time (RT) using a low-order loop

ﬁlter [8]. Additionally, Karjalainen and Järveläinen introduced

the concept of velvet noise, a smooth-sounding sparse random

noise [8]. More recently, Lee et al. [9] and Oksanen et al. [10]

have investigated variants of recursive structures employing

velvet noise. In addition to artiﬁcial reverberation, velvet noise

has been applied recently to audio decorrelation [11], [12], time-

expansion of sounds [13], [14], music synthesis [15], speech

synthesis [16]–[18], and acoustic measurements [19].

A recurring problem in previous velvet-noise-based recursive

algorithms is that a single sequence is ﬁltered and attenuated

over time [8]–[10]. Human hearing is sensitive to repetitions,

and the produced repetitive noise sounds similar to ﬂutter echo,

a well-known problem in room acoustics. The ﬂuttering is

easiest to perceive in percussive sounds. Earlier solutions to

reduce the ﬂutter problem include time-varying randomization

of the impulses, and cross-fading sequences drawn from a

small collection of velvet-noise sequences [8]–[10]. However,

these time-variant techniques lead to another problem: warbling,

which is apparent when a stationary sound is processed using

the reverberation algorithm. Suppressing both the ﬂutter and the

warbling simultaneously seems hard in a recursive velvet-noise

reverberation algorithm.

This paper proposes to hide the repetition in the noisy response

by constructing a velvet-noise sequence from several sequences

that are combined. The positive and negative impulses must not

be allowed to accumulate or cancel each other in the combination

sequence so as not to destroy the advantageous properties of vel-

vet noise. An extended velvet-noisesequence [20] can leave gaps

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

1150 IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 29, 2021

in the sequence at regular intervals so that it can be added to other

similar sequences maintaining the prescribed pulse density. This

work shows that when several repetitive sequences of different

length are combined, the repetitions become inaudible. This

principle is then used to devise a novel recursive reverberation

algorithm. A graphic equalizer is used to control the RT of each

sequence.

The rest of this paper is organized as follows. Sec. II intro-

duces the idea of interleaving velvet-noise sequences to hide

repetitive patterns and describes a listening test to verify its

perceived quality. Sec. III proposes a novel reverberation al-

gorithm that uses interleaved velvet-noise sequences as sparse

FIR ﬁlters. A smearing technique to soften the onset and a

segmented decay technique for smooth approximation of the

exponential decay are also proposed as additions to the basic

algorithm. Sec. IV presents a validation and comparison to

previous methods. Sec. V concludes the paper.

II. INTERLEAVED VELVET-NOISE SEQUENCES

This section introduces an interleaving technique for velvet-

noise sequences, which provides a lossless prototype for the new

reverberation technique. Additionally, this section describes a

listening test to evaluate the perceptual quality of the produced

white noise.

A. Interleaving Extended Velvet-Noise Sequences

Velvet noise consists of 1’s, −1’s, and zeros. The locations

kvn(m)of the non-zero samples in the sequence are determined

kvn(m)=round[mT d+r1(m)(Td−1)],(1)

where m=0,1,2,... is the pulse counter, Tdis the grid size,

and r1(m)is a value produced with a random-number generator

having uniform distribution (0,1) [20]. Another random number

sequence r2(m)is used to select the sign of the impulse, so

that the sample inserted at index kvn(m)is either 1 or −1.

The remainder of the samples in the velvet-noise sequence are

zero. This method places exactly one non-zero sample in every

range of Tdsamples. When the number of non-zero impulses

is sufﬁciently large, or, equivalently, Tdis small enough, velvet

noise sounds similar to white noise. In fact, experiments have

established that it sounds even slightly less rough than Gaussian

white noise when there are at least 2000 impulses per second,

when the sample rate is 44.1 kHz [8], [20], which is also used

in this work.

The interleaving technique is based on the use of extended

velvet-noise sequences (EVN) introduced in [20]. In an EVN,

the range where the impulse can appear is further limited, leaving

speciﬁed times between the impulses always empty.The impulse

locations in an EVN are determined as follows:

kevn(m)=round[mT d+Δr1(m)(Td−1)],(2)

where 0<Δ<1is a scale factor limiting the range where the

impulse can be located [20]. For example, when Δ=0.25,the

impulses can only appear in the ﬁrst quarter of the grid, so that

the remaining 75% of the samples are always known to be zeros.

Fig. 1. Example of delaying and interleaving four extended velvet-noise

sequences (the four upmost subﬁgures) so that impulses never collide in their

sum (bottom). The shaded areas indicate the ranges in which impulses do not

appear in each sequence. Only the non-zero samples are plotted.

The algorithm proposed here employs several EVNs in paral-

lel and combines them so that their non-zero samples never occur

at the same time. This is possible by delaying each sequence by

a different number of sampling intervals so that the non-zero

samples in each sparse sequence occur at different times. Fig. 1

shows an example of four interleaved EVNs. The grid size

of each of them is four times the targeted overall number,

Td=4Td, and Δ=1/4. Here, we have chosen Td=20,so

each EVN has a grid size of ˆ

Td=80samples, but an impulse

can only appear within the ﬁrst 20 samples of the grid range in

each EVN. Furthermore, in Fig. 1, the second, third, and fourth

sequences are delayed by ˆ

Td/4,2ˆ

Td/4, and 3ˆ

Td/4samples,

respectively.

The ﬁnal sequence shown at the bottom of Fig. 1 is the

resulting interleaved sequence in which samples originating

from the four EVNs appear alternately but never collide. The

grid size of the ﬁnal sequence is Td=20.

B. Smooth Noise Generation Using Interleaved Sequences

To produce a long interleaved noise sequence, each EVN can

have a ﬁnite length and they can be repeated indeﬁnitely. Such

a repetitive noise sequence is called frozen noise [21]–[23]. The

EVN lengths must be selected to be different from each other and

to be coprime with each other, so that there is no simple relation

between any two EVN lengths, such as one is twice longer

than the other. This problem is equivalent to that of choosing

the delay-line lengths of comb ﬁlters in traditional delay-based

reverberation algorithms, such as in an FDN [3], to be mutually

incommensurate. Otherwise, a repetitive disturbance similar to

ﬂutter echo appears. A safe option is to use prime-number based

sequence lengths of the form

Li=Ciˆ

Td,(3)

VÄLIMÄKI AND PRAWDA: LATE-REVERBERATION SYNTHESIS USING INTERLEAVED VELVET-NOISE SEQUENCES 1151

Fig. 2. Example of interleaving four ﬁnite-length EVN signals. In the top four

panes, each EVN has a different prime-number based duration and restarts at

the vertical lines. In the bottom pane, the combined boxes show that, except for

the beginning, the EVN start times do not overlap.

where i=1,2,...,M is the sequence index, Mis the number

of sequences combined, Ciare different prime numbers, and

Td=MTd.

The density of the interleaved velvet-noise sequence must be

sufﬁcient so as not to sound rough. We know from previous

studies that a density of about 2000 samples/s is sufﬁcient [8],

[20], and therefore, in this work we have chosen the grid size

to be Td=20, which gives a density of 2205 samples/s at the

sample rate of 44.1 kHz. Each EVN in an interleaved velvet-

noise sequence will then have a grid size of ˆ

Td=20 Msamples

and Δ=1/M .

The example in Fig. 2 shows four repeating EVN signals

interleaved to produce a velvet-noise sequence that does not

repeat for a long time. Here, the chosen primes are C1=97,

C2= 101,C3= 103, and C4= 107, so when they are mul-

tiplied by ˆ

Td=80, the sequence lengths of 176 ms, 183 ms,

187 ms, and 194 ms, respectively, are obtained. The combined

velvet-noise sequence repeats after about 8.6 billion samples, or

54.4 hours, which is obtained by multiplying the four sequence

lengths.

We extensively tested different numbers of interleaved se-

quences Mand various delay-line lengths Li. The repetitions

seem hard to mask using less than four interleaved sequences.

Furthermore, if any of the delay lines is very short, such as

shorter than about 5000 samples or 110 ms, the repetition will

easily become audible.

C. Listening Test

A Multiple Stimuli with Hidden Reference and Anchor

(MUSHRA) listening test [24] was conducted to verify the per-

ceptual qualities of interleaved repetitive velvet-noise samples.

Before the start of the experiment, sufﬁciently many interleaved

velvet-noise sequences were assumed to produce a smooth, or

non-repetitive, sound and would receive a high average score.

The data set used in the listening test was created as sequential

velvet-noise sounds. The stimuli consisted of one to ﬁve inter-

leaved sequences, whose lengths were determined by the use of

consecutive primes from 83 to 139 multiplied by the number

of parallel lines and the grid size of the velvet-noise sequence.

The length of each type of sequence was that of the shortest

delay line determined by a prime number in the range 83–113.

However, the sequences created with the numbers 101, 103, 107

and 109, primes that are very close to one another, were excluded

from the experiment to avoid the audible repetition in the sound

produced with these numbers.

The task in the listening test was to assess the smoothness

of the sound stimuli compared to the reference. The possible

grades ranged from 0 to 100, with 0 given to the most repetitive

sound and 100 to a perfectly smooth stimulus. Additionally,

text descriptions for ﬁve grade ranges were used: Very annoying

(0–20), Annoying (20–40), Slightly annoying (40–60), Percep-

tible, but not annoying (60–80), and Imperceptible (80–100).

A regular, inﬁnitely long, velvet-noise sequence having a

density of 2205 samples/s was used as a reference sound that

was meant to receive the maximum score. The test items were

constructed using interleaved velvet-noise sequences with one

delay line as the low-quality anchor and with two to ﬁve delay

lines as MUSHRA conditions. These six sounds were assessed

in every question. Each page in the MUSHRA test consisted

of samples having the same prime number that determined the

length of the shortest delay line, which, together with the anchor

and the reference, were presented in random order. All sounds

used in the experiment were four seconds long. The test was

carried out using the web audio API-based experiment software

webMUSHRA developed by International Audio Laboratories

Erlangen [25].

The experiment was conducted in sound-proof listening

booths at the Aalto Acoustics Lab using Sennheiser HD-650

reference headphones. In all, 26 people participated in the test.

Five of the results, however, were excluded from the analysis for

giving a score under 100 to the reference item more than three

times. None of the 21 participants whose results were analyzed

reported a hearing impairment. Their average age was 30.7 years

(the standard deviation was 6.1). All the participants were either

students or employees of the Aalto University Department of

Signal Processing and Acoustics, and the majority of them had

prior experience with MUSHRA tests.

1152 IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 29, 2021

Fig. 3. Results of the MUSHRA listening test with 21 subjects. The absolute

mean grades and the 95% conﬁdence intervals are shown for the reference noise,

the anchor, and the interleaved EVN signals produced with differentnumbers of

delay lines (DLs), which is equal to the number of interleaved sequences. The

horizontal dashed lines divide the grading ranges labeled with text descriptions.

The subjects were allowed to adjust the volume of the sound

before starting the test. They were also familiarized with the

task, its structure, and some of the test sounds in a short training

session. The scores of the training session were excluded from

the results. During the test, the subjects were presented with

seven questions, all of which were doubled, resulting in 14

MUSHRA test pages. After completing the listening part of

the task, the subjects were asked to answer questions about the

strategy they used to distinguish between test items and the type

of differences they heard.

The scores granted by the listeners to each sound sample were

averaged based on the number of delay lines in the sequence.

Fig. 3 shows the results with the 95% conﬁdence intervals that

reveal that in general the participants had no difﬁculties in rec-

ognizing the reference and the anchor, which received the score

0(Very annoying) in all trials. They also easily distinguished

between the repetitive and the smooth sounds. The samples

with velvet-noise sequences composed of two delay lines were

considered very repetitive and received the average score 34.7

(Annoying). The stimuli with three delay lines were perceived as

less repetitive, receiving the average score of 61.5 (Perceptible,

but not annoying/Slightly annoying). The sounds containing four

or ﬁve delay lines, on the other hand, scored much higher and

were perceived as almost equally smooth. The average scores

for the stimuli with four and ﬁve delay lines were 82.0 and

88.1 (Imperceptible), respectively, and the conﬁdence intervals

overlapped, as shown in Fig. 3.

After the listening test, the participants were interviewed,

and most of them classiﬁed the stimuli on each page into two

groups: three samples were repetitive and of low quality, and

the remaining three sounded smooth and were very similar to

one another. The subjects described the samples that received

low scores as “engine-like”, “buzzy” or “ﬂutter-like”. On the

other hand, the stimuli consisting of four or ﬁve delay lines

were characterized as hardly distinguishable from the reference

Fig. 4. Structure of the proposed reverberation algorithm based on interleaved

velvet-noise ﬁlters (M=4).

Fig. 5. Structure of transfer function Gi(z)consisting of a delay line and a

loop ﬁlter in a feedback loop. In practice, the delay line of Lisamples can be

shared with the SFIR ﬁlter in the same branch (see Fig. 4).

and requiring more time and effort to assess. The stimuli used

in the listening test are available online at [26].

III. NOVEL REVERBERATION ALGORITHM

This section introduces the new reverberation algorithm,

which is based on the interleaving of velvet-noise signals.

A. Interleaved Velvet-Noise Reverberator

Fig. 4 shows the basic form of the proposed structure, called

the interleaved velvet-noise sequence (IVN) reverberation algo-

rithm, that consists of Mparallel signal-processing branches.

This example has M=4 branches, which appears to be a

sufﬁciently large number according to the listening test of Sec. II.

Each branch includes a feedback structure Gi(z)and a sparse

FIR (SFIR) ﬁlter Si(z), the coefﬁcients of which are samples

taken from different EVNs. The branch impulse responses are

interleaved by delaying them appropriately. The blocks marked

z−Tdin Fig. 4 are delay lines of Tdsamples. When the delay-line

length is set to Td=ˆ

Td/M , it is guaranteed that the non-zero

samples produced by each SFIR ﬁlter will not occur at the same

time, in the same manner as in Fig. 1.

Fig. 5 shows the structure of transfer function Gi(z), which

contains a feedback loop with a delay of Lisamples and a

loop ﬁlter Hloop,i(z). Each SFIR ﬁlter also requires a delay

line of Lisamples, but this delay memory can be shared with

the comb ﬁlter: the SFIR ﬁltering, which is equivalent to the

convolution of the input signal with velvet-noise sequence, is

in practice implemented with multiple output taps, which are

added [11], [13], [27], [28]. The signal passing through the whole

delay length of Lisamples, not processed with the multi-tap

delay system, is the output signal fed to the loop ﬁlter and

thereafter added to the input of feedback loop. Thus, only one

delay line of Lisamples is needed for each branch.

VÄLIMÄKI AND PRAWDA: LATE-REVERBERATION SYNTHESIS USING INTERLEAVED VELVET-NOISE SEQUENCES 1153

Fig. 6. Stereo IVN reverberator is obtained by interleaving the same velvet-

noise sequences a second time in a different order, which is accomplished with

additional delay lines of Tdsamples on the right-hand side. The delays of D1,

D2,andD3samples and gain factors gs1 ,gs2,andgs3 implement the smearing

technique.

The EVN signals can be arranged in all different permuta-

tions, and not only in the order shown in Fig. 4. To obtain

multiple decorrelated outputs, additional delay lines can be used

to interleave the branch impulse responses. Fig. 6 shows one

possible way to obtain two decorrelated outputs, which leads to

a stereo effect. The impulse responses of outputs 1 and 2 consists

of the same interleaved sequences but in opposite order, i.e.,

1-2-3-4 vs. 4-3-2-1.

B. Controlling the Decay Rate

Each branch has its own loop ﬁlter Hloop,i(z)inside the

feedback loop, as shown in Fig. 5. These ﬁlters must be designed

collectively so that they all approximate the same frequency-

dependent RT. One possibility is to devise a target response

for each loop ﬁlter based on either a unit-sample response or

a per-second response. The latter approach is chosen here, so

that each target response is obtained by cascading a fraction

ν=Li/fsof the prototype magnitude response Hprot(ω), when

fsis the sample rate used (e.g., 44.1 kHz):

Htarget,i(ω)=|Hprot(ω)|ν.(4)

Each target response is then approximated using the same ﬁlter-

design technique to obtain commensurate loop ﬁlters. Naturally,

the order of each loop ﬁlter must be sufﬁciently large, so that

they approximate well the target response. Otherwise, some

components in the response will decay faster than the others,

which will lead to a metallic disturbance in the sound [3].

Fig. 7 shows an example with M=4, where the comb ﬁlters

Gi(z)are replaced with a delay loop having a constant gain

factor (i.e., no ﬁltering). The grid size for all EVNs is ˆ

Td=80

samples, which yields a sufﬁcient velvet-noise density of 2205

non-zero samples at the sample rate of 44.1 kHz. The lengths

of the four EVN sequences are again prime multiples of 80

samples: L1=97×80,L2= 101 ×80,L3= 103 ×80, and

L4= 107 ×80 samples. The gain factors controlling the decay

of the four branches are 0.6669, 0.6558, 0.6504, and 0.6396,

respectively, which correspond to a decay of 20log(0.1) =

20 dB per second, or a RT T60 of 3.0 s. Note that the steps

Fig. 7. Four repetitive EVN signals having lengths differing from each other

and various starting times but decaying at the same rate. The decaying steps are

accented with gray and white blocks.

Fig. 8. (Top) The linear and (bottom) logarithmic impulse response obtained

as a sum of the four sequences of Fig. 7. The dashed lines mark exponential

decay with an offset of 0.5 and 5 dB in the top and bottom pane, respectively.

in the different signals of Fig. 7 appear always at different time

instances, since sequences of different lengths are used.

Fig. 8 shows the impulse response obtained by adding the

signals of Fig. 7. The signals in Figs. 7(b), (c), and (d) have been

delayed by 20, 40, and 60 samples, respectively, with respect to

the sequence in Fig. 7(a), i.e. Td=20samples in Fig. 4. For this

reason, the non-zero samples never appear at the same sample

times in Fig. 8(a), but are always interleaved. Thus, the density

of the sum signal is four times that of any of the individual EVN

signals. Additionally, the decay of the logarithmic response in

Fig. 8(b) is approximately linear (i.e., exponential on the linear

scale), because the steps caused by gradual attenuation appear

at different times. This impulse response is produced by the

feedback structure in Fig. 4, having a total of 97 + 101 + 103 +

107 = 408 taps in the four SFIR ﬁlters.

1154 IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 29, 2021

Fig. 9. (a) Synthetic impulse response and its envelope(dashed) produced with

(a) all sequences starting at time zero without segmentation, (b) with smearing

of the starting times, and (c) with both smearing and segmentation.

C. Smearing by Delaying the Sequences

The difference in magnitude between the ﬁrst and second step

is the steepest in the entire signal and may be audible. To provide

smoother changes and fade-in, a smearing technique, which has

most impact on the ﬁrst few steps, is introduced by delaying the

start of all except the ﬁrst EVN signals, as shown in Fig. 6, where

delay lines of D1,D2, and D3samples implement this. In order

to ensure that the non-zero samples will not occur at the same

time, the delays must be multiples of the grid size ˆ

Td.

An example of the delayed sequences is shown in Fig. 7.

Each one of them is delayed by a multiple of D1= 240 sam-

ples, which equals 3ˆ

Td, where the integer multiplier is a free

parameter set manually. Fig. 9 presents the comparison of the

few ﬁrst steps of the impulse response obtained with the IVN

reverberator without (top pane) and with smearing (middle

pane). This method is seen to introduce a fade-in technique,

which requires more operations in other recursive reverberation

algorithms, such as in the FDN [29]–[31]. Different starting

times, however, do not change the fact that all sequences must

approximate the same T60 value. Thus, the gain of each branch

should be reduced proportionally to the delay in its starting time.

This is realized using gain factors gs1,gs2, and gs3 ,shownin

Fig. 6.

D. Segmented Decay

Although the proposed method works best when synthesizing

a very long reverberation tail, shorter impulse responses appear

more frequently in practice. The disadvantage of the basic

method in such cases is the step-like nature of the beginning

of the synthetic response, as seen in Figs. 7 and 8. This problem

Fig. 10. Four EVN signals with segmented decay. The EVN frames accented

with gray and white blocks now decay in three steps, leading to a closer

approximation of an exponential decay than in Fig. 7.

is emphasized, when the RT is very small or the decay is very

fast, leading to big differences between consecutive steps.

To achieve a smoother decay in the early part of the synthetic

reverb, a segmentation method can be applied to each EVN

sequence, as suggested by Alary et al. [11]. The decaying steps

of Lisamples are divided into segments, which are attenuated

in smaller steps by inserting a multiplier between two adjacent

segments. This means that introducing one more segment to the

EVN sequence requires just one multiplication. In the extreme

case, every impulse in the EVN can have its own multiplying

coefﬁcient to obtain a completely smooth exponential decay, but

then the computational efﬁciency is lost.

To avoid audible jumps in loudness between the steps, the

level of the segments within one step should gradually decay

from the initial level of this step to the initial level of the next

one. The smallest number of segments producing a virtually

stepless decay was empirically found to be three. This number

prevents the segments from being too long and does not add

excessive multiplication operations to the computational cost.

Non-uniform segmentation is beneﬁcial in terms of reducing the

risk of audible periodic changes and achieving an exponential-

like decay.

In the example shown in Fig. 10, the lengths of the segments

were set to 25%, 35%, and 40% of the initial step length. To

obtain small differences between consecutive steps in the pre-

sented case, gains for the segments were based on the difference

in the magnitudes of two ﬁrst steps. The ﬁrst segment was left

unaltered, the second was attenuated by 1/3 of the magnitude

difference, and the third one by 2/3 of the magnitude difference.

This way the change between the segments is always the same

within one step.

Since each step is treated separately and the segmentation

does not change the initial level of the EVN sequence, the

overall decay rate of the sum of sequences is not affected by this

operation. The segmentation of each frame into three requires

only two extra multiplications per EVN branch. The bottom pane

of Fig. 9 shows that adding the four EVN sequences having three

VÄLIMÄKI AND PRAWDA: LATE-REVERBERATION SYNTHESIS USING INTERLEAVED VELVET-NOISE SEQUENCES 1155

Fig. 11. (Top) Measured impulse response of the “world’s longest echo” and

(bottom) its spectrogram.

decaying segments each leads to an approximately exponential

decay pattern, i.e. a linear decay on the dB scale.

E. Loop-Filter Design

Historically, the ﬁrst attempt to produce frequency-dependent

reverberation was made by inserting a one-pole lowpass ﬁlter

into a feedback structure [2], [3]. Later, controlling the decay

rate in three independent frequency bands was possible by

introducing biquadratic ﬁlters with adjustable crossover fre-

quencies [32]. In [33], a 13th-order ﬁlter comprising single

bandpass ﬁlters with a second-order Butterworth bandpass ﬁlter

was proposed. Recently, Jot [34], Schlecht and Habets [35],

and Prawda et al. [36] have considered using graphic equalizers

to control the frequency-dependent reverberation time in FDN

reverberators.

The approach adopted in this work used the cascaded graphic

equalizer as proposed in [37] as an attenuation ﬁlter in order to

accurately control the decay rate in ten octave frequency bands.

The prototype per-second response was determined based on the

reference RT valuesand transformed into the target responses for

each of the delay lines es presented in (4). Shifting and scaling of

the magnitude response by the median of gains was included, as

suggested in [36]. The ﬁrst-order high-shelf ﬁlter for attenuating

frequencies above 16 kHz was also inserted in the loop ﬁlter.

IV. VALIDATION AND COMPARISON

This section presents results of synthesizing the reverberation

tail of “the world’s longest echo” and a concert hall response us-

ing the IVN reverberator. The properties and the computational

cost of the proposed method and the FDN reverberator are also

compared.

A. Synthesizing the Tail of the “World’s Longest Echo”

To examine the ability of the proposed algorithm to reproduce

the reverberation tail of a real impulse response, the extreme case

of the world’s longest reverberation was chosen. The sample

Fig. 12. (Top) Synthetic late reverberation and (bottom) spectrogram of the

“world’s longest echo” produced with the IVN reverberator, cf. Fig. 11.

selected for analysis was recorded in tank number 1 at the Inchin-

down oil depository, Ross-shire, Scotland, U.K. [38] and was

obtained from [39]. The average RT of the tank is 1 min 15 s [38].

Since the purpose of the experiment was to control the reverbera-

tion in as wide a frequency range as possible, the values were not

taken directly from [38], where the T20 and T30 were given for

seven and six octave frequency bands, respectively. Moreover,

since the measurements of the world’s longest impulse response

were performed according to [40], the numbers provided are the

result of analysis of several impulse responses, and thus may

vary considerably from the sample chosen for the purpose of

this work. Therefore, the reference RT values were calculated

directly from the impulse response used in the experiment.

The IVN reverberator used for the experiment comprised four

(M=4) delay lines. This number was proven to be sufﬁcient

to obtain smooth, non-repetitive sound, as described earlier in

Sec. II-C. The grid size of the IVN reverberator was set to ˆ

Td=

80, and the lengths of the EVN sequences in samples were L1=

97 ˆ

Td,L2= 101 ˆ

Td,L2= 103 ˆ

Td, and L4= 107 ˆ

Td, which led

to a total of 408 taps in the four SFIR ﬁlters (97 + 101 + 103 +

107). Because of small differences between the magnitude of

the consecutive steps in the EVN sequences, there was no need

to perform the segmentation mentioned in Sec. III-C.

The original impulse response of the “world’s longest echo”

and the corresponding spectrogram are shown in the top and

bottom panes of Fig. 11, respectively. The reverberation tail syn-

thesized with the IVN reverberator is depicted in the upper pane

of Fig. 12, whereas its spectrogram is shown in the bottom pane.

The impulse response synthesized with the proposed algorithm

appears to be smoother than the original sample and its shape is

more regular. However, the spectrograms reveal almost identical

decay characteristics for both responses, with differences notice-

able only in the low frequencies between 100 Hz and 300 Hz.

The impulse response with the reverberation tail created with

the proposed method was also analyzed in terms of the RT values

in octave bands. The obtained T60 values were compared to the

reference and are depicted in Fig. 13. Minor differences between

1156 IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 29, 2021

Fig. 13. Comparison between reference values of the RT of the “world’s

longest echo” and the values obtained with the proposed method.

the measured and obtained values are visible. At low frequencies

below 2 kHz, which are perceptually important, the difference

does not exceed 5% of the target, making the dissimilarity

imperceptible in this respect, according to [41]. Audio examples

of both impulse responses—the original and the one with late

reverberation created using the IVN reverberator—are available

online at [26].

B. Synthesis of Short Reverberation

To test the performance of the algorithm on a practical case

of a short impulse response, the reverberation of a concert hall

in Pori, Finland, was synthesized. The T60 values in this case

stretched from over 2 s in the low frequencies to less than 0.2 s

in the high frequencies. The IVN reverberator’s conﬁguration

was the same as that for reproducing the “world’s longest echo”

described in Sec. IV-A. To synthesize a short reverberation,

however, the fast attenuation forced steep transitions from step

to step in the interleaved sequences. Therefore, the segmen-

tation of the steps was necessary to avoid audible artifacts in

the produced reverberation. Each step was divided into three

parts, as described in Sec. III-C. The effect of the segmentation,

compared to the unsegmented IVN algorithm output, is shown in

Fig. 14. Segmentation makes the transition between the consec-

utive steps more gradual, which results in a smoother sounding

reverberation.

The spectrograms of the impulse responses with the original

and the synthesized late reverberation are depicted in Fig. 15.

The reverberation reproduced using the IVN reverberator fol-

lows the decay characteristics of the measured impulse response

well up to around 8–9 kHz, where it is visibly slower. This is due

to the step-like nature of the IVN reverberation, which means

that the sound decay cannot be shorter than the longest EVN

sequence, in this case 8560 samples, or 194 ms at the sampling

rate of 44.1 kHz. Both impulse responses, together with other

samples of short reverberation tails synthesized with the IVN

reverberator, are available online at [26].

Fig. 14. Segmented decay of SFIR coefﬁcients improves the exponential

shaping of the impulse response in the proposed IVN algorithm.

Fig. 15. Spectrograms of the (a) original and (b) synthetic impulse response

of the Pori concert hall. The RT estimation is shown for both with a black solid

line.

C. Echo Density

Two aspects of the proposed method were compared to an

FDN, which is considered to be a leading artiﬁcial reverberation

algorithm. The ﬁrst was the echo density of the reverberation

tail produced with both methods. To avoid bias in the number of

echoes caused by the smearing created by the attenuation ﬁlters,

they were all removed. This way, only the echoes generated by

the circulation of the delay lines in feedback structures were

counted.

The main difference between the IVN and FDN reverbera-

tors is that, except for the beginning of the signal, the former

algorithm has a ﬁxed number of echoes, which is determined

bythegridsizeTd. Therefore, for the IVN reverberator used

in Sec. IV-A with Td=20and fs=44.1kHz, the number of

impulses per second is 2205. This was proven to produce per-

ceptually smooth and random noise [20]. Thus, after the initial

increase, determined by the delay after which the sequences

VÄLIMÄKI AND PRAWDA: LATE-REVERBERATION SYNTHESIS USING INTERLEAVED VELVET-NOISE SEQUENCES 1157

Fig. 16. Comparison of echo density between FDN reverberators of different

order and the proposed IVN reverberator with four branches.

begin, the echo density in the synthesized late reverberation is

0.05 echoes per sample. The echo density is independent of the

length of the signal and the number of delay lines in the IVN

reverberator’s structure.

On the other hand, the echo density of the FDN reverberator

depends on the order of the structure and accumulates in time.

Fig. 16 shows the normalized echo density for different orders

of the FDN, calculated with the method proposed in [42]–[44],

compared with that of the IVN reverberator. Fig. 17 shows the

ﬁrst 0.1 s of the echo density build-up. A very small number

of impulses in the beginning of the impulse response of the

FDN is usually clearly noticeable as a series of audible clicks

and artifacts. As the build-up of the echo density is directly

proportional to the order of the structure, a reverberator with a

small number of delay lines is usually unsuitable for producing

quality reverberation.

Additionally, the rise in the echo count of the FDN continues

until the impulse response is saturated, i.e., there is an echo at

every successivetime unit [45]. The very dense impulse response

contributes to production of the smooth reverberation tail, which

is similar to white noise. However, this adds a considerable

number of operations to the overall computational cost. At

the same time, [20] proves that such a high echo density is

unnecessary, since smooth noise-likesound can be obtained with

a much smaller number of echoes per sample, when the impulses

are appropriately distributed over time, i.e., never too densely

or sparsely. This is how impulses appear in the velvet-noise

sequences used in the IVN algorithm, which quickly reaches

the perceptually sufﬁcient density, as shown in Fig. 17.

D. Computational Cost

The second aspect assessed here is the computational cost.

The computational efﬁciency of reverberation algorithms is of

great importance, since they are often used in real-time applica-

tions. The cost is usually presented as the number of ﬂoating-

point operations (FLOP) per processed sample, speciﬁed as a

Fig. 17. Comparison of echo density growth in the beginning of the response,

cf. Fig. 16.

TAB L E I

NUMBER OF SFIR FILTER TAPS FOR DIFFERENT NUMBERS OF DELAY LINES IN

THE IVN STRUCTURE

sum of additions and multiplications required by the algorithm

to produce one output sample.

The computational complexity of the proposed IVN rever-

berator depends on the number of EVN sequences, number of

taps in the SFIR ﬁlters, and the complexity of the attenuation

ﬁlter. Adding together the prime numbers Cithat specify the

number of taps in the SFIR ﬁlters, the FLOPs required by the

attenuation ﬁlter for every delay line, the addition per delay

line from each comb ﬁlter (M), two multiplications per branch

that account for segmentation (2 M), M−1multiplications that

adjust the gain for the delay introduced by smearing, and the

M−1additions that form the output signal. The number of

operations for an attenuation ﬁlter consisting of 10 second-order

inﬁnite impulse response sections and a ﬁrst-order shelving ﬁlter

is 53 multiplications and 41 additions, or 94 operations in total.

To account for the fact that the number of taps should be

estimated using unique prime numbers, the number of FLOPs

was determined when there were at least 400 taps in the SFIR

ﬁlters. Of course, this number increases with the number of EVN

sequences, since ﬁnding 18 or more unique primes that add up

to less than 500 is impossible. The required number of taps for

IVN structures with 4 to 20 branches is shown in Table I.

To calculate the total FLOP/sample for an FDN, the formula

proposed in [46] was used. Since the approach of comparing the

number of operations per processed sample was adopted, the

multiplication by the sample rate was omitted.

The costs of both methods were determined for different num-

bers of delay lines, including operations required by the attenu-

ation ﬁlter. The results of the comparison are shown in Fig. 18

and Table II. Both algorithms are almost equal in complexity

1158 IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 29, 2021

Fig. 18. Comparison between the computational cost of the proposed method

(IVN) and the FDN. The conditions in which each method gives smooth

reverberation tail are 4 delay lines and 802 FLOP/sample for the IVN (marked

with a square), and 16 delay lines and 2065 FLOP/sample for the FDN (indicated

with a circle).

TAB L E I I

NUMBER OF OPERATIONS PER OUTPUT SAMPLE FOR IVN

AND FDN REVERBERATORS.THE NUMBER OF DELAY LINES FOR WHICH

ASMOOTH REVERBERATION TAIL IS ENSURED IS EMPHASIZED IN

BOLD FOR EACH METHOD

for 16 delay lines, with the FDN being less costly for small and

the IVN for big numbers of delay lines. However, because the

proposed method works well for as few as four delay lines, the

number of FLOPs required for good-quality late reverberation

is only 802 operations (794 without segmenting). This number

is marked in Fig. 18 with a square on the corresponding curve

as well as emphasized in bold font in Table II.

The smallest FDN order that is sufﬁcient for high-quality

reverberation is still a controversial question. Our recent work

shows that the smallest useful order of the FDN is 16 [47],

whereas Alary et al. point out that an order as high as 32 may

be necessary to achieve sufﬁcient echo and modal densities,

depending on the algorithm implementation [48]. Fagerström

et al. consider the reverberation produced by an FDN of order 32

as sufﬁciently dense and the one synthesized with a 16th-order

FDN as slightly too sparse [49]. For fairness, we choose here

the order 16 as the smallest order for the FDN that is useful

for high-quality audio. Table II indicates that the corresponding

computational cost is 2065 operations per sample, which is

marked with a circle in Fig. 18.

A comparison of the highlighted computational costs of 802

and 2065 operations per sample in Table II shows that the

number of operations required by the IVN reverberator is just

40% of that required by the FDN. This study indicates that the

proposed IVN reverberator can provide a high-quality response

with a much smaller number of parallel systems than the FDN.

This reduces similarly the number of loop ﬁlters needed.

V. DISCUSSION

The proposed algorithm synthesizes the late part of the re-

verberation well in terms of reverberation time values and echo

density. As the sound examples presented online [26] prove, the

synthetic impulse responses are very similar to the original ones,

but there are still perceptible differences.

The evaluation of synthetic reverberation is a complex issue.

On one hand, objective measures, such as reverberation time

or frequency characteristics of the reverberation [50], provide

quantiﬁcation which, to some extent, is relevant to the acoustic

quality of the produced sound [51], [52]. Such parameters,

however, do not consider the whole spectrum of perceptual

aspects of sound within a room, which has been studied for well

over a century [53]. Recent research suggests that there are tens

of attributes associated with small listening rooms only [54] and

at least as many used to describe concert hall acoustics [55]–[58].

Of these, the term “reverberance,” which is used as a descriptor

for the perception of reverberation, is not solely dependent on

the T60 values, in the same way as “diffuseness” is not only

determined by echo density.

On the other hand, numerous studies have compared syn-

thesized reverberation with measured impulse responses [27],

[59]–[62]. Participants of listening tests can usually distinguish

between the target and synthesized sound [27], [59], [60], [62].

This shows that the parameters regulating artiﬁcial reverberation

algorithms are insufﬁcient to impeccably imitate the complexity

of a real-world sound. However, the practical use of artiﬁcial

reverberation hardly ever involves direct comparison between

measured and synthetic impulse responses of the same space.

Thus, plausible yet slightly modiﬁed artiﬁcial reverberation can

still be highly useful in music production and gaming, where it

is used for artistic effect or spatial impression.

VI. CONCLUSION

This paper proposes a novel algorithm for synthesizing late

reverberation by interleaving extended velvet-noise sequences.

Each of the EVNs has one non-zero sample in each MTdsam-

ples, where Mis the number of interleaved sequences and Tdis

the grid size, which in this work was set to 20 samples. Delaying

each EVN by a different number of sampling intervals ensures

that the non-zero samples never occur at the same time when

several parallel sequences are combined. This is a new principle

in audio processing. The results of the listening test presented

in this paper show that interleaving four EVNs having a total

density of 2205 samples/s is sufﬁcient to obtain perceptually

smooth, non-repetitive noise.

VÄLIMÄKI AND PRAWDA: LATE-REVERBERATION SYNTHESIS USING INTERLEAVED VELVET-NOISE SEQUENCES 1159

Each branch of the IVN reverberator includes an SFIR ﬁlter

and a feedback structure containing a delay line and a loop ﬁlter.

Because SFIR ﬁltering is equivalent to convolving the input with

a velvet-noise sequence that can be implemented by adding and

subtracting delayed samples, only one delay line is needed for

each branch of the reverberator. To control the decay rate in

ten octave bands, this work proposes to use an accurate graphic

equalizer as a loop ﬁlter in every branch.

The proposed IVN reverberator is best suited for producing

long reverberation, and therefore the “world’s longest” impulse

response was used for validation. In this case, the perceptually

important RT values for frequencies below 2 kHz deviated by

less than 5% of the target values. For the synthesis of shorter

impulse responses, this paper proposes a segmented decay tech-

nique, that helps to attenuate the velvet-noise sequences in ﬁne

steps, approaching a continuous exponential decay. An example

design showed how accurately a concert hall impulse response

could be reproduced.

The proposed IVN algorithm produces late reverberation

with an optimal echo density that ensures smooth sound and

is economical to compute. The number of FLOPs per processed

sample in the proposed IVN algorithm is about 60% lower

than that of the FDN algorithm synthesizing reverberation of

comparable smoothness.

Future work may investigate how well the different permuta-

tions of the branches of the IVN reverberator are decorrelated.

This will lead to a better understanding how to use the proposed

method in multichannel setups. Future research may also aim at

improving the knowledge of the perception of reverberation.

ACKNOWLEDGMENT

The authors would like to thank Dr. Henri Penttinen for his

help in the early stages of this study as well as Benoit Alary and

Prof. Sebastian Schlecht for helpful comments and discussions.

The authors would also like to thank Luis Costa for proofreading

the manuscript.

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Italy, Jun. 2020, pp. 247–254.

VesaVälimäki(Fellow, IEEE) received the M.Sc. and

D.Sc. degrees in electrical engineering from the

Helsinki University of Technology (TKK), Espoo,

Finland, in 1992 and 1995, respectively.

He was a Postdoctoral Research Fellow with the

University of Westminster, London, U.K., in 1996.

From 1997 to 2001, he was a Senior Assistant (cf. As-

sistant Professor) with TKK. From 2001 to 2002,

he was a Professor of signal processing with Pori

unit, Tampere University of Technology, Pori, Fin-

land. From 2006 to 2007, he was the Head with the

Laboratory of Acoustics and Audio Signal Processing, TKK. From 2008 to

2009, he was a Visiting Scholar with Stanford University, Stanford, CA, USA.

He is currently a Full Professor of audio signal processing and the Vice Dean

for research in electrical engineering with Aalto University, Espoo, Finland.

His research interests include artiﬁcial reverberation, digital ﬁlter design, audio

effects processing, and sound synthesis.

Prof. Välimäki is a Fellow of the Audio Engineering Society and a Life

Member of the Acoustical Society of Finland. From 2007 to 2013, he was a

Member of the Audio and Acoustic Signal Processing Technical Committee

of the IEEE Signal Processing Society, and is currently an Associate Member.

From 2005 to 2009 and from 2007 to 2011, he was an Associate Editor for the

IEEE SIGNAL PROCESSING LETTERS and the IEEE TRANSACTIONS ON AUDIO,

SPEECH AND LANGUAGE PROCESSING.From 2015 to 2020, he was a Senior Area

Editor of the IEEE/ACM TRANSACTIONS ON AUDIO,SPEECH AND LANGUAGE

PROCESSING. He was on the Editorial Board of the Research Letters in Signal

Processing and the Journalof Electrical and Computer Engineering. He was the

Lead Guest Editor of a Special Issue of the IEEE Signal Processing Magazine

in 2007 and of a Special Issue of the IEEE TRANSACTIONS ON AUDIO,SPEECH

AND LANGUAGE PROCESSINGin 2010. He was the Guest Editor of Special Issues

of the IEEE Signal Processing Magazine in 2015 and 2019. He has been the

Guest Editor of Special Issues published in Applied Sciences in 2016, 2018, and

2020. He was the Chair of the 2008 International Conference on Digital Audio

Effects (DAFX) and the Chair of the 2017 International Conference on Sound

and Music Computing. He is currently the Editor-in-Chief of the Journal of the

Audio Engineering Society.

Karolina Prawda received the M.Sc. degree in

acoustic engineering from the AGH University of

Science and Technology, Kraków, Poland, in 2017.

She is currently working toward the Doctoral de-

gree with the Acoustics Lab, Aalto University,Espoo,

Finland. Her research interests include artiﬁcial rever-

beration and variable acoustics.

Since 2018, she has been a Member of the Polish

Section of the Audio Engineering Society.

Publication V

Karolina Prawda, Sebastian J. Schlecht, and Vesa Välimäki. Improved

Reverberation Time Control for Feedback Delay Networks. In Proceed-

ings of the International Conference on Digital Audio Effects (DAFx 2019),

Birmingham, UK, September 2019.

Reprinted with permission.

123

Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

IMPROVED REVERBERATION TIME CONTROL FOR FEEDBACK DELAY NETWORKS

Karolina Prawda, Vesa Välimäki ∗

Acoustics Lab, Dept. of Signal Processing and Acoustics

Aalto University

Espoo, Finland

karolina.prawda@aalto.fi

Sebastian J. Schlecht

International Audio Laboratories†

Erlangen, Germany

sebastian.schlecht@audiolabs-erlangen.de

ABSTRACT

Artiﬁcial reverberation algorithms generally imitate the frequency-

dependent decay of sound in a room quite inaccurately. Previous

research suggests that a 5% error in the reverberation time (T60)

can be audible. In this work, we propose to use an accurate graphic

equalizer as the attenuation ﬁlter in a Feedback Delay Network re-

verberator. We use a modiﬁed octave graphic equalizer with a cas-

cade structure and insert a high-shelf ﬁlter to control the gain at the

high end of the audio range. One such equalizer is placed at the

end of each delay line of the Feedback Delay Network. The gains

of the equalizer are optimized using a new weighting function that

acknowledges nonlinear error propagation from ﬁlter magnitude

response to reverberation time values. Our experiments show that

in real-world cases, the target T60 curve can be reproduced in a

perceptually accurate manner at standard octave center frequen-

cies. However, for an extreme test case in which the T60 varies

dramatically between neighboring octave bands, the error still ex-

ceeds the limit of the just noticeable difference but is smaller than

that obtained with previous methods. This work leads to more re-

alistic artiﬁcial reverberation.

1. INTRODUCTION

Reverberation time is one of the most important parameter used to

determine the acoustic quality of physical spaces. Multiple stud-

ies have been conducted to evaluate the accuracy of perceiving

the changes in the reverberation time for various types of signals.

Seraphim [1] determined the just noticeable difference (JND) of

the reverberation time to be 5%. However, more recent studies

showed that for bandlimited noise the difference is perceivable

only when it exceeds 24% of the target value [2], compared to 5%

to 7% for impulse signals and 3% to 9% for reverberated speech

[3]. The JND of 5% is used in this work to comply with the current

ISO standard [4].

Various algorithms are used to produce artiﬁcial reverberation,

with the Feedback Delay Network (FDN) being currently among

the most popular ones [5–7]. The ﬁrst objective in designing an

FDN is to make it lossless. Attenuation ﬁlters are introduced to

achieve target energy decay. Over time, various types of atten-

uation ﬁlters have been proposed. Initially, a ﬁrst-order lowpass

∗This work was supported by the “Nordic Sound and Music Computing

Network—NordicSMC”, NordForsk project number 86892.

†The International Audio Laboratories Erlangen are a joint institu-

tion of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) and

Fraunhofer Institut für Integrierte Schaltungen IIS.

2019 Karolina Prawda, Vesa Välimäki et al. This is an open-access

article distributed under the terms of the Creative Commons Attribution 3.0 Un-

ported License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original author and source are credited.

inﬁnite impulse response (IIR) ﬁlter was used because of its low

computational cost and ease of design [5, 8]. Later, biquadratic

ﬁlters were introduced allowing to control the decay time in three

independent frequency bands with adjustable crossover frequen-

cies [9]. In [10], a 13th-order ﬁlter comprising single bandpass

ﬁlters as described in [11] and a second-order Butterworth band-

pass ﬁlter was proposed.

The most advanced method of controlling decay time in arti-

ﬁcial reverberation in several frequency bands uses a proportional

graphic equalizer [12]. This method was recently improved by

Schlecht and Habets, who determined the ﬁlter parameters by solv-

ing the nonlinear least-squares problem with linear constraints ap-

proximating the target reverberation-time response directly [13].

This approach offered very accurate control of decay time and en-

sured that the FDN remained stable. However, the computation

of ﬁlter parameters proved to be inefﬁcient, especially in real-time

applications [13].

The present work proposes an accurate method to control re-

verberation time in octave bands utilizing attenuation ﬁlters that

produce small approximation errors. It is an extension to previous

work done by Schlecht and Habets [13]. This paper introduces a

novel graphic equalizer (GEQ) with an additional high-shelf ﬁlter

as an attenuation ﬁlter inside the FDN and presents a weighted-

gain optimization method that acknowledges nonlinear error prop-

agation from ﬁlter magnitude response to reverberation time val-

ues. The paper is organized as follows. Section 2 discusses attenu-

ation ﬁlters and proposes a new design as well as a weighted-gain

optimization method. Section 3 presents case studies in which we

test the proposed method and compare the proposed design to other

solutions in terms of the approximation error as well as computa-

tional cost. Section 4 summarizes the work presented in the paper,

gives conclusions about the results, and proposes ideas for future

research.

2. ATTENUATION FILTER

An FDN is a comb ﬁlter structure with multiple delay lines inter-

connected by a feedback matrix [5]. When designing FDNs, the

ﬁrst step is to make it lossless, ensuring that the energy will not

decay for any possible type of delay [7]. The frequency-dependent

reverberation time can then be implemented by inserting an atten-

uation ﬁlter at the beginning or at the end of each delay line. As

the ﬁlters do not work in relation to one another and are only de-

pendent on their corresponding delay line, they can be analyzed

separately. Instead of the FDN, we can analyze the simpler single-

delay-line absorptive feedback comb ﬁlter, i.e.,

H(z) = 1

1−A(z)z−L,(1)

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

Figure 1: Relationship between (top) gain-per-sample and (bot-

tom) resulting reverberation time for the delay-line length of L=

1000 samples. Red markers indicate the octave bands. The hori-

zontal dashed line in the top ﬁgure is the unit-gain limit, reaching

which would lead to an inﬁnite reverberation time.

where Lis the delay length in samples and A(z)is the transfer

function of the attenuation ﬁlter. For further analysis of the mag-

nitude in dB, the attenuation ﬁlter is given by

AdB(ω) = 20 log10 |A(ej ω )|,(2)

where ω= 2πf /fsis the normalized frequency, fis the frequency

in Hz, and fsis the sampling rate in Hz. Such a ﬁlter should be

designed to approximate the gain-per-sample necessary to obtain

the desired frequency-dependent reverberation time, T60(ω). This

gain in dB is expressed as

γdB(ω) = −60

fsT60(ω),(3)

where T60(ω)is in seconds. The gain is dependent on the delay-

line length, growing proportionally to the number of delay samples

L. As a result, longer delay lines decay faster than short ones. To

obtain the target gain and, as a consequence, the desired frequency-

dependent reverberation time, the following condition should be

met:

AdB(ω) = LγdB (ω).(4)

Fig. 1 illustrates the relation between the gain-per-sample val-

ues of the single-delay-line absorptive feedback comb ﬁlter as pre-

sented in Eq. (1) with the attenuation ﬁlter designed according to

Eq. (2-4) and the resulting T60(ω)values. The delay-line length

was set to L= 1000 samples and the target reverberation time was

set to decrease linearly in octave bands from 7 s at 31.5Hz to 1 s at

16 kHz.

2.1. Graphic equalizer design

The attenuation ﬁlter in the present work is realized with the cas-

cade GEQ, composed of second-order IIR peak-notch ﬁlters pro-

posed by Orfanidis [14] and designed using a method proposed by

Figure 2: Comparison of magnitude responses of the proposed

GEQ with a high-shelf ﬁlter to the proportional graphic equalizer

used in [13]. Top: magnitude responses for individual biquadratic

ﬁlters and a prototype gain of 1 dB for ten frequency bands. Bot-

tom: Single-band proportional gain behavior of the magnitude re-

sponse.

Välimäki and Liski [15], where extra frequency points are added

and one iteration step is used to obtain a highly accurate magnitude

response. The GEQ is also composed of only peak-notch ﬁlters,

as opposed to the usual approach in which shelf ﬁlters are applied

to the highest and lowest frequency bands. Using only peak-notch

ﬁlters improves the symmetry of the magnitude responses of in-

dividual ﬁlters and the accuracy of the equalizer. This results in

the proposed design producing approximation errors of less than

±1 dB for command gains within a range of −12 to +12dB [15].

The top plot of Fig. 2 depicts the magnitude responses of the indi-

vidual biquadratic ﬁlters of the proposed GEQ, with an additional

high-shelf ﬁlter as described in Sec. 2.2., compared to the pro-

portional graphic equalizer from [13]. The approach adopted in

the present paper displays more symmetrical magnitude responses

even for high frequencies. The bottom plot of Fig. 2 presents the

magnitude response of peak-notch ﬁlters for command gains be-

tween −30 and +30 dB.

The transfer function of a GEQ with Mbands is given by

H(ejω ) = G0

m=1

Hm(ejω ),(5)

where G0is the overall broadband gain factor and Hm(ejω)are

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

the frequency responses of equalizing ﬁlters (m= 1,2,3..., M).

The corresponding response in dB can be written as

HdB(ejω ) = g0+

m=1

HdB,m(ejω ).(6)

For the accurate approximation of the reverberation time T60

over a broad frequency range, the command gains are deﬁned for

ten octave bands, having center frequencies ranging from 31.5 Hz

to 16 kHz. This results, however, in the magnitude approaching

0 dB quite dramatically outside the considered frequency range.

The reverberation time approximated for the octave bands below

31.5 Hz and above 16 kHz can appear to be very long, which may

affect whole decay, preventing it from ever reaching −60 dB. To

avoid this situation, we propose that the command gains be ﬁrst

shifted up to decrease their distance from zero and after the gain

optimization, the entire magnitude response be scaled down by

the same amount as for scaling up. This is depicted in the top

and middle panes of Fig. 3. This yields the following changes to

Eq. (6):

HdB(ejω ) = g0+

m=1

(HdB,m(ejω )−g0

M).(7)

In this way, the rise in magnitude at frequencies below 31.5Hz and

above 16 kHz are less steep. The shifting and scaling value can be

set to the median of all command gains, as suggested in [16]. This

also smooths the frequency response, causing very little ripple, as

seen in the ﬁlter response comparison with and without scaling in

Fig. 4.

2.2. High-shelf ﬁlter

In physical room acoustics, the decay time at high frequencies

is usually shorter than at low frequencies, thus making the cor-

responding command gains considerably lower at high frequen-

cies. Therefore, the operation of scaling and shifting the gains by

the median may not be sufﬁcient to prevent the magnitude from

quickly approaching large values for frequencies above 16kHz.

For this reason, we use a ﬁrst-order high-shelf ﬁlter implemented

as suggested in [17] to equalize the problematic high frequencies.

The gain of the shelf ﬁlter is set to the gain of the highest consid-

ered octave band, and the crossover frequency is set to 20.2kHz.

The latter value was experimentally found to introduce the small-

est error in the reverberation time at 16 kHz.

The shelf ﬁlter introduces considerable ripple in the equalizer

response above the frequency range of interest, but since it oc-

curs at very high frequencies and the resulting reverberation time

is much shorter than at lower frequencies, it is assumed to be in-

audible. The response of the GEQ with the shelf ﬁlter is shown in

the bottom pane of Fig. 3.

2.3. Filter-gain optimization

A common ﬂaw in graphic equalizers is the interaction occurring

between neighboring peak-notch ﬁlters, causing the response of

each ﬁlter leak to other center frequencies [15–17]. A K-by-Nin-

teraction matrix Bthat shows this effect and stores the normalized

amplitude response in dB of all Mﬁlters at Kcontrol frequency

points is given by:

Bk,m =HdB,m(ej ωk)/gp,m,(8)

Figure 3: Stages of obtaining the ﬁnal frequency response of the

GEQ for a delay length of 100 ms. (Top) Gains shifted up by their

median value. (Middle) Gains scaled by a constant value. (Bot-

tom) A high-shelf ﬁlter inserted to attenuate frequencies above

16 kHz.

Figure 4: Frequency response of the GEQ for a delay length of

100 ms, shifted and scaled by the median of gains compared to the

response without scaling.

where k= 1,2, ...K are control frequency points, m= 1,2, ...M

are ﬁlter indices, and gp=gp,1, gp,2 , ..., gp,m,T, where .Tde-

notes the transpose, is the vector of prototype dB gains common to

all equalizing ﬁlters. The interaction matrix of the proposed GEQ

for K= 100 and N= 11 is shown in Fig. 5. As a consequence of

leakage, the magnitude response of the equalizer depends on the

values stored in the interaction matrix. Considering that the GEQ

is used as the attenuation ﬁlter in the FDN, Eq. (4) can now be

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

Figure 5: A100-by-11 interaction matrix of the proposed GEQ

that stores the normalized amplitude of each ﬁlter that leaks to

neighboring frequency points.

expressed as

AdB(ω) = Bgp,(9)

where ωis a K×1vector of control frequencies.

2.4. Weighted-gain optimization

The GEQ used in the present work approximates the command

gains strictly within 1 dB. However, in some instances of the rever-

beration time changing dramatically for neighboring octave bands,

the differences in the command gains may be too high for the

resulting ﬁlter magnitude response to follow without producing

much error. Therefore, a method for gain optimization is intro-

duced.

The approximation of the target magnitude response can be

done on the dB scale by minimizing the error norm based on Eq. (4):

kAdB(ω)−LγdB (ω)k2

2.(10)

This approach assumes that the error propagates to the resulting

reverberation time linearly. In reality, the reverberation time is

affected in a nonlinear fashion: a small error in the ﬁlter magnitude

response, when the attenuation is weak and the gain close to 0 dB,

causes much greater changes in the resulting reverberation time

than the same error, when the attenuation is strong and the gain

much smaller than 0 dB [13].

This problem can be overcome by directly minimizing the

squared error in the resulting reverberation time, as suggested ear-

lier in [13]:

E=



AdB(ω)−1

LγdB(ω)



.(11)

Alternatively, we can minimize the relative error between the ﬁlter

and the target reverberation time:

E=



1−AdB(ω)

LγdB(ω)



.(12)

When the weighting matrix Wis deﬁned as

W=diag1

LγdB(ω),(13)

the relative error from Eq. (12) becomes

E=



1−WAdB(ω)



,(14)

where the role of the weighting matrix Wis to mimic the nonlin-

ear behavior with a linear approximation in the sense of emphasiz-

ing the approximation error occurring close to 0dB.

Error minimization was also performed by solving the linear

problem using the Taylor-series approximation presented in [18].

However, since the method operates on a linear scale, as opposed

to the suggested design which operates on the dB scale, no relevant

improvement was observed. Therefore the method utilizing the

Taylor-series approximation was not implemented further in the

present work.

3. EVALUATION

The present work proposes to perform reverberation time approx-

imation using a GEQ with weighted-gain optimization that min-

imizes the relative error between ﬁlter response and target rever-

beration time values. In order to evaluate that algorithm, two case

studies were conducted. The ﬁrst was aimed to reproduce the re-

verberation time of Promenadi Hall, a multipurpose hall located

in Promenadikeskus in Pori. The second was conducted using

a predeﬁned reverberation time that differs considerably between

neighboring octave bands to reveal potential weak spots of the al-

gorithm and provide a valid comparison to the previous work. For

both cases, the algorithm was tested with three lengths of delay

lines: 10 ms, 50 ms, and 100 ms. The performance of the proposed

algorithm was compared to the previous method of reverberation-

time control in FDN presented in [13]. The computational cost

measured in the number of operations per output sample with re-

lation to other graphic equalizers was also examined.

3.1. Promenadi Hall

In the ﬁrst case, the aim was to approximate the reverberation time

of an existing architectural object. The target values were deﬁned

for octave bands and are presented in Table 1. The command gains

for the GEQ were calculated based on these values. Fig. 6 com-

pares the target magnitude response needed to obtain the desired

reverberation time and the response of the GEQ for the delay-line

lengths of 10, 50, and 100 ms.

The target magnitude response is followed by the response of

the ﬁlter very accurately in every octave band it was speciﬁed in.

The magnitude response below 31.5 Hz approaches the median

value for the command gains, which was set as the equalizer’s

broadband gain. The only visible ripple of 0.02 dB, 0.11 dB and

0.23 dB for delay-line lengths of 10ms, 50 ms and 100ms, respec-

tively, occur at very high frequencies, above 16 kHz, and is caused

by the high-shelf ﬁlter.

The resulting reverberation time was calculated based on the

magnitude response of the GEQ by converting it to dB using Eq. (2),

and then using the condition from Eq. (4) to obtain the gain-per-

sample in dB. The values of T60(ω)were acquired based on Eq. (3)

and are depicted in the top plot of Fig. 7 together with target values

from Promenadi Hall. The obtained reverberation follows the be-

havior of the ﬁlter response, approximating the desired values very

closely not only in the octave frequencies, but also in the entire fre-

quency range. Although the values were calculated for three dif-

ferent delay-line lengths, the results do not vary visibly from each

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

Table 1: Reverberation time values and error percentage for octave frequencies for Promenadi Hall. DL stands for delay length.

Center frequency 31.5 Hz 63 Hz 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz 16 kHz

Reverberation time 3.00 s 2.80 s 2.68 s 2.55 s 2.47 s 2.50 s 2.30 s 1.89 s 1.40 s 1.20 s

Error for DL of 10 ms 0.12% 0.03% 0.09% 0.05% 0.02% 0.35% 0.79% 1.00% 1.51% 4.43%

Error for DL of 50 ms 0.12% 0.03% 0.09% 0.05% 0.02% 0.35% 0.79% 1.00% 1.53% 4.44%

Error for DL of 100 ms 0.12% 0.03% 0.09% 0.05% 0.02% 0.35% 0.79% 0.99% 1.56% 4.50%

Figure 6: Target magnitude response and response of the GEQ

with ﬁrst-order high-shelf ﬁlter for the case of Promenadi Hall in

Pori.

other, with the biggest difference between them reaching 0.07%.

This proves that the proposed method works well regardless of the

delay chosen when designing the FDN. This is also conﬁrmed by

the error values shown in Table 1, none of which exceed 5%, mak-

ing the difference unnoticeable. Further evidence for the method’s

efﬁciency to accurately approximate reverberation time is in the

bottom plot of Fig. 7, which shows the difference between the tar-

get and the obtained reverberation time for the whole frequency

range.

The results obtained with the GEQ introduce deviations no

bigger than 5% from the target value and therefore we refrained

from trying to minimize the error.

Figure 7: (Top) Target reverberation time from Promenadi Hall

and the values approximated by the GEQ. (Bottom) The difference

expressed as percent of obtained reverberation time deviation from

target values. RT is reverberation time.

3.2. Artiﬁcial extreme case

The second case was tested with predeﬁned reverberation time val-

ues, which were aimed at being similar to those in [13]. Although

in [13] the reverberation time values for octave bands were gen-

erated randomly, we decided to specify them manually in order to

ensure the same tendency, which provides a good reference point

for comparison of the two methods and is able to reveal shortcom-

ings of the proposed design. The values of the reverberation time

for octave frequencies are shown in Table 2.

The ﬁlters’ magnitude responses obtained based on the desired

decay are presented in Fig. 8. The target response is generally fol-

lowed accurately. However, large differences in the reverberation

between 2 kHz and 4 kHz cause a slight overshoot in magnitude for

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

Table 2: Reverberation time and error on octave frequencies for the artiﬁcial extreme case. Errors exceeding JND of 5.0% are highlighted.

Center frequency 31.5 Hz 63 Hz 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz 16 kHz

Reverberation time 1.00 s 1.00 s 1.00 s 1.00 s 1.00 s 3.00 s 3.00 s 0.25 s 1.00 s 1.00 s

Error for DL of 10 ms 0.12% 0.09% 0.17% 0.07% 3.66% 12.59% 11.47% 0.38% 2.96% 5.80%

Error for DL of 50 ms 0.13% 0.07% 0.22% 0.12% 3.11% 8.79% 15.59% 2.62% 5.42% 4.42%

Errorfor DL of 100 ms 0.15% 0.02% 0.39% 0.50% 1.52% 0.35% 24.41% 6.03% 11.12% 0.77%

Figure 8: Target magnitude response and response of the GEQ for

three different delay-line lengths for the artiﬁcial extreme case.

frequencies between 1 kHz and 2 kHz, which at its highest point

lies very close to zero. This causes a huge increase in the rever-

beration time for those frequencies, which is seen in the top plot

in Fig. 9. When the difference between the target and the approxi-

mated reverberation time is expressed as a percentage into percent-

age, as shown in the bottom plot in Fig. 9, the 5% JND threshold

is exceeded everywhere except for the low frequencies, where the

target decay is the same in neighboring octave bands.

The results in Fig. 9 show the effect of the delay-line length on

the approximation error. The attenuation for the shortest delay line

is the weakest, making all deviations from the target magnitude to

cause much error in the resulting reverberation time values. For

the longest delay-line length, the overshoots between 1 kHz and

2 kHz, as well as around 8kHz, are the smallest.

In order to improve the resulting reverberation time, we ap-

plied the gain-optimization method proposed in Sec. 2.3 by mini-

mizing the error norm in Eq. (14) and using the weighting matrix

Figure 9: (Top) Target reverberation time and reverberation time

obtained with the GEQ for three different delay-line lengths for

the artiﬁcial extreme case. (Bottom) The difference expressed as

a percent of the obtained reverberation time deviation from target

values.

in Eq. (13). The magnitude response of the GEQ is presented in

Fig. 10. Overshoots between 1kHz and 2 kHz were decreased at

the cost of lower accuracy in approximating the target at 4 kHz.

Additionally, some ripple was introduced in the frequency range

between 250 Hz and 500Hz.

The corresponding reverberation time values are shown in

Fig. 11. The values that exceed the target the most were success-

fully reduced. The improvement was made without an unreason-

able increase in error in the reverberation time for less problematic

frequencies. Fig. 11 shows that the deviation in percent from the

target values was the same or less for frequencies over 500Hz. The

error for low frequencies increased slightly, in most octave bands

not exceeding the 5% JND.

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

Figure 10: Target magnitude response and response of the GEQ

for three different delay-line lengths for the artiﬁcial extreme case.

The command gains were weighted according to Eq. (9).

The error-minimization method worked well for every delay-

line length. In the end, all delay lines displayed similar differences

from target values, which is a huge improvement from before the

optimization.

3.3. Comparison with previous method

The proposed method was compared with the design presented in

[13], which solves the nonlinear least-squares problem based on

the T60 least-squares problem, as given in Eq. (11), with additional

constraints on the command gains. The abbreviation TLSCon in

Tables 3 and 4 refers to the results obtained with this method. To

allow direct comparison, the same reverberation time and delay-

line length were chosen. The error percentage in T60 for each

method are presented in Table 3.

The proposed method produces a smaller approximation error

than the solution suggested in [13]. The largest deviation from the

target value occurs at 4 kHz, where an attenuation of −60 dB is

needed. However, the obtained 62.69% error is a huge improve-

ment compared to the TLSCon solution error of 280% for the same

frequency.

3.4. Computational complexity of the proposed design

The GEQ with the ﬁrst-order high-shelf ﬁlter used in the present

work was compared in terms of computational cost with three

Figure 11: (Top) Target reverberation time and reverberation time

obtained with the GEQ for three different delay-line lengths for

the artiﬁcial extreme case after gain weighting. (Bottom) Differ-

ence expressed as a percentage of the obtained reverberation time

deviation from target values. Cf. Fig. 9.

other graphic equalizers: the proportional graphic equalizer

(TLSCon) [13,16], the cascaded fourth-order equalizer (EQ4) [19],

and the high-precision parallel equalizer (PGE) [17, 20]. The val-

ues are given for ﬁlter conﬁgurations as stated in [15] and are pre-

sented together with the number of operations for the proposed

design in Table 4.

The design proposed in the present work requires less compu-

tation than the cascaded fourth-order equalizer and the high pre-

cision parallel equalizer. It needs a few more operations than the

proportional graphic equalizer because the ﬁrst-order high-shelf

ﬁlter has been inserted to process frequencies above 16 kHz.

4. CONCLUSIONS

The present work investigated the effect of using a cascaded GEQ

with a ﬁrst-order high-shelf ﬁlter as the attenuation ﬁlter in the

FDN. In order to evaluate the performance of the proposed de-

sign, two cases, a real-life case of an existing concert hall’s rever-

beration time T60 and an artiﬁcially created extreme case, were

tested. Additionally, weighted-gain optimization was performed

to improve the results. The new weighting matrix emphasizes the

approximation error occurring close to 0dB. The gains are then de-

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Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019

Table 3: Error percentage for T60 approximated using the previous method (TLSCon) and the proposed method.

Center frequency 63 Hz 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz 16 kHz

TLSCon [13] 1.00% 0.00% 3.00% 2.00% 9.00% 32.00% 280.00% 31.00% 5.00%

Proposed method 3.39% 1.44% 8.15% 2.54% 16.98% 49.24% 62.59% 20.69% 4.98%

Table 4: Number of operations per output sample for octave band

equalizers.

Design ADD MUL TOTAL

TLSCon 40 50 90

EQ4 140 150 290

PGE 80 81 161

Proposed 42 52 94

termined by minimizing relative error between the ﬁlter response

and the target reverberation time.

The proposed method was shown to perform an excellent ap-

proximation of the real-life reverberation time values, resulting in

an error between the target and obtained values that is lower than

the JND. When the desired values change dramatically between

neighboring frequency bands, the presented algorithm causes greater

errors in the reverberation time, which can be then considerably re-

duced by the weighted-gain optimization. The study showed that

the proposed method produces a smaller approximation error in

the reverberation time than previous methods, and its computa-

tional cost is low or about the same compared to other designs.

The plans for further development of this work include pro-

viding subjective evaluation of the results as well as incorporating

the proposed attenuation ﬁlter and the weighted-gain optimization

method in other tools for creating artiﬁcial reverberation.

5. REFERENCES

[1] H. P. Seraphim, “Untersuchungen über die Unterschiedss-

chwelle exponentiellen Abklingens von Rauschbandim-

pulsen,” Acta Acustica united with Acustica, vol. 8, no. 4,

pp. 280–284, 1958.

[2] M. G. Blevins, A. T. Buck, Z. Peng, and L. M. Wang, “Quan-

tifying the just noticeable difference of reverberation time

with band-limited noise centered around 1000 Hz using a

transformed up-down adaptive method,” in Proc. Int. Symp.

Room Acoustics (ISRA), Toronto, Canada, June 9–11, 2013.

[3] M. Karjalainen and H. Järveläinen, “More about this rever-

beration science: Perceptually good late reverberation,” in

Proc. 111th Audio Eng. Soc. Conv., New York, USA, Sept.

21–24, 2001.

[4] ISO, “ISO 3382-1, Acoustics – Measurement of room acous-

tic parameters – Part 1: Performance spaces,” Tech. Rep.,

2009.

[5] J. M. Jot and A. Chaigne, “Digital delay networks for de-

signing artiﬁcial reverberators,” in Proc. 90th Audio Eng.

Soc. Convention, Paris, France, Febr. 19–22, 1991.

[6] V. Välimäki, J. D. Parker, L. Savioja, J. O. Smith, and J. S.

Abel, “Fifty years of artiﬁcial reverberation,” IEEE Trans.

Audio Speech Lang. Process., vol. 20, no. 5, pp. 1421–1448,

Jul. 2012.

[7] S. J. Schlecht and A. P. Habets, “On lossless Feedback Delay

Networks,” IEEE Trans. Signal Process., vol. 65, no. 6, pp.

1554–1564, Mar. 2017.

[8] J. Moorer, “About this reverberation business,” Computer

Music J., vol. 3, no. 2, pp. 13–28, 1979.

[9] J. M. Jot, “Efﬁcient models for reverberation and dis-

tance rendering in computer music and virtual audio reality,”

in Proc. Int. Computer Music Conf., Thessaloniki, Greece,

Sept. 1997.

[10] T. Wendt, S. van de Par, and S. D. Ewert, “A

computationally-efﬁcient and perceptually-plausible algo-

rithm for binaural room impulse response simulation,” J.

Audio Eng. Soc., vol. 62, no. 11, pp. 748–766, Nov. 2014.

[11] M. Holters and U. Zölzer, “Parametric high-order shelving

ﬁlters,” in Proc. 14th European Signal Processing Confer-

ence (EUSIPCO), Florence, Italy, Sept. 4–8, 2006.

[12] J.-M. Jot, “Proportional parametric equalizers–Application

to digital reverberation and environmental audio processing,”

in Proc. 139th Audio Eng. Soc. Conv., New York, USA, Oct.

29–Nov. 1, 2015.

[13] S. J. Schlecht and A. P. Habets, “Accurate reverberation time

control in Feedback Delay Networks,” in Proc. Digital Au-

dio Effects (DAFx-17), Edinburgh, UK, Sept. 5–9, 2017, pp.

337–344.

[14] S. J. Orfanidis, Introduction to Signal Processing, Rutgers

Univ., Piscataway, NJ, USA, 2010.

[15] V. Välimäki and J. Liski, “Accurate cascade graphic equal-

izer,” IEEE Signal Process. Lett., vol. 24, no. 2, pp. 176–180,

Feb. 2017.

[16] R. J. Oliver and J. M. Jot, “Efﬁcient multi-band digital audio

graphic equalizer with accurate frequency response control,”

in Proc. 139th Audio Eng. Soc. Conv., New York, USA, Oct.

29–Nov. 24, 2015.

[17] V. Välimäki and J. Reiss, “All about audio equalization: So-

lutions and frontiers,” Applied Sciences, vol. 6, no. 5, May

2016.

[18] B. Bank and V. Välimäki, “Robust loss ﬁlter design for dig-

ital waveguide synthesis of string tones,” IEEE Signal Pro-

cess. Lett., vol. 10, no. 1, pp. 18–20, Jan. 2003.

[19] M. Holters and U. Zölzer, “Graphic equalizer design using

higher-order recursive ﬁlters,” in Proc. Int. Digital Audio

Effects (DAFx-06), Montreal, Canada, Sept. 18–20, 2006, pp.

37–40.

[20] J. Rämö, V. Välimäki and B. Bank, “High-precision parallel

graphic equalizer,” IEEE/ACM Trans. Audio Speech Lang.

Process., vol. 22, no. 12, pp. 1894–1904, Dec. 2014.

DAFX-8

Publication VI

Karolina Prawda, Silvin Willemsen, Stefania Seraﬁn, and Vesa Välimäki.

Flexible Real-Time Reverberation Synthesis with Accurate Parameter Con-

trol. In Proceedings of the International Conference on Digital Audio Effects

(DAFx 2020), Vienna, Austria, September 2020.

Reprinted with permission.

133

Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020

FLEXIBLE REAL-TIME REVERBERATION SYNTHESIS WITH ACCURATE PARAMETER

CONTROL

Karolina Prawda ∗

Acoustics Lab

Dept. of Signal Processing and Acoustics

Aalto University, Espoo, Finland

karolina.prawda@aalto.fi

Silvin Willemsen, Stefania Seraﬁn

Multisensory Experience Lab

Dept. of Architecture, Design & Media Tech.

Aalborg University, Copenhagen, Denmark

{sil,sts}@create.aau.dk

Vesa Välimäki

Acoustics Lab

Dept. of Signal Processing and Acoustics

Aalto University, Espoo, Finland

vesa.valimaki@aalto.fi

ABSTRACT

Reverberation is one of the most important effects used in audio

production. Although nowadays numerous real-time implementa-

tions of artiﬁcial reverberation algorithms are available, many of

them depend on a database of recorded or pre-synthesized room

impulse responses, which are convolved with the input signal. Im-

plementations that use an algorithmic approach are more ﬂexible

but do not let the users have full control over the produced sound,

allowing only a few selected parameters to be altered. The real-

time implementation of an artiﬁcial reverberation synthesizer pre-

sented in this study introduces an audio plugin based on a feed-

back delay network (FDN), which lets the user have full and de-

tailed insight into the produced reverb. It allows for control of

reverberation time in ten octave bands, simultaneously allowing

adjusting the feedback matrix type and delay-line lengths. The

proposed plugin explores various FDN setups, showing that the

lowest useful order for high-quality sound is 16, and that in the

case of a Householder matrix the implementation strongly affects

the resulting reverberation. Experimenting with delay lengths and

distribution demonstrates that choosing too wide or too narrow a

length range is disadvantageous to the synthesized sound quality.

The study also discusses CPU usage for different FDN orders and

plugin states.

1. INTRODUCTION

Artiﬁcial reverberation is one of the most popular audio effects. It

is used in music production, sound design, game audio, and movie

production to enhance dry recordings with the impression of space.

The development of digital artiﬁcial reverberation started nearly 60

years ago [1], and since then various improvements as well as dif-

ferent techniques have been developed [2]. The designs available

nowadays can be roughly divided into three groups: convolution

algorithms, delay networks, and physical room models [2, 3, 4].

The methods involving physical modeling simulate sound prop-

agation in a speciﬁc geometry. Due to their high computational

cost, though, they are used mostly in off-line computer simulations

of room acoustics [3]. Recent developments in hardware and soft-

ware technologies have also allowed computationally expensive

simulations, such as those based on 3-D ﬁnite-difference schemes,

to run in real time [5].

∗This work was supported by the “Nordic Sound and Music Computing

Network—NordicSMC”, NordForsk project number 86892.

under the terms of the Creative Commons Attribution 3.0 Unported License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the

original author and source are credited.

The techniques convolving the input signal with a measured

room impulse response (RIR) produce rich, high-ﬁdelity reverber-

ation. However, since the RIR samples serve as the coefﬁcients of

a ﬁnite impulse-response (FIR) ﬁlter, with which the dry signal is

ﬁltered, the computational cost is high, especially for long RIRs.

Another group of artiﬁcial reverberation algorithms is based

on networks of delay lines and digital ﬁlters. The ﬁrst example

of such reverberators was introduced by Schroeder and Logan [1],

who used feedback-comb-ﬁlter structures to create a sequence of

decaying echoes. A similar architecture using allpass ﬁlters was

also proposed to ensure high echo density without spectral col-

oration. The development of such structures led to the invention of

feedback delay network (FDN) algorithms, which can be regarded

as a “vectorized” comb ﬁlter [2]. The FDN, as used in its current

form, was presented in the work of Jot and Chaigne [6, 7].

Over the years, many real-time implementations of artiﬁcial

reverberation algorithms have been developed. The designs that

use a convolution-based approach, however, depend on measured

or pre-synthesized RIRs convolved with the signal, which are col-

lected in groups of presets [3, 8, 9, 10]. Such Virtual Studio Tech-

nology (VST) plugins allow modifying the reverberation by modu-

lating, damping or equalizing the available RIRs. The possibilities

are, however, limited by the size of the RIR databases and there-

fore prove to be relatively inﬂexible.

Algorithmic reverb plugins that are based on delay network

designs are both computationally efﬁcient and easily modulated,

thus providing more ﬂexibility and freedom in producing reverber-

ated sounds [4, 11]. The available designs vary between simple so-

lutions allowing the user to change only a few parameters [12] and

complex architectures with an elaborate interface enabling control

over a wide range of variables [13]. Many of those plugins, how-

ever, still remain ambiguous about the reverberation they synthe-

size, allowing the user to set only the broadband decay parameter,

and rely on presets based on the types of rooms they are supposed

to imitate (e.g., Bright Room or Dark Chamber [14]). Usually,

they also lack the information about the reverberation algorithm

they use and its elements.

The present work proposes a real-time implementation of an

FDN algorithm with accurate control over the reverberation time

(RT) in ten octave frequency bands in the form of an audio plu-

gin. The graphical user interface (GUI) gives a thorough insight

into the attenuation ﬁlter’s magnitude response, corresponding RT

curve, and resulting impulse response (IR). The plugin also pro-

vides several possibilities to control the elements of the FDN struc-

ture, such as the feedback matrix and delay lines. It gives the user

a full view of the decay characteristics and quality of the synthe-

sized reverberation. The study also presents the effect that the type

and size of the feedback matrix and the lengths and distribution of

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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020

the delay lines have on the produced sound and the algorithm’s

performance.

This paper is organized as follows: Section 2 presents the

theory behind the FDN, and Section 3 shows the GUI of the im-

plemented plugin, describes the functionalities and user-controlled

parameters of the reverberator, presents the code structure and dis-

cusses the real-time computation issues. Section 4 shows and dis-

cusses results regarding the echo density produced by the imple-

mentation and the CPU usage of the plugin. Finally, Section 5

summarizes and concludes the work.

2. FEEDBACK DELAY NETWORK

Figure 1 presents a ﬂow diagram of a conventional FDN, which is

expressed by the relation:

y(n) =

cisi(n) + dx(n),(1a)

si(n+Li) =

Ai,je

hi(n)sj(n) + bix(n),(1b)

where y(n)and x(n)are the output and input signal, respectively,

at time sample n,si(n)is the output of the ith delay line, and

Ai,j is the element of an N-by-Nfeedback matrix (or scattering

matrix) A, through which all the delay lines are interconnected.

Parameters biand cisymbolize input and output coefﬁcients, re-

spectively, dis the direct-path gain, and e

hi(n)is the attenuation

ﬁlter of the ith delay line.

When designing an FDN, a common practice is to ﬁrst ensure

that the energy of the system will not decay for any possible type

of delay. Therefore, the matrix Ashould be unilossless [15]. To

obtain a speciﬁc frequency-dependent RT, each of the delay lines

must be cascaded with an attenuation ﬁlter, which approximates

the target gain-per-sample expressed by

γdB(ω) = −60

fsT60(ω),(2)

where T60(ω)is the target RT in seconds, ω= 2πf /fsis the

normalized angular frequency, fis the frequency in Hz, and fs

is the sampling rate in Hz. In order to ensure that all delay lines

approximate the same RT, the gain-per-sample for each of them

must be scaled by a respective delay in samples L. This implies

that the target magnitude response of the attenuation ﬁlter in dB is

deﬁned as follows:

AdB(ω) = LγdB (ω).(3)

In order to provide an accurate approximation of the target

RT, and therefore to closely follow the AdB, the attenuation ﬁlter

used in the FDN implementation in the present study is a graphic

equalizer (GEQ), which controls the energy decay of the system in

ten octave bands, with center frequencies from 31.25 Hz to 16 kHz.

The equalizer is composed of biquad ﬁlters [16] and designed with

the method proposed by Välimäki and Liski [17] with later modiﬁ-

cations, such as the scaling by a median of gains and the adding of

a ﬁrst-order high-shelf ﬁlter as proposed in [18]. The GEQ mag-

nitude response for the ith delay line is expressed in dB as

HdB,i(ejω ) = g0+

m=1 HdB,i,m(ejω )−g0

M,(4)

ℎ



ℎ





ℎ











Filter

EQDelayLine



EQDL



EQDL



⋮



−





















−































(



)





(



)

⋮

(



)



(



)





(



)





Figure 1: Flow diagram of an FDN with Nequalized delay lines

and their Moctave-band biquad ﬁlters shown in detail. See Sec-

tions 2 and 3.5 for more details.

where g0is the broadband gain factor, HdB,i,m are the magni-

tude responses of the band ﬁlters, and m= 1,2, ..., M is the

frequency-band index with Mcontrolled frequency bands. The

time-domain representation e

hi(n)of e

HdB,i(ejω )is used in Eq. (1b).

3. IMPLEMENTATION

This section describes the real-time implementation of late rever-

beration synthesis using an FDN and a modiﬁed GEQ as the atten-

uation ﬁlter. The algorithm has been implemented in the form of an

audio plugin in C++ using JUCE, an open-source cross-platform

application framework [19]. The plugin can be downloaded from

[20], and an explanatory demo video can be found in [21].

3.1. Control over RT Values

In the present implementation, the modiﬁed GEQ attenuation ﬁlter

allows controlling the RT values in ten frequency bands. In order

to utilize the whole potential of the ﬁlter, the GUI of the plugin is

equipped with ten vertical sliders, one for each frequency band, as

depicted in Fig. 2. By changing the value of each of the sliders, the

user is able to change the RT value for the corresponding frequency

band from 0.03 s to 15s with a 0.01-s step.

Since too large a difference between two consecutive RT val-

ues can cause instability [18, 22], two extra modes are imple-

mented for better control: the All Sliders and the Smooth modes.

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Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020

(a) The attenuation ﬁlter’s response (red line) and the corresponding RT

curve (black line). No preset is selected, and the Smooth button is pressed.

(b) Reverberator IR. The Fix coeffs button has been pressed,and the preset’s

drop-down menu and sliders have been disabled (see Sec. 3.6).

Figure 2: GUI of the implemented FDN plugin.

The modes are activated by pressing the respective buttons, as in-

dicated in Fig. 2a, with the corresponding buttons on the GUI be-

ing highlighted in green. If a mode is activated when the other is

active, the latter will deactivate. The All Sliders mode allows the

user to set all the RT values to be the same by changing the slider

position in one of the frequency bands.

When the Smooth mode is activated, changing the value of one

RT will also adjust the RT in the other frequency bands. RT values

of bands closer to the band that is changed are more affected than

other RT values via the formula

T60[m] = T60,init [m] + T60[mc]−T60,init [mc]ϵ|m−mc|,(5)

where mcis the index of the currently adjusted slider, m= 1,2, ...,

Mis the slider number, T60 and T60,init are the ﬁnal and initial RT

values, respectively, and ϵ= 0.6is a heuristically chosen scaling

factor.

Five typical reverberation presets were created: Small Room,

Medium Room,Large Room,Concert Hall, and Church. The ﬁrst

three presets are based on the measurement results presented in

[23], whereas RT values for the last two are taken from [24]. All

examples are available in a drop-down list in the top part of the

GUI. If one of the sliders is changed, “– no preset –” is displayed

in the drop-down list, as shown in Fig. 2a.

The Impulse button at the bottom right of the GUI empties

the delay lines and feeds a Dirac delta into the system so that the

impulse response of the reverberator is produced as an output.

3.2. Response Plotting

The window in the upper-half part of the GUI displays plots that

inform the user about the state of the plugin. As seen in Fig. 2a, the

GUI can display the RT curve (black) and the corresponding mag-

nitude response of the attenuation ﬁlter (red), which are plotted in

real time based on the values set by the sliders. This provides the

user with an insight into the actual decay characteristics of the syn-

thesized reverberation, which may differ from the user-deﬁned RT

values. This happens due to the limited ability of the attenuation

ﬁlter in following the target RT curve, especially when the differ-

ences between values set for the neighboring frequency bands are

big [18, 22]. Very extreme differences may lead to the ﬁlter’s mag-

nitude response reaching or exceeding 0 dB, which results in the

system’s instability. This state is signaled by the background color

of the window changing to light red. For the response, only one

delay line is used to retain real-time plotting. Due to the fact that

the attenuation ﬁlter adopts smaller values for shorter delay-line

lengths, the shortest delay line is chosen as it exhibits instability

sooner than the others.

The Show IR button located in the top right of the window al-

lows the user to toggle between the RT curve and ﬁlter’s response

plots and the reverberator’s IR plot, which is shown in Fig. 2b. As

opposed to the response, the longest delay line is used to calcu-

late the IR. Even though the effect of the scattering matrix, and

with that the effect of other delay lines, are not included, using the

longest delay line has been proven empirically to give a good in-

dication of the audible IR. The values displayed on the x-axis are

determined by the average slider value, i.e., a shorter reverb time,

results in a more detailed plot of the earlier seconds of the IR. Fur-

thermore, not every sample is drawn, but 1,000 data points spread

over the plot-range.

3.3. Choice of Delay Lengths and Distribution

Although FDN-based reverbs are nowadays among the most pop-

ular algorithmic reverbs, there is no clear rule on how to choose

the lengths of the delay lines [25, 26]. The common practice is

to choose the number of samples that are mutually prime and uni-

formly distributed between the maximum and minimum lengths to

avoid clustering of echoes [26].

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Figure 3: The advanced settings window.

Through the Advanced Settings window shown in Fig. 3, the

distribution of delay-line lengths can be chosen through a drop-

down menu from four options: Random,Gaussian,Primes, and

Uniform. Whenever an option is selected, the delay-line lengths

are randomly generated based on the distribution selected and rounded

to the nearest integers. The generation can be repeated by clicking

on the Randomize button. Furthermore, the minimum (500 sam-

ples) and maximum (10,000 samples) delay-line lengths can be

controlled; the minimum difference between the two has been set

to 100 samples. Moreover, there is an option to have the lengths

pre-deﬁned for each distribution so that the plugin will have the

same behavior every time it is used. The minimum and maximum

delay-line lengths have been empirically set to 1,500 and 4,500

samples, respectively (~30–100 ms at fs=44.1 kHz).

3.4. Choice of Feedback Matrix

The choice of the feedback matrix is crucial for the FDN algo-

rithm to work correctly. The popular matrix types used in FDN

implementations that fulﬁll the requirement of being unilossless

are Hadamard [27], Householder [27], random orthogonal, and

identity matrices [28]. Where the ﬁrst three are chosen to en-

hance speciﬁc properties of the algorithm, e.g., density of the im-

pulse response, the identity matrix, however, reduces the FDN to

a Schroeder reverberator, or a parallel set of comb ﬁlters [6, 28].

The plugin presented in this study allows the user to choose be-

tween these four matrices through a drop-down menu and to learn

about the differences in the sound obtained by changing this part

of the FDN reverberator. Additionally, the order of the FDN, and

thus the size of the feedback matrix, can be changed. The avail-

able options are 2, 4, 8, 16, 32, and 64, which can be chosen from

a menu.

In the case of the Householder matrix type, the implementa-

tion of matrices of different sizes vary. For all orders except for

16, the matrix is constructed using following the formula:

AN=IN−2

NuNuT

N,(6)

where uT

N= [1, . . . , 1], and INis the identity matrix [27]. The

matrix of order 16, on the other hand, following [29], is con-

structed using the recursive embedding of matrix of order 4:

A16 =1

2





A4−A4−A4−A4

−A4A4−A4−A4

−A4−A4A4−A4

−A4−A4−A4A4





.(7)

As a result, the matrix of order 16 consists of the same values,

differing only in their sign.

3.5. Code Structure

The plugin is divided into two main components that run on dif-

ferent threads at different rates. Firstly, the DSP component run-

ning at 44,100 Hz (audio rate), is structured in the same fashion

as shown in Fig. 1. An FDN class contains the scattering matrix

A, vectors band cthat scale the inputs and outputs of each delay

line (marked as biin Eq. (1b) and ciin Eq. (1a), respectively, and

in the current implementation all set to 1), and Ninstances of the

EQDelayLine class. This class, in turn, contains a delay line of

length Li(implemented as a circular buffer) and Minstances of

the Filter class. This class does all the low level computation and

contains the ﬁlter states and coefﬁcients bi,m and ai,m of the ith

delay line and the mth octave band.

Secondly, the GUI component running at 5 Hz is responsible

for the graphics and control of the FDN. Apart from the controls,

this component contains the Response class that is used to draw the

RT and gain curves and the IR shown in Figs. 2a and 2b. The ﬁl-

ter coefﬁcients necessary for drawing the curves are updated at the

aforementioned rate. This calculation also provides information

about the stability of the FDN and is used to trigger the light-red

background denoting instability. The Response class also contains

a single instance of the EQDelayLines class that is used to calcu-

late the IR.

Communication from the GUI to the DSP component happens

at a 5-Hz control rate, which has been found to be a great trade-off

between speed and quality of control. When changing any of the

non-RT controls, the GUI triggers ﬂags that are outside of the pro-

cess buffer (512 samples) to avoid the manipulation of parameters

when sample-by-sample calculations are being made.

3.6. Real-time Considerations

The components of the plugin requiring most computations are the

(re-)calculation of the ﬁlter coefﬁcients and the plotting of the re-

sponses. Even though the ﬁlter coefﬁcients only need to be recal-

culated when the sliders’ values are changed, it is good practice for

a plugin to have the same CPU usage when its values are changed

as when its values are static to prevent unexpected spikes in the

CPU usage. Instead, a Fix coeffs (coefﬁcients) button has been

implemented that, when clicked, will deactivate the preset’s drop-

down menu and the sliders (as shown in Fig. 2b). Furthermore,

the plugin will stop recalculating the plots and ﬁlter coefﬁcients,

greatly decreasing CPU usage (see Sec. 4.2). The CPU usages of

both threads are shown at the top of the plugin.

When any change is made to the FDN, be it the order, delay-

line distribution or length, the delay lines and ﬁlter states are set to

zero to prevent any unwanted artifacts. Only the RT control works

in real time without emptying the delay lines and ﬁlter states.

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0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

Time (s)

DL number

(a) Distribution of delay-line outputs for the option Primes.

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

Time (s)

DL number

(b) Distribution of delay-line outputs for the option Uniform.

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

Time (s)

DL number

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

DL number

0 0.2 0.4 0.6 0.8 1

Time (s)

DL number

(d) Distribution of delay-line outputs for the option Gaussian.

Figure 4: Distribution of the outputs of 32 delay lines (without scattering) for the options (a) primes, (b) random, (c) uniform, and (d)

Gaussian, and the length range (top) pre-deﬁned lengths (1,500–4,500 samples), (middle) lengths randomized over the entire range (500–

10,000 samples), and (bottom) lengths randomized over a narrow range (5,000–6,500 samples). Each dot marks the time when the given

delay line outputs a sample.

4. RESULTS AND DISCUSSION

This section presents results regarding the echo density produced

by and CPU usage of the plugin.

4.1. Echo Density

To achieve smooth reverberation, a sufﬁcient echo density, i.e., the

number of echoes per time unit produced by the algorithm and

their distribution [26], should be obtained. Echo density is affected

by a few factors, such as the lengths and the distribution of the

delay lines, the type of the feedback matrix [30] and its size, all of

which are discussed below.

4.1.1. Delay Lengths

The choice of delay-line length-distribution can help avoid more

than one sample appearing at the system’s output at the same time

and a clustering of the echoes, since both of these phenomena

lower the echo density. Additionally, the range over which the

delay-line lengths are chosen also affects the quality of the synthe-

sized sound. The distribution of delay-line outputs over time, with-

out a scattering matrix (i.e., an identity feedback matrix is used),

is shown in Fig. 4 for all the options available in the plugin. In the

case of the randomized selection of the delay-line lengths (middle

and bottom panes of Figs. 4a–4d), the results show one of the pos-

sible conﬁgurations. The delay-line lengths used in the examples

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were sorted in ascending order.

The top panes of Figures 4a–4d show the outputs of the pre-

deﬁned delay lines, which depict the typical behavior of the FDN

algorithm. The outputs become more diffused over time, mak-

ing the reverb smoother. It should be noticed, however, that when

using Uniform distribution, the chosen range is divided into por-

tions proportional to the FDN order, and the delay-line lengths are

chosen from such “bands”. This makes the consecutive delay lines

differ by a similar number of samples, and the possibility of output

samples overlapping or clustering is higher than with other distri-

butions. Choosing the Gaussian option, on the other hand, draws

the delay-line lengths from the normal distribution with the mean

being the midpoint between the range’s boundaries. This results in

choosing the lengths closer to the mean more often than those fur-

ther from it, as depicted in Fig. 4d, potentially causing clustering

of echoes and slowing down the increase of the echo density.

The distribution of outputs presented in the middle panes show

that when the delay-length range is very wide, the output is dif-

fused from the beginning. Since such decay is rarely met in reality,

it is useful when recreating only speciﬁc spaces [31]. Additionally,

very short delay lines create clusters of echoes and a huge portion

of the output samples overlap. They do not contribute to the in-

crease of echo density, but nevertheless add to the computation.

Such clusters are well visible in Figs. 4a and 4c. Moreover, the

attenuation applied to the short delay lines is usually small, and

therefore closer to 0 dB, which makes them more prone to causing

the system’s instability.

On the other hand, very long delay lines (10,000 samples trans-

lates to about 0.23 s for the 44.1-kHz sample rate) may not produce

a meaningful contribution to the synthesized reverb for low RT val-

ues. However, such long delay lines still add to the computation,

since the order of the FDN, and at the same time, the size of the

feedback matrix needs to be equal to the number of delay lines.

Using a very narrow range over which the delay-line lengths

are distributed results in clusters of samples arriving at the output

within a very short time, as seen in the bottom panes of Figs. 4a–

4d. Between the consecutive clusters, however, relatively long

silences occur. The synthesized reverberation tail diffuses very

slowly. Regardless of whether the delay-line lengths are chosen to

be prime, random, distributed normally or uniformly, choosing too

narrow a range results in low sound quality with clearly audible

segmentation and in the effect’s behavior resembling more that of

a single delay line than a reverb.

4.1.2. Feedback Matrix

The normalized echo densities for all types of matrices available

in the plugin were calculated, following the method presented in

[32, 33, 34], for orders 2–64 and the delay lines selected randomly

from the range between 1,500 and 4,500 samples (the same set of

delay lines was used for all calculations). To avoid bias caused by

the smearing of echoes due to the ﬁltering, the attenuation ﬁlters

were not used in the calculations. The results are presented in

Fig. 5 which generally show that the echo density increases faster

with a higher FDN order than with a lower one.

When matrices of size 2 and 4 are used, the number of echoes

in the output of the reverberator increases slowly and may never

reach saturation, i.e., the moment when there is an echo at ev-

ery successive time unit [26]. Therefore, these low orders do not

produce smooth sound. In the case of an FDN of order 8, the

echo density build-up is slow, which results in audible artifacts in

Table 1: CPU usage for all FDN orders in the cases of unﬁxed

(plotting IR and EQ) and ﬁxed coefﬁcients.

FDN CPU usage (%)

Order Unﬁxed (IR) Unﬁxed (EQ) Fixed

2 18.4 11.0 3.1

4 19.8 12.0 5.4

8 22.7 15.2 7.9

16 28.6 22.2 13.3

32 46.1 40.2 30.4

64 110.5 100.1 92.5

synthesized reverbs for as long as one second. Thus, a matrix of

size 16 is the smallest that increases the number of echoes quickly

enough so that the resulting sound is perceived as smooth for all

matrix types (except for the identity matrix). For the Hadamard

and random matrices, a further rise in the size accelerates the echo

density build-up, as evident in Fig. 5c and 5d.

Interestingly, the Householder matrix excels with the order of

16 using the recursive embedding of Eq. (7). This can be explained

by the fact that for all other orders, the implementation follows

Eq. (6), which produces matrices in which the difference between

the diagonal and the rest of the elements grows proportionally to

the order. Effectively, this makes the FDN approach a bank of

decoupled comb ﬁlters, which results in high variability of echo

density for orders 32 and 64, as seen in Fig. 5b, leading to audible

artifacts in the reverberation. For the matrix of order 16, however,

the echo density increases fast and remains high once saturation is

reached.

Because the identity matrices produce a very low echo den-

sity that does not increase with time, as seen is Fig. 5a, they are

not well ﬁtted for the FDN. Reverberation synthesized using such

matrices is always low-quality. Being also an identity matrix, the

Householder matrix of order 2 should be avoided as well.

4.2. CPU Usage

Table 1 shows the CPU usage for all implemented FDN orders for

three different plugin-states: unﬁxed coefﬁcients plotting the IR,

unﬁxed coefﬁcients plotting the EQ, and ﬁxed coefﬁcients (plot-

ting and recalculation of ﬁlter coefﬁcients disabled). The perfor-

mance has been measured on a MacBook Pro with a 2.2 GHz Intel

i7 processor using Xcode’s time proﬁler [35].

For all plugin states, the CPU usage increases exponentially

with the FDN order. Furthermore, ﬁxing the coefﬁcients, and thus

disabling the plotting and ﬁlter-coefﬁcient calculation, greatly de-

creases the plugin’s CPU usage. Comparing this to the unﬁxed EQ

case, an additional ~8.0% is added to the CPU usage, and when

plotting the IR versus the EQ, an additional ~7.5% is added to the

usage. This value, however, depends on the average reverb time

used. For testing, the Concert hall preset was used, which requires

calculating 2.5 s of sound for the IR plot. With a higher average

slider value, and thus a longer IR to be calculated and plotted, the

CPU usage also increases.

The smallest useful FDN order is 16, as stated in Sec. 4.1.2.

Table 1 shows that this order, or even 32, is unlikely to cause audi-

tory drop-outs, especially when the coefﬁcients are ﬁxed.

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0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

0.2

0.4

0.6

0.8

Echo density

Order 2

Order 4

Order 8

Order 16

Order 32

Order 64

(a) Identity matrix.

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

0.2

0.4

0.6

0.8

Echo density

(b) Householder matrix.

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

0.2

0.4

0.6

0.8

Echo density

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

0.2

0.4

0.6

0.8

Echo density

(d) Random orthogonal matrix.

Figure 5: Normalized echo densities for four types of feedback matrices and different FDN orders.

5. CONCLUSIONS

The present study introduces the FDN-based artiﬁcial reverber-

ation synthesis plugin. The implementation allows control over

the decay characteristics of the sound in ten octave bands in real

timeand plots the corresponding RT curve, the attenuation ﬁlter’s

magnitude response, and the IR. Additionally, users can explore

different setups of the FDN by changing the type and size of the

feedback matrix, and the lengths and distribution of the delay lines.

Experiments with the delay-line lengths and their distributions

suggest that these parameters should always be used in a balanced

manner, that suit the target reverberation. A wrong choice may

result in the creation of clusters of output samples and a low echo

density, which is undesirable in a reverberator. Choosing the lengths

over a narrow range results in low-quality, segmented sound, which

diffuses slowly. Picking the right distribution of delay-line lengths

is also important.

The ability to choose from among different FDN orders shows

that the lowest useful order for high-quality sound processing is

16, as it a sufﬁciently provides fast echo density build-up to ob-

tain smooth reverberation without audible artifacts. Shifting be-

tween feedback matrix types proves that the identity matrix, even

though it is lossless, should not be used in such applications, since

the produced sound is ﬂuttery. It also shows that, in the case of

the Householder matrix, implementation affects the reverberation.

Results show that using recursive embedding when constructing

the Householder matrix increases the echo density in the produced

reverberation.

6. ACKNOWLEDGMENTS

This work was initialized, when Karolina Prawda made a Short-

Term Scientiﬁc Mission to the Aalborg University Copenhagen

from October 28 to November 15, 2019.

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DAFx.8

Publication VII

Karolina Prawda, Vesa Välimäki, and Stefania Seraﬁn. Evaluation of Accu-

rate Artiﬁcial Reverberation Algorithm. In Proceedings of the 17th Sound

and Music Computing Conference (SMC 2020), Turin, Italy, June 2020.

Reprinted with permission.

143

Evaluation of Accurate Artiﬁcial Reverberation Algorithm

Karolina Prawda, Vesa V¨

alim¨

aki

Acoustics Lab,

Dept. Signal Processing and Acoustics

Aalto University, Espoo, Finland

karolina.prawda@aalto.fi

Stefania Seraﬁn

Multisensory Experience Lab,

Dept. Architecture, Design and Media Technology

Aalborg University, Copenhagen, Denmark

sts@create.aau.dk

ABSTRACT

Artiﬁcial reverberation algorithms aim at reproducing the

frequency-dependent decay of sound in a room that is per-

ceived as plausible for a particular space. In this study,

we evaluate a feedback delay network reverberator with a

modiﬁed cascaded graphic equalizer as an attenuation ﬁlter

in terms of accurate reproduction of measured impulse re-

sponses of three rooms with different decay characteristics.

First, the late reverb is synthesized by the proposed method

and mixed with the early reﬂections separated from the

original signal. The synthesized and measured signals are

compared in terms of their decay characteristics and re-

verberation time values. The experiment shows that the

proposed reverberator design reproduces real impulse re-

sponses well, although the decay-rate error exceeds the

just noticeable difference of 5% in many cases. Addition-

ally, perceptual qualities of the synthesized sounds were

assessed through a listening test. Four qualities were tested

for three room impulse responses and three kinds of stim-

uli. The results show that for the qualities reverberance,

clarity, and distance, on average 75–79% of participants

noticed only a slight or no difference between the mea-

sured and synthetic reverbs. Similar results were obtained

for the speech and signing voice stimuli and the reverbera-

tion of lecture room and concert hall.

1. INTRODUCTION

Reverberation is considered to be one of the most impor-

tant sound qualities of physical spaces. It is also used in

virtual environments to make them sound and feel more

real. Therefore, many algorithms that aim at synthesiz-

ing reverberation are used nowadays, with Feedback De-

lay Networks (FDNs) being among the most popular due

to ﬂexibility in design and computational efﬁciency [1,2].

To make the artiﬁcial reverberation sound perceptually

plausible, i.e., logical and probable for the particular space,

the energy decay must be frequency-dependent, which can

be achieved by inserting attenuation ﬁlters into the algo-

rithm. Over time, various types of such ﬁlters have been

2020 Karolina Prawda, Vesa V¨

alim¨

aki et al.

This is an open-access article distributed under the terms of the

Creative Commons Attribution 3.0 Unported License, which permits unre-

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proposed, starting from a ﬁrst-order low-pass inﬁnite im-

pulse response (IIR) ﬁlter [3], to biquadratic ﬁlters, which

allowed to control the decay in few frequency bands [4], to

high-order ﬁlter designs [5].

Advanced control over frequency-dependent reverbera-

tion is achieved by using a proportional graphic equalizer

(GEQ). This idea was ﬁrst proposed by Jot [6] and later

improved by Schlecht and Habets [7] to enhance the sys-

tem’s accuracy while ensuring its stability. Recent work

by Prawda et al. [8] suggests to use a modiﬁed cascaded

GEQ with shifted and scaled frequency response and a

ﬁrst-order high-shelf ﬁlter inserted at high frequencies to

further increase the accuracy of the reverberation approxi-

mation.

The artiﬁcial reverberation should primarily be perceptu-

ally plausible. Therefore, various types of perceptual eval-

uation techniques were proposed to assess different quali-

ties of synthesized reverberation. Czy ˙

zewski [9] proposed

several criteria for assessing concert hall reverberation. The

listening tests together with the objective evaluation com-

paring synthetic late reverberation and a measured impulse

response were described in [5, 10–12]. The two studies,

[11, 12], are of special interest to the present work since

they focus on evaluating FDN reverberators.

This paper presents an objective as well as a perceptual

evaluation of the accurate reverberation design proposed

in [8]. The study compares impulse responses measured

in rooms with various reverberation characteristics with

synthesized versions of the same signals. The evaluation

includes objective measures, such as frequency-dependent

energy decay, and listening tests examining the perceptual

qualities of signals for different types of sound stimuli.

The paper is organized as follows. Section 2 describes

the algorithm used to synthesize room impulse responses.

Section 3 presents the target signals and shows the results

of an objective evaluation of the artiﬁcial reverberation al-

gorithm. Section 4 describes the listening test and reports

on its results and their statistical analysis. Section 5 dis-

cusses the results. Finally, Section 6 summarizes the work

presented in the paper, concludes on the ﬁndings, and pro-

poses ideas for future research.

2. ARTIFICIAL REVERBERATION ALGORITHM

The FDN algorithm with the modiﬁed GEQ that was pre-

sented in [8] proved to work well for a simpliﬁed case in

which one delay line (single-delay-line absorptive feed-

back comb ﬁlter) was analyzed. However, that type of

FDN is unusual in practice, as it does not provide enough

echo and modal density. Usually, a high-order system is

required to obtain smooth reverberation without audible ar-

tifacts [13]. Therefore, a commonly used FDN comprising

of 16 delay lines was adopted in this work.

Another decision concerning the design of the algorithm

was the choice of the feedback matrix. In principle, the

stability of an FDN is achieved, when the matrix is uniloss-

less, i.e., it does not cause any loss of energy for any type

of delay when no attenuation is introduced in the system

[14]. To fulﬁll the above-mentioned requirement, in the

present work a 16th-order Householder matrix is used. It is

created by recursively embedding the fourth-order House-

holder matrix

A4=1

2





1−1−1−1

−1 1 −1−1

−1−1 1 −1

−1−1−1 1





(1)

for each entry in a matrix of identical structure [4, 12]:

A16 =1

2





A4−A4−A4−A4

−A4A4−A4−A4

−A4−A4A4−A4

−A4−A4−A4A4





.(2)

2.1 Attenuation ﬁlters

To obtain a frequency-dependent reverberation, attenua-

tion ﬁlters must be inserted either at the beginning or at

the end of every delay line. They should be designed to

approximate the same target gain-per-sample in dB, which

is given by

γdB(ω) = −60

fsT60(ω),(3)

where T60(ω)is the reverberation time in seconds, ω=

2πf /fsis the normalized frequency, fis the frequency in

Hz, and fsis the sampling rate in Hz. For all the delay lines

to approximate the same reverberation time, the attenua-

tion should be proportional to the number of unit delays

in samples L, such that the attenuation ﬁlter’s magnitude

response in dB is expressed as

AdB(ω) = LγdB (ω).(4)

The attenuation ﬁlter controls the decay rate of the syn-

thetic response in a broad enough frequency range to make

it perceptually the same as the measured room impulse re-

sponse (RIR). To obtain such similarity, this study used

a cascaded GEQ that can regulate the reverberation time

in ten octave bands, having their center frequencies from

31.5 Hz to 16kHz [15].

To smooth the equalizer’s magnitude response and to avoid

any unwanted increase in T60(ω)below and above the fre-

quency range of interest, the gains for all the frequency

bands were ﬁrst shifted up (boosted) by their median value

and then scaled down (attenuated) by the same number.

The more detailed explanation of those operations is pre-

sented by Prawda et al. [8]. The equalizer’s ﬁnal response

in dB is given by:

HdB(ej ω) = g0+

m=1

(HdB,m(ej ω)−g0

M),(5)

where m= 1,2, ..., M is the number of controlled fre-

quency bands, g0is the broadband gain factor (set to the

median of all gains), and HdB,m are the frequency responses

of equalizing ﬁlters.

Additionally, the ﬁrst-order high-shelf ﬁlter was inserted

in the GEQ above 16 kHz to ensure a correct energy decay

for the high frequencies. The gain of the ﬁlter was set to

the gain of the highest peak-notch ﬁlter and the crossover

frequency, i.e. the frequency at which the gain is the arith-

metic mean of the extreme gains in dB [16], ﬁxed at

20.2 kHz, as proposed in [8].

2.2 Early reﬂections

In this study, the FDN was intended to synthesize the late

part of the impulse response and therefore the decision was

made to obtain the early reﬂections from a suitable portion

of the measured RIR and mix it with the FDN’s output. In

this approach, the challenge is to ﬁnd the correct truncation

point to capture the right amount of early reﬂections.

There exist a few approaches discussing the issue of RIR

truncation that suggest studying the skewness and kurto-

sis of the windowed signal [17], the changes in the RIR’s

phase over time [18], or the echo density proﬁle [19–21].

In this study the method suggested by Stewart and San-

dler [22] was adopted. It assumes that the RIR has an

approximately Gaussian distribution in the time domain,

with a standard deviation of the group of samples being

the measure of the spread of samples deﬁned as

σ=pE(x2)−E(x)2,(6)

where E(x)is the expected value of x. In a normal distri-

bution one-third of the samples lies outside and two-thirds

inside of one standard deviation from the mean, but in case

of early reﬂections, more samples lie within one standard

deviation. Therefore, the truncation point is found by ana-

lyzing the ratio of samples outside and inside the standard

deviation.

To observe the ratio of samples, the RIRs of interest were

windowed with a 20-ms rectangular window with no over-

lap. The length of the window was described in [12, 23]

as favorable, since a shorter one would not provide enough

samples to perform reliable calculations, whilst a longer

window could lead to choosing the truncation point too

early or too late. The ratio’s threshold was set to 30%,

meaning that when at least 30% of samples lie outside one

standard deviation, the point after the window is the correct

truncation point. After truncating, the right half of a 32-

sample long Hanning window was applied to the early re-

ﬂections part of the RIR to fade the signal energy smoothly

to zero, as suggested in [12].

3. OBJECTIVE EVALUATION

In order to test the proposed method of producing reverber-

ation, an evaluation was performed. In the objective part,

Center frequency 31.5 Hz 63 Hz 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz 16 kHz

Meas. RIR RT (s) 1.62 0.81 0.76 0.73 0.63 0.48 0.41 0.36 0.27 0.19

Synth. RIR RT (s) 1.13 1.00 0.79 0.74 0.67 0.56 0.43 0.42 0.33 0.30

Error (%) 30.25 23.46 3.95 1.37 6.35 16.67 4.87 16.67 22.22 57.89

(a) Ofﬁce room, short reverberation, cf. Fig. 1a.

Center frequency 31.5 Hz 63 Hz 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz 16 kHz

Meas. RIR RT (s) 1.20 0.95 0.71 0.78 0.85 0.88 0.87 0.87 0.62 0.39

Synth. RIR RT (s) 1.10 0.89 0.75 0.78 0.86 0.93 0.87 0.88 0.67 0.53

Error (%) 8.33 6.32 5.63 0.00 1.18 5.68 0.00 1.15 8.06 35.90

(b) Lecture room, medium reverberation, cf. Fig. 1b.

Center frequency 31.5 Hz 63 Hz 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz 8 kHz 16 kHz

Meas. RIR RT (s) 2.09 2.08 2.03 2.06 2.03 2.10 1.98 1.6 0.68 0.18

Synth. RIR RT (s) 2.33 2.10 2.11 2.06 2.12 2.21 1.99 1.77 0.83 0.27

Error (%) 11.48 0.96 3.94 0.00 4.43 5.23 0.51 10.63 20.06 50.00

Table 1: Reverberation time for measured and synthetic RIRs and the modeling error in octave frequencies for the three

tested reverbs. Errors exceeding JND of 5.0% are highlighted. Cf. Fig. 1.

three RIRs measured in different venues were chosen to be

reproduced with the proposed algorithm:

1. Short reverberation – RIR of an ofﬁce room [24];

2. Medium reverberation – RIR of a lecture room [24];

3. Long reverberation – RIR of a concert hall in Pori,

Finland.

The reverberation time (RT) values of the chosen RIRs

were calculated in ten octave bands with center frequencies

from 31.5 Hz to 16 kHz with the energy decay curve eval-

uation method [25, 26]. To ensure that all the delay lines

create a meaningful contribution to the synthesized rever-

beration, the delay-line lengths were randomized over the

range between 10 ms and 100ms.

The RT values of RIRs produced with the FDN were

compared to the target values to check whether the differ-

ences between them were exceeding 5%, which is the just

noticeable difference (JND) for the RT of an acoustic im-

pulse response [25, 27]. The spectrograms of both sets of

impulse responses were also analyzed.

3.1 Results

Figure 1 presents the spectrograms of the measured (top)

and synthetic (bottom) RIRs, respectively, for all the test

cases, and the Table 1 shows the RT values for center fre-

quencies of the octave bands and the error between the tar-

get and obtained RT.

For all three RIRs, the errors are biggest for the high-

est frequencies, 8-16 kHz, where the reverberation synthe-

sized with the proposed algorithm is longer than the mea-

sured values. The differences are also considerable in the

31.5-Hz band. For the short and medium reverbs the ob-

tained values are lower than the target ones, whilst for the

concert hall the RT is longer.

The short reverberation of the ofﬁce proved to be the most

problematic case in the objective evaluation. The JND of

5% is not exceeded in only a few bands, as shown in the

Table 1a. The error for 500 Hz, however, is higher than the

threshold by less than two percentage points.

The best results were obtained for the medium reverbera-

tion of the lecture room. Although the differences between

the target and obtained values are lower than the JND for

only a few bands, as presented in the Table 1b, the er-

ror exceeds 10% only at 16 kHz. In two bands, 125 Hz

and 1 kHz, the JND is passed by less than one percentage

point, whilst in 63 Hz the difference is only 1.32 percent-

age points larger than the threshold of noticeability. The

spectrograms of the measured and synthesized RIRs shown

in Fig. 1b. The target decay is modeled accurately for low

and mid frequencies from around 100 Hz up to 6kHz. The

only exception is the slight lack of energy at about 300 Hz,

from 0.55 s on.

In the case of the concert hall, that the error between

target and obtained values is smaller than 5% in ﬁve fre-

quency bands, 63-500 Hz and 2 kHz whilst for 1 kHz the

JND is exceeded only by 0.23%, as shown in the Table 1c.

As depicted in the Fig. 1c, the decay of the measured signal

is well reproduced by the synthetic one in mid frequencies,

however, there is a visible overshoot in energy over 5 kHz.

The modeled decay is noticeably shorter than the original

one at about 300 Hz.

4. SUBJECTIVE EVALUATION

In addition to the objective evaluation of the algorithm per-

formance, a subjective evaluation in a form of a listening

test was conducted in order to assess the perceptual quali-

ties of the synthetic reverberation.

(a) Ofﬁce

(b) Lecture room

Figure 1: Spectrograms of the (top panes) measured and

(bottom panes) synthetic RIRs of the three test cases.

4.1 Stimuli and test setup

Prior to the listening test, the assumption was made that as-

sessing the perceptual qualities of raw impulse responses

would prove difﬁcult for participants. Therefore, the test

sounds were created by adding reverberation to the follow-

ing anechoic recordings:

1. Speech – A 3-s sample of a female saying “The juice

of lemons makes ﬁne punch” [28];

2. Singing voice – A 49-s sample of a male singing

[29], which was truncated to 10 s;

3. Guitar – A 49-s sample of guitar music [29], which

was truncated to 4 s.

The truncation was performed in order to avoid tiring par-

ticipants with overly long stimuli.

Every sample was convolved with each of the measured

RIRs and also reverberated using the approach described

in previous sections of this paper. The samples convolved

with the measured RIRs were used as references, whilst the

remaining sounds were the test items. Each question of the

test comprised of one reference sound and a respective test

sample. The task was to determine how big the audible dif-

ferences in the test sounds in comparison to the reference

sound were within the scope of four qualities: reverber-

ance, clarity, distance, and coloration. The answers were

presented in the form of a Likert scale, with “No differ-

ence”, “Slight difference”, “Clear difference”, and “Strong

difference” as possible responses.

The listening test was conducted in the anechoic chamber

of the Aalborg University Multisensory Experience Lab,

using Sennheiser HD-600 headphones. The test was car-

ried out by using the web-based experiment software web-

MUSHRA developed by International Audio Laboratories

Erlangen [30]. Before the test, the subjects were allowed to

adjust the volume of the sound, which remained the same

during the experiment. Since it is known that the loudness

affects the perception of reverberance [31], the subjects

were advised to keep the volume at a high level, and not

reduce it below the default setting. It was also conﬁrmed

that all participants knew and understood the terminology

used during the evaluation. They were familiarized with

the task in a short training session, which was not included

in the results.

Twelve people participated in the test. The answers of

two of them, however, were dismissed due to hearing im-

pairment reported in the post-test questionnaire. The av-

erage age of the participants whose results were analyzed

was 29.8 years (SD = 5.6). All the participants were either

students or employees of Aalborg University in Copen-

hagen. Many of them had previously participated in similar

listening tests.

4.2 Listening test results

The results of the listening test are presented separately for

the three reverbs (short, medium, and long) with the fur-

ther division based on the type of stimulus (speech, singing

Speech Singing Guitar

Type of stimulus

100

Answers granted (%)

Reverberance

Clarity

Distance

Coloration

No difference

Slight difference Clear difference

Strong difference

Figure 2: Listening test results for the short reverberation.

Speech Singing Guitar

Type of stimulus

100

Answers granted (%)

Reverberance

Clarity

Distance

Coloration

No difference

Slight difference Clear difference

Strong difference

Figure 3: Listening test results for the medium reverbera-

tion.

voice, and guitar music) and assessed quality (reverber-

ance, clarity, distance, and coloration). To ensure the ease

of interpretation of the results the number of responses

given to each question was converted to a percentage.

Figure 2 shows the distribution of answers for the short

reverberation. Most subjects perceived the differences be-

tween the reverberance of the test and reference sounds as

slight or did not notice any difference at all, regardless of

the type of stimulus. In the guitar music sample, 70% of

the participants chose the “No difference” answer and only

a small percent of “Clear difference” responses. Clarity

was evaluated similarly, however, it received fewer “No

difference” and more “Clear difference” answers.

For the speech stimulus, over 60% of the subjects chose

the “No difference” answer. In cases of the singing voice

and guitar music more “Clear difference”, and “Strong dif-

Speech Singing Guitar

Type of stimulus

100

Answers granted (%)

Reverberance

Clarity

Distance

Coloration

No difference

Slight difference Clear difference

Strong difference

Figure 4: Listening test results for the long reverberation.

ference” answers were given. For speech stimulus, dis-

tance received over 60% of “No difference” responses and

no “Strong difference” ones. Coloration received the largest

number of “Clear difference” responses for all stimuli. When

accessing the reverberated singing voice, however, over

50% of participants picked the “No difference” answer.

The listening test results for the medium reverberation are

presented in Fig. 3. The distribution of answers is similar

for each stimulus, with the answer “Slight difference” be-

ing chosen most of the time – between 30% and 60% for

each quality. The “No difference” response was granted

almost as frequently, from 20% to 40% times in each ques-

tion, except for reverberance in case of speech stimuli.

For this reverberation, the “Clear difference” grades were

granted no more than 30% of the time. Strong differences

between the test and reference sounds were noticed mostly

for guitar music but by no more than 20% of the partici-

pants. Three qualities for the speech stimulus – distance,

clarity, and coloration - two for singing voice – distance

and coloration – and reverberance for the guitar music did

not receive any “Strong difference” answers.

The results for the long reverberation of the concert hall

are presented in Fig. 4. For the singing voice, the “No dif-

ference” answer is the most prominent, being chosen be-

tween 30% and 60% of the time, depending on the assessed

quality. For speech the percentage of the “No difference”

and the “Slight difference” grades was similar, between

20% and 40%. Two out of the four qualities, reverber-

ance and coloration for speech, and clarity and coloration

for singing voice, did not receive any “Strong difference”

responses.

In the case of guitar music, the distribution of the “No

difference” and the “Slight difference” responses was the

most uneven. Reverberance and distance were many times

granted the “No difference” response, but both of them also

received “Strong difference” answers. 50% of participants

reported slight differences between the clarity of the as-

sessed samples. The most “Clear difference” and “Strong

difference

Slight

difference

Clear

difference

Strong

difference

Type of answer

Answers granted (%)

Short reverb

Medium reverb

Long reverb

Figure 5: Listening test results analyzed for the RT values.

difference” answers were given to the coloration.

4.3 Statistical analysis

The results of the listening test were further analyzed ac-

cording to the RT, quality, and stimulus. Figs. 5, 6, and

7 present the average percentage of each type of answer

granted to the sounds depending on the factors mentioned

above. The mean grade is marked with a dot of the respec-

tive color with bars showing the 95% conﬁdence intervals.

The smaller dots with less opacity present the percentage

of each type of response granted in each question of the

listening test.

4.3.1 Reverberation time

Figure 5 presents the average percentage of each type of

answer granted to the sounds depending on their RT val-

ues. It shows that the participants chose the “Slight differ-

ence” and “No difference” answers most frequently. The

combined average percentages for those two answers were

73%, 74%, and 72% for short, medium, and long reverber-

ation, respectively.

The analysis reﬂects the tendencies observed in the ob-

jective evaluation, where the algorithm performed better

for long and medium reverberation than for the short one.

It shows that the more accurately modeled RIRs are per-

ceived as more similar to the measured ones even when the

JND between target and modeled RT values is exceeded in

some frequency bands.

4.3.2 Quality type

The analysis based on the quality type is given in Fig. 6. In

the case of reverberance, clarity, and distance there are sig-

niﬁcant differences between the mean percentage of “Slight

difference” and “Clear difference” answers, and between

the “Clear difference” and the “Strong difference”. For

coloration, only the mean percentage of the “Strong differ-

ence” answers was signiﬁcantly different from the others.

The analysis shows that in most cases, the “Slight Dif-

ference” or the “No difference” responses were chosen.

difference

Slight

difference

Clear

difference

Strong

difference

Type of answer

Answers granted (%)

Reverberance

Clarity

Distance

Coloration

Figure 6: Listening test results analyzed for the type of

perceptual sound quality.

difference

Slight

difference

Clear

difference

Strong

difference

Type of answer

Answers granted (%)

Speech

Singing voice

Guitar

Figure 7: Listening test results analyzed for the type of the

stimulus.

The combined average percentage of those two types of re-

sponses was 79%, 75%, and 78% for reverberance, clarity

and distance, respectively. Distance was perceived as the

quality that was the most similar in the test and reference

sounds, with the “No difference” response granted in 42%

questions on average. Fig. 6 shows that the results con-

cerning coloration are the most equivocal, indicating that

the algorithm still needs improvement to accurately repro-

duce that quality of sound.

4.3.3 Stimulus type

The results analyzed according to the type of stimulus are

presented in Fig. 7. The subjects gave the most consistent

grades for questions concerning speech, which resulted in

all adjacent means for all types of responses for that stim-

uli to be signiﬁcantly different. In the case of the singing

voice, the mean percentage of answers “No difference”

and “Slight difference” are similar, whilst for the guitar,

only the mean percentage of “Strong difference” answers

are signiﬁcantly different from the other three. For speech

the “No Difference” answer was picked 30% of the time,

whilst slight dissimilarities between the references and test

sounds were noticed in 48% of questions. The respective

values for singing voice are 35% and 44%. For guitar mu-

sic the “No difference”, “Slight difference”, and “Clear dif-

ference” were chosen around 30% of the time.

The analysis proves that the algorithm works well when

reproducing the frequency range of the human voice, how-

ever, in the case of a guitar, with a lower frequency range,

the improvement is needed to obtain better accuracy.

5. DISCUSSION

There are a few explanations as to why the proposed ar-

tiﬁcial reverberation algorithm reproduces the target RT

most inaccurately in the high frequencies. One of them

may be that the GEQ used as the attenuation ﬁlter works

best when the gains are within the range of ±12 dB. Many

times the attenuation required to obtain target RT goes be-

yond the lower bound of that range, especially when long

delay lines are used to approximate short reverbs, as pre-

sented in Fig. 8.

Another reason is that the high-shelf ﬁlter, which reduces

overshoot in the decay for frequencies above 16kHz, also

introduces ripple in the ﬁlter’s magnitude response, as was

reported in [8]. This phenomenon is shown in Fig. 8, where

the drop and rise in the magnitude are present at high fre-

quencies in all three attenuation ﬁlters.

Similarly, the undershoot in the RT values observed in the

31.5 Hz band may be the consequence of shifting and scal-

ing of the attenuation ﬁlter’s magnitude response. Those

operations create a decrease in the magnitude response for

very low frequencies, as shown in Fig. 8, which may lower

the algorithm’s accuracy.

6. CONCLUSIONS

The present work studied the ability of the FDN with the

modiﬁed cascaded GEQ as the attenuation ﬁlter to repro-

duce the measured RIRs accurately. The evaluation was

conducted both by comparing the RT values and decay

characteristics of the original and synthetic RIRs and by

the means of the listening test.

In the objective assessment the RIRs produced with the

proposed method replicated the target decay best in the

mid-frequency range between 125 Hz and 2 kHz. For each

RIR type, the biggest dissimilarities between the target and

obtained RT values occurred in the 16kHz band. The re-

sults show that the proposed design performs best when

long reverberation times are modeled, and that small differ-

ences between the RT values in the neighboring frequency

bands produce more accurate approximation.

The listening test showed that for the three types of re-

verberation and stimuli, when the four qualities of sound

were assessed, the subjects mostly perceived only slight

differences between the sounds convolved with measured

and synthetic RIRs. Many times the “No difference” an-

swer was chosen as well. The differences were easiest to

notice for the coloration in case of the sound quality and

30 100 300 1000 3000 10000

Frequency (Hz)

-40

-30

-20

-10

Magnitude (dB)

Short reverb

Medium reverb

Long reverb

Figure 8: The attenuation ﬁlter’s magnitude response for

the three test cases of short, medium, and long reverbera-

tion and a delay-line length of 90 ms.

the guitar music in terms of the stimulus. However, further

testing is needed to establish whether the test sounds were

truly indistinguishable from the references.

Accurately reproducing impulse responses of real spaces

with parametric artiﬁcial reverberation is still a difﬁcult

task. The present study shows that with the proposed FDN

design in many cases it is possible to trick the human per-

ception into not noticing dissimilarities between the orig-

inal and artiﬁcially produced signals. However, this study

suggests that the FDN reverberator can still be improved

by further developing the accuracy of the attenuation ﬁlter.

Additionally, future work may look into the choice of the

feedback matrix and delay lengths, which were not consid-

ered in this work.

Acknowledgments

This work was supported by the “Nordic Sound and Music

Computing Network—NordicSMC”, NordForsk project

number 86892. This work was initialized when Karolina

Prawda made a Short-Term Scientiﬁc Mission to the Aal-

borg University Copenhagen on October 28 – November

15, 2019.

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