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Abstract
The reproduction of acoustics is one of the key challenges in spatial sound reproduction. In order
to reach high levels of realism and immersion, reproduced signals must be processed through a set
of filters accounting for real-life phenomena. From the scattering and absorption of sound on walls
to its diffraction around objects and our head, the propagation of sound waves in a room creates
a complex sound field around a listener. While recreating all of the underlying parts of sound
propagation is not yet within our reach for real-time applications, spatial sound techniques aim to
minimize the computational cost by focusing the efforts on the most perceptually critical
components.
In this dissertation, the reproduction of reverberant sound fields is investigated through the
development of analysis methods, reverberation algorithms, and decorrelation techniques. A
particular emphasis is given to the perception and reproduction of directional characteristics
present in late reverberation.
Most reverberation algorithms consider that the sound energy is evenly distributed across space
in all directions, after an initial period of time, since the diffusion of energy leads to more
homogeneous and isotropic sound fields. However, previous work has demonstrated that
insufficient diffusion in a room leads to anisotropic, and directional, late reverberation.
In this dissertation, a complete framework is proposed for the objective and subjective analysis
of directional characteristics as well as a novel delay-network reverberation method capable of
producing direction-dependent decay properties. The reverberator is also expanded to offer
efficient frequency- and direction-dependent processing. The proposed algorithm contains all the
required elements for the auralization of reverberant sound fields, which may be modulated in
real-time to support six-degrees-of-freedom sound reproduction.
The decorrelation of audio signals, which occurs naturally during the diffusion of energy in a
sound field, is another important aspect of sound reproduction. This dissertation considers the
use of velvet-noise sequences, a special type of sparse noise signals, as decorrelation filters and
offers a method to optimize their characteristics. Velvet noise is also proposed for the
reproduction of an existing impulse response using a small set of time- and frequency-dependent
information, along with a reverberator using velvet noise to improve the echo density of a delay
network reverberator.
Overall, the results contained in this dissertation offer new insights into the perceptual
ramifications of reverberant sound fields containing directional characteristics and their
reproduction. The methods presented bring applications in the context of immersive sound
reproduction, such as in virtual and augmented reality.
Keywords Acoustics, reverberation, decorrelation, digital signal processing, signal analysis
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Preface
The research work detailed in this dissertation has been carried out in the
Acoustics Lab, part of the Department of Signal Processing and Acoustics
at Aalto University. This research was part of the ICHO project (Immersive
Concert at Home, September 2016-2020) funded by the Academy of Finland
(Aalto University project no.13296390) and the work was part of the activi-
ties of the “Nordic Sound and Music Computing Network—NordicSMC”,
NordForsk project number 86892. Part of the work of this dissertation was
conducted at the Institut de Recherche et Coordination Acoustique/Mu-
sique (IRCAM) (UMR STMS IRCAM-CNRS-Sorbonne Université) in Paris
during a research visit in the Autumn of 2018. The research visit was
funded by the Foundation for Aalto University Science and Technology.
The main research question that stemmed from the funding project was
how to best reproduce the acoustics of a concert hall on immersive media
platforms such as virtual reality. One of the main research objective was
to develop new efficient reproduction methods using velvet noise. Towards
these goals, the perceptual characteristics of reverberant sound fields were
studied along with decorrelation techniques and artificial reverberation
methods.
I would like to thank my supervisor, Prof. Vesa Välimäki, for sharing his
vast expertise in the field and for all of his guidance throughout this Ph.D.,
as well as his unwavering trust in allowing me to pursue my research
objectives. I would also like to thank my thesis advisors, Dr. Archontis
Politis and Prof. Sebastian J. Schlecht, for their invaluable contributions
to this work as well as for their support and guidance throughout my
research. I also thank the pre-examiners, Prof. Huseyin Hacıhabibo˘
glu
and Prof. Sylvain Marchand, for their comments that helped improve this
dissertation, and Dr. Jean-Marc Jot for agreeing to be my opponent.
For hosting me at their facilities at IRCAM in Paris and for a fruitful col-
laboration, I thank Markus Noisternig, Pierre Massé, Thibault Carpentier,
and Olivier Warusfel. I also thank Luis R. J. Costa, whose mastery of the
English language was invaluable when publishing the work contained in
this dissertation.
1
Preface
I thank my many colleagues at the Acoustics Lab for all the profound
conversations and their general awesomeness throughout my time at
Aalto. Doing a Ph.D. is a complex journey, let alone doing it in a land
that intermittently dips in complete darkness. Having proximity to such
a stellar group of colleagues who can encourage, challenge, inspire, and
support each other is immensely valuable. For these reasons, I thank
Prof. Tapio Lokki, Prof. Ville Pulkki, Prof. Lauri Savioja, Abraham, Aki,
Aksu, Alec, Aleksi, Alessandro, Andrea, Antti, Catarina, Christoph, Craig,
Dimitri, Eloi, Étienne, Eero-Pekka, Fabiàn, Georg, Henna, Henri, Ilkka,
Jan, Janani, Janne, Javier, Jon, Jose, Juhani, Juho, Julie, Jukka, Jussi,
Karolina, Lauri, Lauros, Leo, Leonardo, Michael, Nils, Otto, Pablo, Pedro,
Petteri, Raimondo, Rapolas, Ricardo, Sebastian, Sakari, Stefan, Symeon,
Taeho, Thomas, Vasileios, and Winfried...
I also thank my close friends in the research community who’s work has
been inspiring, Brian, Julian, Michele, Stefan, and Thanasis. Finally, I
would like to thank my family who has always been so supportive of my
endeavours and especially my parents, Sylvie and Pierre, for being such
immovable pillars of support.
Helsinki, August 18, 2021,
Benoit Alary
2
Contents
Preface 1
Contents 3
List of Publications 5
Author’s Contribution 7
List of Abbreviations 9
List of Symbols 11
1. Introduction 13
2. Reverberant Sound Field 17
2.1 Sound Propagation ....................... 17
2.2 Statistical Properties ..................... 21
2.3 Diffuse Field .......................... 27
2.4 Properties of Reverberant Sound Fields ........... 28
2.4.1 Objective Measures . ................ 28
2.4.2 Perception ...................... 29
2.5 Reproduction of Sound Fields ................. 30
2.5.1 Ambisonics ...................... 31
2.5.2 Binaural Reproduction ............... 33
3. Decorrelation 35
3.1 Overview ............................ 35
3.2 Objective Evaluation ...................... 36
3.2.1 Cross-Correlation . . ................ 36
3.2.2 Coherence ...................... 37
3.2.3 Covariance Matrix . . ................ 37
3.2.4 Incoherence Profile . ................ 38
3.3 Decorrelation Methods ..................... 39
3
Contents
3.4 Velvet-Noise Decorrelation .................. 40
4. Artificial Reverberation 43
4.1 Historical Overview ...................... 43
4.2 Captured Impulse Response . . . .............. 44
4.3 Virtual Acoustics ........................ 46
4.3.1 Geometric Methods ................. 46
4.3.2 Wave-based Methods ................ 47
4.4 Artificial Noise ......................... 48
4.5 Delay Networks ......................... 50
4.5.1 Velvet-Noise Feedback-Delay Network ...... 57
5. Hybrid Reverberation Methods 59
5.1 Geometrically-Informed Reverberators ........... 59
5.2 Binaural Reverberation .................... 60
5.3 Digital Waveguide ....................... 63
5.4 Ambisonics Delay Networks . . . .............. 64
5.5 Directional Reverberation ................... 65
5.5.1 Analysis ........................ 66
5.5.2 Reproduction . . . .................. 68
5.5.3 Directional Feedback Delay Network ....... 69
5.5.4
Frequency-Dependent Directional Feedback De-
lay Network ..................... 70
6. Summary of Main Results 75
7. Conclusion 79
References 81
Errata 97
Publications 99
4
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.
I
Benoit Alary, Archontis Politis, and Vesa Välimäki. Velvet-noise decor-
relator. In Proceedings of the 20th International Conference on Digital
Audio Effects, Edinburgh, UK, pp. 405–411, September 2017.
II
Sebastian J. Schlecht, Benoit Alary, Vesa Välimäki, and Emanüel A.
P. Habets. Optimized velvet-noise decorrelator. In Proceedings of the
21st International Conference on Digital Audio Effects, Aveiro, Portugal,
pp. 87–94, September 2018.
III
Vesa Välimäki, Bo Holm-Rasmussen, Benoit Alary, and Heidi Maria
Lehtonen. Late reverberation synthesis using filtered velvet noise. Ap-
plied Sciences, vol. 7, no. 5, May 2017.
IV
Jon Fagerström, Benoit Alary, Sebastian J. Schlecht, and Vesa Välimäki.
Velvet-noise feedback delay network. In Proceedings of the 23rd Interna-
tional Conference on Digital Audio Effects, Vienna, Austria, pp. 219–226,
September 2020.
V
Benoit Alary, Pierre Massé, Sebastian J. Schlecht, Markus Noisternig,
and Vesa Välimäki. Perceptual analysis of directional late reverberation.
Journal of the Acoustical Society of America, vol. 149, no. 5, pp. 3189–
3199, May 2021.
VI
Benoit Alary, Archontis Politis, Sebastian J. Schlecht, and Vesa Väli-
mäki. Directional feedback delay network. Journal of the Audio Engi-
neering Society, vol. 67, no. 10, pp. 752–762, October 2019.
5
List of Publications
VII Benoit Alary and Archontis Politis. Frequency-dependent directional
feedback delay network. In Proceedings of the 2020 IEEE Interna-
tional Conference on Acoustics, Speech and Signal Processing (ICASSP),
Barcelona, Spain, pp. 176–180, September 2020.
6
Author’s Contribution
Publication I: “Velvet-noise decorrelator”
The present author developed and implemented the method, produced all
of the figures, and wrote most of the manuscript. The second author had
the original idea for the method and wrote part of the introduction.
Publication II: “Optimized velvet-noise decorrelator”
The author provided the original implementation for velvet-noise decorrela-
tion. The first author had the idea for the improved method, implemented
it, and wrote most of the article. The present author helped validate and
fine-tune the method. He also planned, implemented, and conducted the
two perceptual studies detailed in the article and helped produce Figure 6.
The author wrote part of the introduction and most of Section 5. The
author co-presented the paper in the conference.
Publication III: “Late reverberation synthesis using filtered velvet
noise”
The author was responsible for improving the initial implementation of
the algorithm. The author planned, implemented, and conducted the
perceptual study. The author participated in the writing of Section 7 and
produced Figures 5, 10, 11, and 16. He also produced the data used in
Tables 2, 3, 4, and 5.
7
Author’s Contribution
Publication IV: “Velvet-noise feedback delay network”
The author planned the research project with the co-authors. The present
author also assisted and provided research directions to the first au-
thor throughout the development, implementation, and evaluation of the
method. The author wrote the abstract, co-wrote the introduction and parts
of the manuscript. He produced Figure 8 and helped produce Figures 6
and 7.
Publication V: “Perceptual analysis of directional late reverberation”
The author planned the research project with the co-authors. The present
author implemented the method and produced all of the figures except
for Figure 1. He also wrote most of the manuscript, except for the back-
ground Sections 2.A and 2.C-D. The author implemented and conducted
the perceptual study.
Publication VI: “Directional feedback delay network”
The author produced the original idea for the method and developed the
algorithm. He wrote most of the manuscript and produced all of the
figures. The second author developed the recursive spatial transform and
wrote Sections 1.3, 1.4, as well as Sections 3.2-3.4. The co-authors helped
formalize the method and the terminology used in the manuscript.
Publication VII: “Frequency-dependent directional feedback delay
network”
The author implemented the method, produced all of the figures, and wrote
the manuscript. The co-author contributed to the original idea, provided
critical feedback, and proofreading.
8
List of Abbreviations
ACN Ambisonics channel number
AP all-pass filter
DFDN directional feedback delay network
DIR directional impulse response
DirAC directional audio coding
DWG digital waveguide
EDC energy-decay curve
EDD energy-decay deviation
EDR energy-decay relief
FDN feedback-delay network
FDTD finite-difference time-domain
FFT fast Fourier transform
FIR finite impulse response
FLOPS floating-point operations
FOA first-order Ambisonics
HRTF head-related transfer function
IACC interaural cross-correlation
IEDD interaural energy-decay deviation
IFFT inverse fast Fourier transform
IIR infinite impulse response
ILD interaural level difference
ISM image-source method
IR impulse response
ITD interaural time difference
SDM spatial decomposition method
SH spherical harmonics
9
List of Abbreviations
SIR spatial impulse response
SIRR spatial-impulse-response rendering
SMA spherical microphone array
SNR signal-to-noise ratio
VBAP vector-base amplitude panning
VNS velvet-noise sequence
10
List of Symbols
Arecirculating matrix
cspeed of sound
Ddistance
ffrequency
fssample rate
fSchroeder Schroeder frequency
gattenuation coefficient
Gdirectional gain matrix
Gabsorbent filter
himpulse response
hnoise noise signal
Hcoloration filter
jimaginary unit
kwavenumber
Knumber of plane waves
Ldlength of a discontinuity
Lnumber of spherical harmonics
(Lx,L
y,L
z)room dimensions
nsample number
(nx,n
y,n
z)mode number
psound pressure
Pnumber of directional signals
Qnumber of plane waves
rradius
Rreflected factor
sAmbisonics signal
11
List of Symbols
s
˜plane-wave signal
Ssurface
Sαequivalent absorptive area
ttime
Tdirectional weighting transform
T60 reverberation time
vimpulse vector
Vvolume
wbeamforming weight
xinput signal
xmultichannel input signal
youtput signal
ymultichannel output signal
Y(φ, θ)spherical harmonics coefficient
y(φ, θ)spherical harmonics vector
Yspherical harmonics matrix
Zwall impedance
αabsorption coefficient
δDirac delta function
ΔHdifferential filter
γcross-correlation
Γiacoustic admittance
Γcovariance matrix
∇Laplace operator
θmax maximum angle
(φ, θ)spherical coordinate
φlphase lag
ρdensity parameter
ρ0characteristic impedance of air
ωfrequency band
12
1. Introduction
The reverberation of sound plays a crucial role in our perception of a space:
from the first few reflections, giving us a perceptual hint of the proximity
of walls, to the late reverberation, giving us some intuition on the overall
shape and materials in a room [
1
]. As such, the reverberation properties
of a room are determined by its shape, the absorbing material it contains,
and the scattering properties of its surfaces [
2
]. Although the acoustics
of a concert hall are carefully crafted to ensure a pleasing and enveloping
listening experience, many other types of rooms exists with rich and varied
acoustic properties creating a unique reverberation in each room [3].
Artificial reverberation is an important aspect of sound reproduction,
which strives to efficiently reproduce the perceptual qualities of sound
waves propagating in a room [
4
]. Several artificial reverberation methods
exist, and the choice of one method over another is usually determined
by its application. For instance, music production requires algorithms
with good tuning capabilities and an aesthetically pleasing reverbera-
tion, whereas architectural acoustics benefits from very accurate acoustics
simulations, which often carry a heavy computational cost.
The emergence of immersive applications, such as virtual and augmented
reality, creates a specific set of requirements [
5
,
6
]. Due to the limited
computing resources on these platforms, the sound reproduction methods
used need to be efficient and highly adaptive to the movement of a listener
and a sound source. In augmented reality, where real sounds overlap
with virtual ones, the reverberation needs to be compatible with other
spatial sound reproduction methods to provide accurate and perceptually
transparent reverberation [6].
This dissertation focuses on the directional properties of late reverber-
ation in the context of immersive sound reproduction. More specifically,
an objective and subjective analysis framework is introduced to study the
directional characteristics observed in late reverberant sound fields and
evaluate their importance in the context of sound reproduction (Publication
V). Artificial reverberation algorithms, capable of producing directional
decay characteristics, are also presented (Publication VI, VII). In this work,
13
Introduction
Aesthetics Acoustics
Perception
Figure 1.1. Key objectives of artificial reverberation algorithms.
directional reverberation refers to reverberant sound fields characteristics,
such as the decay rate, that are not uniform in all directions. Novel use
of sparse noise sequences are also proposed in the context of artificial
reverberation and the decorrelation of audio signals.
When developing artificial reverberation algorithms, different objectives
may be realized [
4
]. Here, we describe the main purposes of reverberation
methods along three main axes (Figure 1.1). Some algorithms focus more
on the aesthetic qualities, in which case the reverberation is controlled
through a set of parameters to adjust the reverberation. Some methods
aim to accurately simulate the acoustics of a room [
7
]. Such methods may
reproduce the fundamental behavior of waves propagating or approximate
this process by using key physical properties of a room, such as its size and
the materials it contains.
Perception is another goal which may define the characteristics of a
reverberator aiming to produce realistic sounding reverberation based on
perceptual characteristics, such as the decay rate of different frequencies
[
8
]. These methods offer some flexibility and may be informed by physical
properties to estimate their perceptual characteristics [
9
], but they may
also be tuned aesthetically. The artificial reverberation methods considered
in this work are mainly from this last category.
The methods presented in this dissertation have two main objectives. The
first is to propose highly efficient algorithms, suitable for use in interactive
applications with limited computing resources. The second objective is to
analyze and reproduce key perceptual aspects of a reverberant sound field,
namely, how the energy is distributed spatially during the reverberation.
These two objectives allow the realisation of reverberation algorithms
suitable for applications in virtual and augmented reality, which requires
an efficient six-degrees-of-freedom spatial sound reproduction [10].
Throughout this dissertation, the figures illustrating key measures of a
reverberant sound field were produced using a spatial impulse response
(SIR) recorded using a fourth-order Ambisonics microphone in the facilities
of the Acoustics Lab at Aalto University. The captured room is a reverber-
14
Introduction
6.17 m
3.60 m
8.69 m
Figure 1.2.
Dimensions of the captured room used to illustrate acoustical properties
throughout this dissertation.
ant chamber with variable acoustics called Arni, but at the moment of the
capture, in May 2019, it was being renovated. For this reason, it had ex-
posed concrete walls and no furniture in the room. This provided a unique
opportunity to capture the acoustics of a shoe-box room with no absorption
or scattering beyond its rigid painted walls. The room has the following
dimensions: 6.17 m wide, 8.69 m long, and 3.6 m high (Figure 1.2). The
same data is used in Section 5.5.2 to reproduce the directional properties
of the decay using an existing SIR.
The introductory part of this dissertation is structured around four main
topics: the physical and statistical properties of sound propagation, the
decorrelation of audio signals, artificial reverberation, and directional
reverberation. Chapter 2 introduces some of the necessary background
theory on sound propagation to discuss the nature of reverberant and
diffuse sound fields. Statistical measures are reviewed as well as the
analysis of captured impulse responses and spatial audio reproduction
methods.
In Chapter 3, we discuss the decorrelation of audio signals in the context
of sound reproduction. A brief overview of measures and existing decorre-
lation methods is given along with contributions made to this topic using
short sequences of sparse noise called velvet noise (Publication I, II).
In Chapter 4, we give an overview of the history of artificial reverberation
methods used to reproduce acoustics. Virtual acoustics and noise-based re-
verberator are covered, and an extensive review of delay-network methods
is given. The chapter also includes two novel uses of velvet noise in the
context of artificial reverberation (Publication III, IV).
Chapter 5 is focused on hybrid reverberation algorithms and their use
for spatial sound reproduction. This chapter also highlights key contribu-
tions made in this dissertation, which includes an analysis framework for
the evaluation of directional characteristics in a reverberant sound field
(Publication V) and two reverberation algorithms capable of reproducing
these decay characteristics (Publication VI, VII). Finally, the work of this
dissertation is summarized and future research directions are explored in
the conclusion.
15
2. Reverberant Sound Field
When a sound source emits energy in an enclosed space, the energy radiates
outwards and propagates in the surrounding environment. At any given
time, if we sample the sound pressure at different points in a room, we will
find signals at various phases carrying different amounts of energy, and
together these pressure points form a sound field [
11
]. As such, a sound
field may be described as a portion of space with acoustical disturbances.
In spatial sound reproduction, a sound field commonly refers to a set of
points captured by a spherical microphone array (SMA).
Propagating waves eventually interact with the various surfaces con-
tained in a room, and these surfaces absorb and scatter the energy when
the waves are incident on it. This interaction quickly increases the density
of waves contained in the sound field while decreasing the overall amount
of energy, thus forming a reverberant sound field. The propagation of
sound in an enclosed space can also be separated into two distinct time
periods, the early part, where reflections are individually salient, and the
late part, where the sound field contains a high density of reflections that
are better described using statistical methods.
In this chapter, the fundamental principles of sound propagation are
reviewed along with the essential statistical properties used in the study
of room acoustics. The concept of diffuse fields, the implications they carry,
and the measures to define them are also examined. Finally, methods used
for the reproduction of sound fields are discussed.
2.1 Sound Propagation
The propagation of sound energy through mechanical waves, which in-
cludes longitudinal and transverse waves, follows the general laws of
thermodynamics to transmit energy from particle to particle in a medium.
In free space, an omnidirectional sound source emits a wave expanding
spherically and causes the displacement of air particles.
However, this displacement is susceptible to a varied range of interfer-
17
Reverberant Sound Field
ence such as the diffraction of waves around obstacles, the absorption of
energy when reflecting off materials, the scattering of waves in multiple
directions when reflecting off irregular surfaces.
In practice, many waves propagate simultaneously, and the cumulative
effect produced by the displacements they cause makes it very challenging
to assess acoustic phenomena as a whole. For this reason, to study the
acoustics of rooms, it is useful to break down this complex system into
smaller parts that can be observed in controlled conditions.
The first step in describing the behavior of waves in a three-dimensional
space, is to use the basic equations to describe the propagation. A funda-
mental equation, known as the wave equation, describes the propagation
of sound in free field, formulated using the following partial differential
equation [11]:
∇2p=1
c2
∂2p
∂t2,(2.1)
where
c
is the speed of sound,
∇2
corresponds the second derivative of
the spatial components called the Laplace operator, and
p
is the sound
pressure.
In the frequency domain, waves are described through a linear partial
differential equation, independent of time, called the Helmholtz equation
[11]
∇2pˆ=−k2pˆ,(2.2)
for a given wavenumber k.
In an open space, an ideal omnidirectional sound source emits vibrations
that expand spherically, spreading energy outwards. The wave equation
for the resulting spherical wave is given by [12]
∂2p
∂r2+2
r
∂p
∂r =1
c2
∂2p
∂t2,(2.3)
where
r
is the radius of the spherical wave and
t
is time,which may also be
written as [2]
∂2p
∂x2+∂2p
∂y2+∂2p
∂z2=1
c2
∂2p
∂t2,(2.4)
where x,y, and zare the Cartesian coordinates.
When the radius is large enough with respect to the wavelength, the
observed curvature of the wavefront is almost flat. For this reason, the
propagation of sounds is best described through a set of plane waves
traveling in three-dimensional space.
On their own, these equations only represent waves traveling in a free
field. In room acoustics, boundaries must be introduced into these wave
equations to account for sound absorption [13].
18
Reverberant Sound Field
Figure 2.1.
An incoming plane wave (in black) and its reflection (in blue), adapted from
from [14].
In an enclosed space, a wave will eventually reach a wall, which will
absorb some of the energy and reflect the rest. The magnitude of energy
and phase of the reflected wave is determined by the impedance of the wall
Z
, a complex number, and the characteristic impedance of air
ρ0
. For a
flat surface of infinite dimensions, the reflection factor corresponds to the
fraction of reflected energy and is defined as [2]
R(θ)=Zcos θ−ρ0c
Zcos θ+ρ0c,(2.5)
where
θ
is the angle between the normal vector of the wall and the incident
wave.
Here, the reflection factor
R(θ)
, also a complex number, has a real value
constrained between
[−1,1]
, based on the incident angle (
0<θ< π
2
) and
the wall impedance. If the wave vector is perpendicular to the wall (
θ=0
),
Equation 2.5 becomes
R(0) = Z−ρ0c
Z+ρ0c,(2.6)
and at a theoretical 90◦angle, parallel to the wall, the equation becomes
R(π
2)=−1,(2.7)
hence, when
ρ0c<Z
, the energy being reflected
|R(θ)|
increases with the
angle. Using the reflection factor, the absorption at the wall boundary is
αb(θ)=1−|R(θ)|2,(2.8)
which illustrates that the energy absorbed during the reflection of single
plane wave is dependent on the incident angle of the wave.
19
Reverberant Sound Field
Figure 2.2.
Normalized low-order two-dimensional eigenmodes (
nz=0
). The top row
illustrates axial modes, which involves two facing walls, while the bottom row
illustrates tangential modes, involving two pairs of facing walls.
However, if the surface of the wall is not regular, the reflection angle
is dependent on the wavelength of a sound wave. If we consider a dis-
continuity of length
Ld
and an incident wave with wavelength
λ>2Ld
,
the discontinuity will not substantially influence the wave. However, a
wave having a smaller wavelength
λ<2Ld
will be reflected at the angle of
the discontinuity, and a wave with
λ≈2Ld
will be scattered in multiple
directions [15].
In practice, a very irregular surface will cause a wave with different
wavelengths to reflect in many directions. As such, the reflection of sound
energy in a room is both angle- and frequency-dependent. The scattering
of waves promotes the diffusion of sound energy in a room throughout the
propagation.
Near a flat surface, an incoming plane wave interacts with its reflected
wave in the specific region where the two overlap, illustrated in Figure 2.1.
In a room with many flat surfaces, the overlapping of waves from all
directions causes an increase of potential energy in the reverberant sound
field in areas further away from the center [14].
In a rectangular room, when the distance between two walls is a multiple
of a wavelength being propagated, the reflected energy combines either
constructively or destructively to introduce stable energy patterns, called
standing waves or room modes. This also occurs when higher-order re-
flection paths, combining different walls, form more modes. As such, the
sound pressure in three-dimensional space can also be described as a linear
20
Reverberant Sound Field
combination of eigenmodes.
Individual eigenmodes are described through an eigenfunction for a given
mode number
(nx,n
y,n
z)
. In a rectangular room with rigid boundaries, the
eigenfunctions are calculated using [2]
pnxnynz(x, y, z)=cosnxπx
Lxcos nyπy
Lycos nzπz
Lz,(2.9)
where
(Lx,L
y,L
z)
are the dimensions of the room, and
(x, y, z)
are coordi-
nates along these dimensions. The wavenumber
k
of a given mode and its
associated frequency fcan be calculated from [2]
knxnynz=πnx
Lx2
+ny
Ly2
+nz
Lz2
,(2.10)
and fnxnynz=c
2πknxnynz,(2.11)
respectively. Therefore, higher modes resonate at higher frequencies. In
Figure 2.2, low-order two-dimensional eigenfunctions are illustrated using
the dimensions of the shoe-box room described in Chapter 1. The areas
with larger
|p|
values represent a larger potential energy. Blue and red
represent, respectively, a negative and positive amplitude.
Figure 2.3 illustrate higher-order modes. To activate a mode in a room,
a wave oscillating at the mode frequency needs to be introduced. A re-
verberant sound field comprises of a superposition of all the activated
modes.
2.2 Statistical Properties
As discussed in the previous section, the reflection of an individual plane
wave is determined by its frequency, its incidence angle, as well as the
impedance and shape of the reflective surface. In a reverberant sound
field, many reflected waves combine together to form a dense distribution
of plane waves.
Having many circulating plane waves from many directions means that
the energy becomes more uniformly distributed as time progresses. Hence,
it is more practical to leverage the stochastic nature of reverberant sound
fields rather than attempting to follow the cumulative impact of individual
reflected plane waves. This section contains an overview of the statisti-
cal properties of a reverberant sound field used to describe the acoustic
properties of a room.
In the early stage of reverberation, individual echoes are sparse in time.
For this reason, early reflections are particularly critical in the perception
of acoustics [
1
]. However, in the later stage of sound propagation, the near
infinite number of reflected plane waves circulating implies a high density
21
Reverberant Sound Field
Figure 2.3. Normalized higher-order two-dimensional eigenmodes (nz=0).
22
Reverberant Sound Field
of echoes. In a room, the number of echoes per second for a given time
t
is
estimated as [16]
ρe(t)=4πc
3
Vt2,(2.12)
where
V
is the volume of the room. In artificial reverberation, the echo
density is an important consideration and must rise rapidly to ensure a
natural sounding reverberation [17].
In the frequency domain, the reverberant sound field is perceived as a
superposition of room modes. The density of superposing modes increases
with the frequency. The modal density for a frequency
f
can be estimated
using [16, 2]
ρm(f)=4πV
c3f2,(2.13)
for a frequency
f
given in hertz. Schroeder suggested that above a certain
frequency
fSchroeder
,
ρm(f)
is sufficiently high that the individual mode be-
comes perceptually indistinguishable [
18
,
19
,
20
]. The cross-over frequency
where this transition occurs is estimated from the volume and decay time
through [18]
fSchroeder ≈2000T60
V,(2.14)
where
T60
, a measure of reverberation, is given in seconds and
V
is the
volume in cubic meters. The
T60
, called reverberation time, represents the
time required for the sound energy to decay by 60 dB.
T60
may be estimated based on the dimensions of the room and the
absorbing properties on its surfaces. Sabine formulated the reverberation
time as [21]
T60Sabine =0.163 V
S α,(2.15)
where
V
is the volume of the space in m
3
,
S
is the total surface area, and
α
is the average absorption per m
2
. Here, the absorption considered is also
known as the random-incidence absorption coefficient [
22
] and is based
on statistical measurements performed in a reverberant room. The term,
0.163, from this equation is derived from (24 ln(10)/c) [2].
This decay time estimation formula was later revised after it was discov-
ered that rooms with highly absorptive materials were underestimated.
The revised formula given by Eyring is [23]
T60Eyring =0.163 V
−Sln (1 −α),(2.16)
which introduces extra terms when solving the natural logarithm using a
23
Reverberant Sound Field
Taylor-series expansion defined by
ln (1 −α)=α+α2
2+α3
3+α4
4+···,(2.17)
whereas T60Sabine only considers αin the denominator.
In modern versions of
T60Sabine
and
T60Eyring
, an extra term is also added
to the denominator (
+4Vα
a
), where
αa
represents air absorption. Air
absorption becomes an important factor to consider in bigger rooms [2].
Both
T60Sabine
and
T60Eyring
make the assumption that the absorption is
evenly distributed in the space, which can lead to an estimation error in
certain rooms [
24
,
25
,
26
,
27
]. Other methods account for a less uniform
distribution of absorption. For instance, the Fitzroy formulation groups
parallel pairs of walls to give [28]
T60Fitzroy =0.163 V
SSx
−Sln(1 −αx)+Sy
−Sln(1 −αy)+Sz
−Sln(1 −αz),
(2.18)
where
Sx
represents the total area of the side walls,
Sy
the area of the floor
and ceiling, and
Sz
that of the front and back walls, while
αx
,
αy
, and
αz
are their respective average absorption coefficients.
When considering the perception of late reverberation in a room, an
important parameter to consider is the critical radius
rc
. This radius
represents the distance at which the direct sound energy is equivalent to
the reverberant energy. This measure has an implication, for example, on
speech intelligibility for example. The critical radius can be calculated
using [29]
rc=Sα
V,(2.19)
where
Sα
is called the equivalent absorptive area, a value representing the
total absorption in a room calculated using [29]
Sα=
i
Siαi,(2.20)
where
Si
is the surface area of the room and
αi
is the corresponding
absorption coefficient.
For the statistical analysis of a room, it is also possible to capture an
impulse response (IR) by playing a short transient sound or a sine sweep
through a loudspeaker placed in the room [
30
,
31
]. A broad-spectrum
stimulus will ensure more room modes will resonate, yielding a more
accurate frequency response of the reverberant sound field. In the time
domain, and for a single microphone-loudspeaker pair, we define the IR as
h(t)=
∞
k=1
akδ(t−τk)+hnoise(t),(2.21)
24
Reverberant Sound Field
0 0.5 1 1.5 2 2.5 3 3.5 4
-60
-40
-20
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-12
-10
-8
-6
-4
-2
0
Figure 2.4.
(a) Captured IR in the time-domain and (b) the first 100 ms of the captured
IR, where sparse reflections can be observed.
where
ak
is the amplitude of an individual reflection,
τk
is the correspond-
ing delay, and
δ
is a Dirac delta function.
hnoise(t)
represents a noise
sequence introduced when capturing an IR. Figure 2.4 illustrates
h(t)
for
the example captured room, and in Figure 2.4(b) individual reflections are
shown.
To analyze the behavior in the frequency domain, a fast Fourier transform
(FFT) is performed on the captured IR, resulting in the frequency response
in Figure 2.5. In Figure 2.5(b), individual low-frequency modes are shown.
The individual modes identified in Figure 2.5(b) correspond to the modes
predicted using the dimensions of the shoe-box room in Equation 2.9 and
illustrated in Figure 2.3.
As illustrated in Figure 2.4, the density of impulses tends to be much
higher during the late reverberation stage. In an IR, the late part tends to
look very similar to a random noise sequence. As such, the echo density
of a captured IR may be estimated from how much it deviates from the
distribution of Gaussian noise, representing ideally dense and stochastic
late reverberation [32, 33].
To analyze the properties of the decay, the energy-decay curve (EDC) [
34
],
an integral of energy from a moment in time tuntil ∞, is defined as
EDC(t)=∞
t
h2(τ)dτ, (2.22)
which is expanded as a time-frequency surface using the energy-decay
25
Reverberant Sound Field
50 100 200 500 1000 5000
-30
-20
-10
0
10
50 60 70 80 90 100 110 120
-30
-20
-10
0
10
Figure 2.5.
(a) Frequency-domain analysis of the IR captured in the shoe-box room and
(b) low-frequency analysis, showing the individual modes calculated from
Equation 2.9 and illustrated in Figure 2.3.
relief (EDR) [35]
EDR(t, ω)=∞
t
h2(τ,ω)dτ, (2.23)
where individual curves are calculated for different frequency bands ω.
In Figure 2.6, the EDR of the captured shoe-box room is analyzed. From
this figure, we observe that the decay properties are frequency-dependent
and also that the presence of noise hides a portion of the exponential
decays. Hence, when recording an IR, the noise captured from the room
and the recording equipment hides some of the decay occurring below the
noise threshold. The decay rates contained in these curves are suitable to
estimate frequency-dependent reverberation times [
T60(ω)
], but in noisy
IR, the reverberation time may be estimated from shorter segments of
exponential decay (T10,T20,T30, ...) [12].
Since the EDC and EDR formulas integrate the noise energy contained
in an IR, further considerations are necessary to interpret the data beyond
the noise floor. In [
36
], different strategies to minimize the impact of noise
on EDC analysis are explored. The decay rates in the beginning of the
IR for different frequencies [
37
] and directions [
38
] may also be used to
replace the noisy part of SIR with decaying noise, which is beneficial in
sound reproduction using captured IRs.
26
Reverberant Sound Field
Figure 2.6. EDR from a captured IR in third octave bands.
2.3 Diffuse Field
A diffuse sound field, or diffuse field for short, is a theoretical sound field
in which the energy is ideally distributed. This specific state may arise if
the reflection of sound energy in a room creates a near infinite amount of
uncorrelated plane waves and reaches a statistically uniform distribution.
The precise definition of a diffuse field varies throughout the literature,
but it is commonly defined through three key preconditions: homogeneity,
isotropy, and incoherence [14, 2, 39].
Homogeneity refers to a state where the mean energy density is the same
at all locations in a space. In this state, the energy is perfectly uniform at
every point in the sound field.
Isotropy implies that the energy coming from all directions for a given
point is equivalent. From a statistical standpoint, this means that the
probability for a wave to be incident from a specific direction is the same
in all directions. From the homogeneity principle, a diffuse field is also
isotropic at all locations. A sound field which is not isotropic is called
anisotropic.
Finally, incoherence means that individual wavefronts have only a small
amount of correlation between them. This implies that the phase of dif-
ferent frequency components is distributed randomly between reflected
waves. This precondition is important in order to avoid constructive and
destructive interference that arises from highly correlated sound waves,
which in turn, would break the homogeneity and isotropy.
When all three preconditions are met, we obtain a statistically diffuse
state. A diffuse field is useful, for instance, for measurements of absorption
materials in a reverberant room. However, building a room with sufficient
diffusion to create such a diffuse field requires special care [
40
,
41
,
42
]. To
27
Reverberant Sound Field
evaluate the characteristics of reverberant rooms, recent work employs
SMAs to analyze and validate the isotropy in a room [43, 44, 45, 46, 47].
In practice, these ideal conditions are never fully realized. Physical at-
tributes such as parallel walls, uneven distribution of absorbing materials,
and the presence of non-scattering surfaces can all impact negatively on
the diffusion of energy in a room [
48
,
14
,
40
,
2
]. Nonetheless, in some
applications, the diffuse-field conditions are not required to be fully met. In
such applications, any reverberant sound field is assumed to be sufficiently
diffuse, which allows sufficient simplifications for practical tasks such as
speech enhancement and de-reverberation [49].
In [
14
,
50
], Waterhouse demonstrates how a rigid wall creates interfer-
ence patterns in the sound field, which leads to up to 6 dB differences in
the potential energy between the center of the room and its corners [
14
].
For this reason, it is recommended to measure the absorption coefficient of
materials at the center of a reverberant chamber. Using simulated acous-
tics, the distribution of incident directions of individual reflections may
be analyzed with good precision to demonstrate the anisotropy in simple
rooms, such as a shoe box room [
51
,
52
]. In [
53
], the direction-of-arrival
of late reflections in a concert hall were analyzed using captured and sim-
ulated impulse responses, and the study determined that the direction
of late reverberation is influenced by both the shape of a room and the
distribution of absorption material in a room.
Recently, new measures were proposed to quantify the amount of aniso-
tropy in a captured spherical sound field by measuring the difference
between different spherical harmonics components [
44
,
45
,
46
]. In [
54
], a
perceptual study demonstrated that the differences in the energy decay
profiles can be perceived above a certain threshold.
2.4 Properties of Reverberant Sound Fields
A diffuse field carries important implications and as such, it is useful
to determine whether a sound field is diffuse or not. In this section,
we briefly review two objective measures and key subjective studies to
characterize the properties of a reverberant sound field in the context of
sound reproduction.
2.4.1 Objective Measures
Diffuseness is a common term used for various measures estimating
whether the conditions for a diffuse field are satisfied [
55
]. However, only
one characteristic of a diffuse field is usually evaluated, and for this reason,
the choice of a diffuseness measure is determined by some assumptions
on the sound field. For instance, some definitions use the acoustic energy
28
Reverberant Sound Field
[
56
,
57
], while others define diffuseness using acoustic intensity [
2
,
58
].
These measures determine that a sound field is diffuse by assessing its
isotropy.
In another approach, the covariance of a sound field is analyzed to mea-
sure the coherence between spherical harmonics (SHs) [
55
]. In the context
of this dissertation, a key benefit of using spatial coherence to rate dif-
fuseness is that it makes no assumptions on the isotropy of a reverberant
sound field [38].
Generally, it is considered that a reverberant sound field transitions
from a highly directional energy distribution during the early stage of
reverberation to a more isotropic and incoherent distribution during the
later stage. The moment of this transition, called mixing time
tmix
,isof
particular interest when studying sounds fields, since different assump-
tions are made before and after this moment. This is similar to
fSchroeder
in
Equation 2.14, which defines the boundary of salient room modes.
One definition of
tmix
looks at the echo density and the moment where
ρe(t)
(Equation 2.12) reaches a perceptual threshold where individual
reflections are indistinguishable [
59
]. One limitation of this approach
is that while individual reflections may not be perceptually salient, an
anisotropic sound field may still maintain more overall energy in specific
directions throughout the decay, resulting in a different perceptual cue.
In the context of spatial sound reproduction, another approach is to
define
tmix
as the moment where a specific threshold of diffuseness is
met [
60
]. This allows the use of a covariance analysis method which is
suitable for the study of sound fields that may be incoherent but anisotropic
(Section 3.2.4).
2.4.2 Perception
Whereas analysis method such as diffuseness and mixing time may be
important as objective measures of a reverberant sound field, subjective
evaluation is essential to determine the appropriate level of details re-
quired in the design of reproduction methods.
The importance of early reflections on the perception have long been
demonstrated [
61
]. Specifically, they play a key role in the spatial impres-
sion of a concert hall, which may be characterized by the ratio of lateral
and non-lateral energy in the first 80 ms of sound propagation [
62
]. This
ratio has been found to be a good descriptor of the spatial impression of
a concert hall, thus demonstrating the need for good lateral reflections
in concert halls [
63
,
1
]. The perceptual impact of the spatial distribution
of early reflections was also studied from the perspective of musicians on
stage [64].
The importance of the early reflections is also echoed in sound repro-
duction, with the development of recording techniques preserving their
29
Reverberant Sound Field
characteristics in stereo reproduction [
65
], and more recently their re-
production was assessed in the context of virtual reality and a listening
experiment found that an accurate reproduction of the first six reflections
was optimal [66].
However, late reverberation also plays an integral role in the perception
of acoustics. In [
67
], the impact of late reverberation on the sense of
envelopment was studied, and a perceptual test showed that envelopment
decreases a subject’s sensitivity to changes in the apparent source width of
the sound field. Many other studies have also demonstrated the positive
correlation between envelopment and late reverberation [
3
,
68
], which is
also influenced by the distance between a listener and a sound source [
69
].
In a different perceptual study, the reproduction of late reverberation
in binaural recordings was investigated using a lecture hall with a short
reverberation time [
70
]. In the experiment, the late reverberation tail
was substituted with another one captured from different source and
microphone locations in the same room. The listeners were not able to
correctly detect the substitution if it occurred after the first 40 ms in the
binaural IR.
Another listening experiment examined the connection between objective
and subjective measures of mixing time in a binaural reproduction system
[
71
]. In this experiment, the early part of the reverberation, before the mix-
ing time, was updated to follow the head movement of a participant while
the late reverberation remained static. The results of the perceptual study
showed that, for the reproduced halls, the sound field was perceptually
isotropic after the estimated tmix.
However, other studies have indicated that, in conditions such as double-
slope reverberation [
72
,
73
] or if the differences in decay energy for dif-
ferent directions reach a specific threshold [
54
], a reverberant sound field
becomes perceptually anisotropic. This suggests that, since isotropy is
determined by the diffusion properties of a room [
53
], the perception of
anisotropy also varies from one room to another. This is explored in Publi-
cation V, detailed in Section 5.5.1.
2.5 Reproduction of Sound Fields
A reverberant sound field contains many plane waves incident from all di-
rections. In this section, we review Ambisonics methods, which decompose
a spatial sound field into plane waves incident from many directions, and
binaural reproduction, which considers the physiological properties of a
listener’s head to reproduce a sound field through headphones.
30
Reverberant Sound Field
2.5.1 Ambisonics
The reproduction of sound fields is crucial for multichannel sound repro-
duction. Ambisonics, is a popular term used to describe a set of meth-
ods, originally developed in the 1970s to encode a sound field using an
SMA [
74
,
75
,
76
,
77
,
78
]. Ambisonics methods were later generalized to
higher order SMAs through the use of SH functions, suitable for the cap-
ture and reproduction of sound fields using a higher spatial resolution
[79, 80, 81, 82, 83].
In higher order Ambisonics, a sound field is expressed by performing
a plane wave expansion to obtain coefficients of a set of orthonormal SH
basis functions covering the surface of a sphere. To obtain SH signals,
a sound field is first sampled from
K
individual plane wave signals
si
,
incident to (φi,θ
i)in spherical coordinates, and is encoded using
s(n)=
K
i=1
si(n)y(φi,θ
i),(2.24)
where
n
is the index of an audio sample and
y(φ, θ)
corresponds to a vector
containing the SHs for a given direction defined as a vector of coefficients
y(φ, θ)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
Y1(φ, θ)
Y2(φ, θ)
.
.
.
YQ(φ, θ)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(2.25)
for a set of
Q=(L+1)
2
coefficients and a maximum order
L
. The SH
coefficients in vector
y(φ, θ)
are placed in a pre-determined order, following
one of the existing conventions to ensure the portability of Ambisonics
signals during sound reproduction [84].
In the Ambisonics channel number (ACN) convention, the index number
corresponds to
i=l2+l+m+1
where
l
and
m
are the SH order and degree,
respectively, with
0≤l≤L
and
−l≤m≤l
[
84
]. Individual SH coefficients
are commonly defined as [85]
Ylm(φ, θ)=2−δm0(2l+1)
4π
(l−|m|)!
(l+|m|)!ym(φ)Pl|m|(cos θ),(2.26)
where
δ
is the Kronecker delta,
Pl|m|
are the associated Legendre polyno-
mials, and ymis
ym(φ)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
sin |m|φm<0,
1m=0,
cos mφ m > 0.
(2.27)
31
Reverberant Sound Field
Once encoded, an Ambisonics signal is converted back to individual plane
waves through
s
˜(n)=YTs(n),(2.28)
where s
˜(n)contains a set of plane wave signals
s
˜(n)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
s˜1(n)
s˜2(n)
.
.
.
s˜K(n)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(2.29)
and Yis a matrix defined as
Y=y(φ1,θ
1),y(φ2,θ
2),··· y(φK,θ
K),(2.30)
which performs a plane wave decomposition in the direction of each SH
vector y(φi,θ
i).
For sound reproduction, an Ambisonics signal is decoded to a set of chan-
nels by using a beamformer pointing in the direction of each loudspeaker.
A beamformer is defined as
y(n, θ, φ)=
Q
i=1
si(n)wi(θ, φ),(2.31)
where
wi
contains the individual SH weights for a given direction. A simple
hypercardoid beamformer is obtained when wi(θ, φ)=Y
i(θ, φ). The shape
of a beamformer can be modified to modify the shape of the main lobe
as well as the side and back lobes [
82
,
86
]. To analyze and reproduce
spherical reverberant sound fields, a SIR is captured using an Ambisonics
microphone and decoded to a set of directional impulse responses (DIR) for
reproduction.
In certain applications, it is necessary to manipulate an Ambisonics
signal [
87
,
88
,
89
,
90
]. Specifically, we may want to alter the energy
of an Ambisonics signal in different directions [
91
,
92
]. If we consider
G=diag(g(φ1,θ
1),g(φ2,θ
2), ..., g(φK,θ
K))
, a diagonal matrix containing a
set of directional gain values, we formulate the transformation matrix
T
as
T=YGY
T,(2.32)
where
Y
is the output SH matrix and
T
is a matrix which is used to apply
energy weighting to an Ambisonics signal. Depending on the smoothness of
the distributed gains in
G
, spatial aliasing may occur, which is minimized
by increasing the order of the encoding matrix Y[91, 92].
32
Reverberant Sound Field
2.5.2 Binaural Reproduction
Due to the reflection and diffraction of sound waves around the human
head and the ears’ pinnae, the perceived spectrum of sounds reaching
our ears is altered based on their incident angles. Using binaural sound-
reproduction techniques [
93
], a spherical sound field is encoded into a
stereo signal by using a set of head-related impulse responses (HRIR) or
head-related transfer functions (HRTF) in the frequency domain, which
will reproduce the interaural time difference (ITD) [
94
,
95
,
96
], interaural
level difference (ILD) [
94
,
95
], and the spectral cues of sound reaching our
ears at different angles [93].
In the context of a reverberant sound field, a key characteristic to consider
in binaural sound reproduction is the perceived interaural cross-correlation
(IACC). A formal definition of cross-correlation is given in Section 3. The
IACC corresponds to the frequency-dependent cross-correlation between
the two ears, defined as [97, 98]
γIACC =
t1
t0
xL(t)xR(t+τ)dt
t1
t0
xL(t)2dt
t1
t0
xR(t)2dt
,(2.33)
where
xL
and
xR
are the two channels of a binaural signal and
τ
represent
a time lag. In a diffuse field, the IACC approximates a sinc function along
the frequency axis [
99
]. However, in practice, the shape of a listener’s head
will greatly impact the IACC in a diffuse field [93].
33
3. Decorrelation
Since one of the primary characteristics of reverberant sound field is the
low-coherence between individual plane waves, the decorrelation of audio
signals relates to the natural process of sound diffusion occurring during
sound propagation [
2
]. For this reason, a multichannel reverberator should
act as a decorrelator (Section 5.1). However, in some applications, it is
useful to control the decorrelation independently from the reverberation,
which requires the use of decorrelation methods that do not introduce
reverberation.
In this context, cross-correlation refers to a measure of the similarities
between two signals. Coherence is a frequency-dependent measure of
these similarities, and covariance is used in the context of multichannel
sound reproduction. The decorrelation of audio signals discussed in this
section refers to signal processing methods that are used to modify a
signal’s waveform slightly, ideally just its phase response, to enhance the
perceptual spatial attributes during multichannel reproduction.
3.1 Overview
As seen in Equation 2.5, the phase of a reflected wave is a function of the
impedance of a wall material, and uneven surfaces will scatter a wave in
different directions depending on its wavelength. As such, decorrelation
occurs naturally due to the impact of reflection, scattering, and absorption
of sound altering the phase of propagating waves.
In signal processing terms, an effective decorrelator should randomize
the group delay of a signal while minimizing the spectral distortions [
100
].
Furthermore, the modification of the phase should be restricted in order not
to exceed the frequency-dependent perceptual threshold of phase shifting
[101] to avoid echoes and the smearing of transients.
Decorrelation is used in multichannel sound reproduction to reduce the
comb-filtering effect encountered when multiple channels produce the
same signal and recombine slightly out of phase in a sound field. For
35
Decorrelation
stereo headphones reproduction, highly correlated signals are perceived
as coming from inside the listener’s head. For this reason, up-mixing, a
common application of decorrelation, is used to convert between different
channel-based configurations during sound reproduction [93].
In a sound field, some signals should be highly correlated, such as the
direct sound and early reflections, whereas late reverberation must be
well decorrelated. For this reason, when decorrelation is performed on a
continuous signal encoded in a multichannel format, such as Ambisonics,
an analysis step is required to separate the diffuse part of the signal from
the non-diffuse part, and the decorrelation is applied only to the diffuse
part. This approach is used to improve the reproduction of Ambisonics
[58, 102, 103].
Another use for decorrelation is to extend the perceived size of a sound
source when mono signals are reproduced on a loudspeaker array. If the
same signal is used on multiple loudspeakers, the incident direction of the
source is perceived from a small area between the loudspeakers [
100
,
104
].
On the other hand, this attribute is beneficial in panning techniques, such
as the vector-base amplitude panning [105].
3.2 Objective Evaluation
To evaluate the correlation of audio signals, several measures are used
depending on if we are interested in the correlation of signals over time,
frequency, or space.
3.2.1 Cross-Correlation
The amount of correlation between two signals
x
and
y
over time is mea-
sured through the cross-correlation, calculated as
γxy(l)=
N
n=1
x(n+l)y(n),(3.1)
where
l
is the lag. As such, two out-of-phase signals may still be highly cor-
related if, for instance, one is a delayed copy of the other. When evaluating
decorrelation, the zero-lag cross-correlation
γxy(0)
is of particular interest
to measure the correlation at a specific moment. The normalized zero-lag
cross-correlation is given by
γxy =
N
n=1
x(n)y(n)
N
n=1
x(n)2N
n=1
y(n)2
,(3.2)
which is bounded between
[0,1]
. A lower
γxy
value indicates dissimilarities
36
Decorrelation
between the two signals.
3.2.2 Coherence
Coherence is a frequency-dependent measure of cross-correlation. In prac-
tice,
x(n, ωc)
and
y(n, ωc)
, the discretized signals given
ωc
, the center fre-
quency for a frequency band in a filterbank. Zero-lag coherence is defined
as
γxy(ωc)=
N
n=1
x(n, ωc)y(n, ωc)
N
n=1
x(n, ωc)2N
n=1
y(n, ωc)2
.(3.3)
3.2.3 Covariance Matrix
Covariance is another related statistical measure. In the context of multi-
channel sound reproduction, covariance is commonly used to measure the
correlation between pairs of channels. For this purpose, a covariance ma-
trix built from the sample covariance of every pair of channels, is defined
as
Γx=
N
n=1
x(n)x(n)T
N,(3.4)
where vector
x
is a signal containing multiple channels and
N
is the
number samples.
As discussed in Section 2.5.1, in Ambisonics, the beamformer used to
extract directional signals from Ambisonics signals contains side and back
lobes (Equation 2.31). These lobes and the width of the main lobe introduce
correlation between the output signals. As such, the covariance matrix is
a useful tool to evaluate various output configurations in the context of
Ambisonics sound reproduction.
In Figure 3.1, we evaluate the covariance for different output configu-
rations. Here, the input signal is the SIR of the example shoe-box room
(Section 1), encoded in first-order Ambisonics. For each configuration eval-
uated,
K
output channels are uniformly distributed around a sphere. The
diagonal line in the normalized matrix is the autocovariance between a
channel and itself and will always have a value of 1. In Figure 3.2, the
same analysis is performed using the SIR encoded in fourth-order Ambison-
ics as input. These figures illustrate the the limitation of using low-order
Ambisonics signals to output on a high-order multichannel loudspeaker
system. In Publication V (Section 5.5.1), the coherence matrix is used to
validate the reproduction method used in a perceptual study.
37
Decorrelation
1234
1
2
3
4
24681012
2
4
6
8
10
12
4
8
12
16
20
24
28
32
36
4
8
12
16
20
24
28
32
36 0
0.2
0.4
0.6
0.8
1
Figure 3.1.
Covariance matrix of a first-order Ambisonics SIR decoded to
K
output chan-
nels, uniformly distributed around a sphere.
1234
1
2
3
4
24681012
2
4
6
8
10
12
4
8
12
16
20
24
28
32
36
4
8
12
16
20
24
28
32
36 0
0.2
0.4
0.6
0.8
1
Figure 3.2.
Covariance matrix of a fourth-order Ambisonics SIR decoded to
K
output
channels, uniformly distributed around a sphere.
3.2.4 Incoherence Profile
The coherence, or its opposite, incoherence, may be used as a measure of
the diffuseness of a reverberant sound field (Section 2.4). As discussed
previously, during the propagation of sound, the increase of mutually uncor-
related reflected plane waves in a sound field should lower the coherence.
To analyse the evolution of coherence in a spatial sound field, we measure
the incoherence profile from a multichannel signal.
For this purpose, we first reformulate the zero-lag cross-correlation using
the covariance matrix as
γab =Γab
√ΓaaΓbb
,(3.5)
where
a, b
are two different channels. The coherence between channels
γab
is then averaged for the different pairs of channels using to obtains a
coherence profile γx.
Finally, we set the incoherence profile to
1−γx
on a short-time moving
window to obtain a time-dependent measure that may be used to estimate
the evolution of diffuseness in a reverberant sound field.
In Publication V (Section 5.5.1), a measure of incoherence is used to
estimate the mixing time of captured SIRs. A more formal measure of
spatial coherence is given in [55].
38
Decorrelation
−g
x(n)z−m1y(n)
g
Figure 3.3. Flow diagram of an AP filter [16].
3.3 Decorrelation Methods
Introduced in the context of artificial reverberation [
16
], the all-pass filter
(AP) has good properties for the decorrelation of signals [
100
]. The AP
consists of combining a negative feed-forward path to a positive feedback
path of a delay-line using the same absolute attenuation gain
|g|
for both
paths (Figure 3.3). The combination of the two paths creates a zero and
a pole, mirrored on opposite sides of the unit circle, which yields a flat
magnitude response.
The transfer function of an AP filter is [16]
H(ω)=e−jωτ 1−gejwτ
1−ge−jwτ ,(3.6)
where
ω
is the angular frequency,
g
is the attenuation,
τ
is the delay time,
and
j
is the imaginary unit. From this equation, we obtain the magnitude
response for the filter
|H(ω)|=1
, implying that the input and output
signal energy is the same at all frequencies. What is caused by the filter is
the frequency-dependent phase lag
φl(ω)=ωτ + 2 arctan gsin ωτ
1−gcos ωτ ,(3.7)
which results in an uneven distribution of phase shifts.
Since the AP filter modifies the phase of a signal without affecting its
magnitude spectrum, it is a useful tool for decorrelation. However, to
ensure sufficient decorrelation, multiple AP filters are cascaded, and their
delay components should be kept short to avoid smearing of transients
[
100
]. In larger sets of cascaded APs, the smearing of transients, causing a
chirp-like effect, may be leveraged as a sound synthesis method [
106
], or in-
dividual AP characteristics may be optimized to obtain better decorrelating
properties [107, 108].
Sequences of random white noise also have a flat spectrum and random
phase, which have been shown to provide good decorrelation when used
as a filter [
100
]. In this approach, an input signal is convolved with a
short sequence of decaying white noise. Here again, the length of the noise
sequence should be kept relatively short to avoid undesired artefacts on
transients and the introduction of a reverberation effect. The challenge
of decorrelating signals containing transients has led to the development
39
Decorrelation
Figure 3.4. Example of a velvet-noise sequence.
of decorrelation methods where the transient elements of signals are
separated before applying decorrelation [109].
The Fourier transform of a signal is a complex variable, which can be
represented in the polar form using the magnitude and phase at each fre-
quency. As such, this is a convenient domain to perform phase modification
without changing the magnitude [
110
,
111
]. This decorrelation method
may be performed at minimal computational cost in applications where
signals are already processed in the time-frequency domain [
110
]. Recently,
physically-informed decorrelation filters were proposed to account for the
specific phase relationship between loudspeakers in a room [112].
3.4 Velvet-Noise Decorrelation
In Publication I (p.99), the use of velvet noise, a special type of random
sparse noise introduced in [
113
], is proposed as a decorrelation method
(Fig. 3.4). With properties similar to white noise, velvet-noise sequences are
suitable to randomize the phase of a signal without significantly impacting
its spectrum. Since velvet noise only contains a few non-zero elements, it
has the benefit of reducing the computational cost of the convolution [
113
].
To form velvet noise, a sequence is divided into windows of fixed length,
where each window contains exactly one impulse at a random location.
This guaranties a smoother distribution of impulses compared to other
types of sparse noise and prevents larger gaps from occurring. Here, the
density parameter controls the window length, and the specific location
and sign of each non-zero impulses is randomized separately.
A velvet-noise sequence (VNS) is created using a density parameter
ρ
,
which determines the length of a windowing function that constrains the
locations of individual impulses. Since every window contains exactly
40
Decorrelation
Figure 3.5.
(a) A decaying VNS (black) and same sequence after the optimization process
(blue) and (b) the magnitude spectrum of both sequences.
one impulse, the maximum distance between two impulses is
2ρ−1
. The
location of the mth impulse is formulated as [114]
v(m)=mfs
ρ+r,(3.8)
where
represents the rounding operation,
fs
is the sample rate, and
r
is
a random value constrained to be within a window through [114]
r∈0,fs
ρ−1 ,(3.9)
and a separate random process determines the sign of each impulse. Thus,
the sequence contains values that are either
−1,0,
and
1
. In Publication I,
several variants of decaying velvet-noise sequences are proposed in order
to better preserve transient signals.
While velvet noise has a relatively flat spectrum [
114
], the short decaying
sequences introduced for decorrelation in Publication I are not perfectly
flat. For this reason, in Publication II (p.109), the use of an optimisation
function is proposed to slightly modify the location of individual impulses
and minimize the spectral coloration of a sequence.
Using this optimization method, a large set of sequences is generated
from which we find the best pairs for use as decorrelation filters. In
Figure 3.5, we see the decaying VNS of the original method, in black, and
the sequence resulting from the optimization process, in blue.
Once a set of sequences is obtained, they are further analyzed to rate their
overall flatness and to find the pairs with the lowest coherence. As such,
a compromise between a flat response and low coherence is required. In
Figure 3.6, we see the impact of the optimization process on the coherence.
41
Decorrelation
Figure 3.6. (a) Two optimized VNSs and (b) their coherence.
In Publication II (p.109), two perceptual studies were conducted to eval-
uate the perceived flatness of the sequences and the performance of the
method in the context of stereo up-mixing. Overall, the method has been
found to provide decorrelation comparable to white noise while using be-
tween 76% and 87% fewer floating-points operations (FLOPS) compared to
a segmented FFT-based convolution used for white-noise decorrelation.
42
4. Artificial Reverberation
The propagation of sound energy in a room produces a large number of
echoes with increasingly decaying amplitudes over a short period of time
(Equation 2.12). While the early reflections are considered perceptually
salient [
1
], the high density of reflections during the late stage of propaga-
tion is perceived as its own complex auditory cue called reverberation.
Methods capable of producing such a reverberation effect are used in a
diverse range of applications with various requirements [
115
,
116
,
117
,
118
,
119
]. In music production, for aesthetic reasons, it is desirable to produce
ideally dense and rich reverberant sound fields with tuning capabilities
that may go beyond what can be experienced in real life [
118
]. Another
application for reverberation is the precise simulation of acoustics during
the design phase of new architectural structures [120].
In this chapter, we focus primarily on methods capable of reproducing
realistic reverberant sound fields at low computational cost in the context
of spatial and immersive sound reproduction, which includes virtual and
augmented reality. We begin with a brief historical summary of early
reverberation methods. This is followed by an overview of reproduction
techniques, which includes capturing room IRs as well as simulating IRs
using virtual acoustics methods or artificial noise.
An in-depth review of delay-based reverberators is given in Section 4.5,
followed by a detailed survey on how these methods have evolved into com-
plex modern approaches capable of producing multichannel sound fields for
real-time interactive applications. Contributions made in this dissertation
to the topic of noise-based and delay-based artificial reverberation are
given in their respective sections.
4.1 Historical Overview
Artificial reverberation is an audio effect with a rich history that continued
to evolved throughout the last century, constantly adapting to the growing
capabilities and needs of emerging technologies [
4
,
121
]. Artificial reverber-
43
Artificial Reverberation
ation can be used to mimic anything from the precise frequency-dependent
energy decay of a room to imaginary spaces with ethereal acoustics, thus
providing a vast set of tools to creatively control the aesthetic sense of
space in an audio production.
Today, more immersive applications such as augmented reality requires
a new level of realism from artificial reverberators to allow reproduced
sounds to blend with existing spaces. For efficiency, many artificial re-
verberation techniques focus primarily on the reproduction of the more
perceptually salient aspects of room reverberation.
The first artificial reverberation method relied on physical spaces, called
reverberant rooms [
122
,
4
]. A reverberant signal was captured by using
a loudspeaker and a microphone placed inside the room to playback the
dry signal and record the reverberant sound from the room. To obtain an
aesthetically pleasing reverberation effect, these rooms had to be carefully
designed to promote the diffusion of energy and avoid the fluttering echoes
caused by parallel walls. The ratio between of the wet reverberant signal
and the original dry signal was adjusted during post-production.
While using a reverberant room can yield a natural-sounding effect, the
need for a physical space for this approach has some obvious limitations.
For instance, the absorbing materials inside the room need to be altered in
order to modify the decay properties of the space.
For this reason, other media such as metallic plates and springs were
later used as they provide interesting reverberation effects in a more
compact form. Although the plates were still relatively bulky, the property
of metals offers unique reverberation properties. This explains why both
plates and springs are still used today and why they have been recreated
digitally for more practical use [123, 124, 125].
4.2 Captured Impulse Response
As covered in Section 2.1, reverberation can be reduced to an FIR filter
containing the response of a specific room (Equation 2.21). So a modern
alternative to using reverberant chambers is to capture the IR of an exist-
ing room. To record an IR, a simple technique is to generate a very sharp
transient sound in the room, such as a balloon pop (Section 2.2).
A captured IR is later used as an FIR filter to convolve with any dry
sound [126]. In the time domain, convolution is defined as
y(t)=
∞
n=1
x(n)h(t−n),(4.1)
where
x
is a dry signal and
h
is the impulse response. Obviously, time-
domain convolution leads to a prohibitive computational cost as
h
becomes
longer. A more efficient approach is to perform the convolution in the
44
Artificial Reverberation
frequency domain, where convolution is defined as
y= IFFT(FFT(x)FFT(h)),(4.2)
which requires both the input and IR to be first converted to the frequency
domain through an FFT, followed by an inverse fast Fourier transform
(IFFT). The FFT/IFFT operations introduce latency, but existing methods
allow partitioning the convolution into shorter time windows, making the
issue of latency negligible in practice [126, 127].
To capture an IR, another approach is to use a frequency sweep as a
stimulus in the room. This ensures all modes in the room (Equation 2.10)
are activated and captured, preferably with the same amount of energy
[
30
,
31
]. The signal recorded using the sweep must first de-convolved to
obtain the IR.
The presence of ambient noise during the recording process and the
noise introduced by the recording equipment, are carried into the captured
IR and perceived as an infinite reverberation tail during reproduction.
To limit the impact of noise, it is recommended to capture IRs using a
logarithmic sine sweep signal, and to increase the length of the sweep to
improve the signal-to-noise ratio (SNR) [31].
Since the late reverberation tail is similar to white noise, it is also
possible to synthesize it artificially using Gaussian noise [
128
]. De-noising
techniques require the detection of a noise threshold to determine the
optimal part to be replaced and a spectral envelope informed by frequency-
dependent decay rates, which may also be applied to SIRs [37, 38].
When an IR is captured, the specific location of the loudspeaker and mi-
crophone determines the source-listener location during reproduction. In
an interactive application, where both the sound sources and the listener
are moving, many source-listener locations need to be captured. However,
in a diffuse field, this can be simplified since the late reverberation is
uniform in all locations. When the late reverberant sound field is homoge-
neous, only the early reverberation is location-dependent. For this reason,
it is common to combine a static late IR with dynamic early reflections
obtained by interpolating the early reverberation from a set of IRs [
129
]
or generated using geometric acoustics [
130
]. However, for heterogeneous
sound fields, dynamic reproduction requires many recording locations and
interpolation between the IRs at those points [131, 132].
In multichannel sound reproduction, a microphone array, such as an
SMA, is used to record a SIR. During reproduction, a mono sound source is
convolved with the captured SIR, and the loudspeaker location during the
capture determines the incident direction of the reproduced sound source.
For better performance, segmenting a SIR into shorter FIR filters has been
proposed, allowing the use of HRTFs with lower spatial resolution for late
segments in a binaural reproduction [
133
], and lower-order encoding in
Ambisonics reproduction [134].
45
Artificial Reverberation
One shortcoming of Ambisonics reproduction occurs when lower order
Ambisonics is reproduced on a high-order loudspeaker array, which pro-
duces highly correlated output signals. However, as covered in Section 2.3,
a key characteristic of late reverberant sound fields is that they produce
low coherent signals. To overcome the coherence limitation of Ambisonics,
several parametric enhancement methods have been developed.
With the spatial impulse-response-rendering method (SIRR)[
135
], a first-
order Ambisonics SIR is analyzed using short-time windows and a filter
bank to find the dominant direction for each time window and frequency
band. This yields a set of time- and frequency-dependent sparse IRs that
can be used in reproduction. A similar approach, the spatial decomposition
method (SDM) [
136
], uses broadband analysis and extends the analysis
to arbitrary microphone arrays. Using parametric analysis methods, the
direction of arrival of early reflections can be estimated [137, 138].
In some applications, instead of using a room IR, a continuous signal is
recorded for reproduction, implying that both the sound source and the
reverberation in the room are captured. Here, to enhance the spatial coher-
ence during the reproduction of recorded Ambisonics signals, an analysis
process extracts the reverberant part of the signal and a decorrelation
filter (Section 3) is used to minimize its coherence [
58
]. This approach
was recently extended to higher order Ambisonics [
139
] and more advance
parametric methods [140, 141].
4.3 Virtual Acoustics
Capturing IRs requires access to a physical space, good recording condi-
tions, and potentially many recording points. For these reasons, it may be
desirable instead to simulate numerically the propagation of sound in a
room. The field of acoustic simulation, called virtual acoustics, is largely
divided into two families of methods, geometric and wave-based methods
[5].
4.3.1 Geometric Methods
In geometric acoustics [
5
,
142
], the behavior of sound waves are approxi-
mated to a set of ideal reflection paths from a sound source to a listener.
This offers a realistic rendering of the propagating of high frequencies when
abstracting more complex elements such as diffraction [143, 13, 144, 5].
The image source method (ISM) considers each wall as a rigid surface,
and the algorithm simulates reflections by using virtual sources that are
mirror image positions (Fig. 4.1), from which the time-delay and incident
angle of individual reflections are calculated [
13
]. With the ISM, the
maximum reflection order determines the complexity of the approximation
46
Artificial Reverberation
Figure 4.1. Image sources from reflected rooms in [13].
and thus the computational cost.
Ray tracing is another geometry-based method that sends rays through-
out a virtual geometry and finds possible reflection paths between a source
and a listener [
143
,
15
]. The computational cost is minimized by limiting
the number of rays and reflection surfaces used to sample the space.
While neither ISM nor ray tracing inherently simulate the full physical
characteristics of sound propagation, they efficiently produce specular
reflections simultaneously approximating the stochastic nature of the
late reverberation [
145
]. Geometric methods can be combined with more
complex analytic methods to simulate specific aspects of the propagation
such as scattering [
15
] and edge diffraction [
146
,
147
]. Ray tracing can
also be generalized into the more efficient beam-tracing method, which
takes advantage of the spatial coherence of a geometric scene to expand the
rays into a larger convex polyhedral shape that splits into smaller beams
when multiple intersecting objects are detected [
148
,
149
,
150
,
151
]. The
radiance transfer method [
152
], inspired from light-propagation modeling
techniques, estimates the properties of a reverberant sound field that are
used to inform other reverberation methods [153, 154, 155].
4.3.2 Wave-based Methods
To simulate the physical properties of sound propagation as acoustical
waves, wave-based methods perform numerical simulations which approxi-
mate the wave-behavior of sound transmitting in a medium [
5
]. Compared
to geometric methods, wave-based simulations are inherently well suited
for the simulation of diffraction, which plays an important role in the prop-
agation of low frequencies when a scene contains objects and openings.
47
Artificial Reverberation
One of these methods, called finite-difference time-domain (FDTD) [
156
],
solves the wave equation (Equation 2.1) on a fixed grid of points using
finite difference operators. The simulation results in sound pressure values
for each point that generates a sound field that evolves over time. However,
the discretization of wave propagation on a fixed grid has limitations due
to the inherent numerical errors of difference operators, such as staircase
effects at boundaries [
157
] and numerical dispersion in high frequencies
[158].
Another approach, the finite-volume time domain, performs a similar
task using a set of flux terms to define the particle velocities at the bound-
aries of a three-dimensional cell volumes [
159
]. Similarly, finite-element
methods commonly operate in the frequency domain to solve the Helmholtz
equation (Equation 2.2) [5].
Due to their potential for a high level of accuracy, wave-based methods are
suitable for applications, such as architectural acoustics, where accuracy
is a priority over efficiency [
7
]. However, for real-time applications, such
as virtual and augmented reality, the computational cost of wave-based
methods becomes prohibitive. Nonetheless, it is also possible to perform
the simulation offline to generate a dataset that can used in real-time to
inform a parametric reverberation algorithm [66].
4.4 Artificial Noise
During the propagation of sound waves, the density of echoes increases
exponentially over time (Equation 2.12), and due to this high density, an
IR eventually acquires a distribution of impulses over time similar to
stochastic noise (Figure 2.4) [
160
,
161
,
162
,
17
,
163
,
144
]. Based on this
attribute, artificial reverberation algorithms were developed leveraging
the use of random-noise sequences. This section explores this family of
methods in more detail.
Random signals, such as Gaussian noise, have a flat frequency spec-
trum and random phase. To produce a reverberation effect, a frequency-
dependent envelope is used to control the decay and create an artificial IR.
This approach was used in [
128
] to replace the segment of a recorded IR
containing background noise. In this model, an EDR analysis (Eq. 2.23)
is performed to extract a frequency-dependent envelope from an earlier
segment of the IR, that is then used to filter Gaussian noise to replace the
IR after the noise floor. Just as for regular captured IRs, a convolution
operation is used to convolve the input signal with the noise and to create
a reverberated signal.
Other types of random noise offer similar characteristics with a lower
computational cost. In particular, a category of noise known as sparse
noise, containing a large proportion of zero values, have been shown to
48
Artificial Reverberation
x(n)z−m1VNS1H1(z)
g1
y(n)
z−m2VNS2H2(z)
g2
.
.
..
.
..
.
..
.
..
.
..
.
.
z−mNVNSNHN(z)
gN
Figure 4.2.
Flow diagram of the segmented velvet-noise reverberator proposed in [
165
].
The coloration filters
Hi,j
are designed to match the magnitude response from
a segment of an existing IR. The final output
y
also goes through a series of
cascaded all-pass filters (omitted in this figure).
retain a perceptually flat spectrum on the condition that a sufficient density
of non-zero values is maintained to avoid audible artefacts [
164
]. Since
the noise sequence is sparse, the zero elements are ignored to efficiently
compute the convolution in the time domain with no latency.
In [
164
], a recursive sparse pseudo-random FIR filter and a low-pass filter
is proposed to replace late reverberation, whereas the early reflections are
handled by a parallel signal path using an FIR filter. The pseudo-random
signal only contains the values
−1
, 0, and 1, and the output is filtered
with an exponentially decaying envelope before recirculating again. A
density parameter controls the probability of having a non-zero element in
a given sample. Finding the optimal density is an important aspect of this
type of reverberator since it has a direct impact on the computational cost.
An average density of at least 2000 impulses per second has been found
necessary for this type of recursive sparse-noise reverberator [164].
In [
113
], a similar approach was used with a new type of sparse noise
called velvet noise. As described in Section 3.4, a velvet-noise sequence
is divided into windows of fixed length, each containing one impulse at
a random location (Figure 3.4). The segmentation guarantees a more
uniform distribution of impulses over time and prevents larger gaps from
occurring, a potential issue in fully random sparse noise. This formulation
of sparse noise was found to be perceptually smoother than Gaussian noise
in a listening experiment, which inspired the name of the method [
113
].
A more in-depth perceptual study was later conducted to investigate the
minimum density of non-zero elements required to obtain perceptually
smooth velvet noise, which was found to be around 2000 impulses per
second [114].
More complex reverberators using velvet noise were developed later.
Several methods are proposed in [
166
] to make use of multiple noise
sequences and prevent the audible periodicity that occurs when using a
single recursive noise sequence. In [
165
], the reverberation is split into
49
Artificial Reverberation
x(n)z−m1VNS1ΔH1,2(z)
g1
y(n)
z−m2VNS2ΔH2,3(z)
g2
.
.
..
.
..
.
..
.
..
.
..
.
.
z−mNVNSNΔHN(z)
gN
Figure 4.3. Flow diagram of the segmented velvet noise reverberator presented in Publi-
cation III. The cascaded filters
ΔHi(z)
are designed to match to difference in
coloration between segments. The final output
y
also goes through a series of
cascaded all-pass filters (omitted in this figure).
N
segments over time. Each segment is delayed and filtered by a VNS
and a coloration filter. The coloration filters
Hi
are designed to reproduce
the spectral envelope from a corresponding segment in an existing IR
(Figure 4.2). Due to the filtering, the computational cost remains close to
that of a segmented FFT-based convolution. However, the required memory
is much smaller. VNSs can also be interleaved to create natural sounding
reverberation at an even lower computational cost [
167
]. Since it was first
introduced, velvet noise has also inspired a variety of new applications
beyond reverberation, such as decorrelation (Section 3.4), time-stretching
[168, 169], and 1-bit sound [170].
Improving on the previous model presented in [
165
] (Figure 4.2), a new
formulation of the segmented velvet-noise reverberation was presented
in Publication III (p.119). The reverberator is modified to use a cascaded
filter structure, that reduces the required number of impulses and allows
further computational saving. The filters are designed as differential filters
ΔHi,j (z)
corresponding to the spectral differences between two segments
of the segmented impulse response (Figure 4.3). The segmentation is also
improved by using low-order linear-prediction analysis to identify the ideal
locations of the segment boundaries. A listening test is also performed to
compare the recorded IR to an artificial one generated using the method.
The results showed that the recorded IR was easily identifiable, especially
when the stimuli have sharp transients, but the differences were rated as
small on average.
4.5 Delay Networks
Back in the middle of the 20th century, electro-mechanical devices, such as
magnetic disks and tapes, were used as delay units in sound reproduction.
In such devices, a signal is converted through electro-mechanical means
50
Artificial Reverberation
x(n)z−m1y1(n)
z−m2y2(n)
z−m3y3(n)
z−m4y4(n)
g
Figure 4.4.
Flow diagram of an early multichannel reverberator using a recirculating
multi-tap delay [
115
]. In this figure, the multi-tap delay is illustrated using
separated delay lines for illustrative purposes.
and is stored on the delay unit for a determined amount of time before
being sent to a loudspeaker. By mixing the delayed signal back to the input,
the signal is repeated endlessly, which creates a simple reverberation effect
[171].
A common application of these early artificial reverberators was to en-
hance the perceived stereo image from venues equipped with a loudspeaker
system. One of the first delay-based reverberation methods used a single
delay unit, acting as a ring buffer, and the output from different points on
the delay unit, or taps, were taken to feed different channels (Figure 4.4).
The last delay-tap was used with a potentiometer to control the signal
recirculating to the input of the delay unit [115].
This technique lowered the coherence between different output channels
in the room, an important characteristic of a reverberant sound field
(Section 2.3). However, one issue of this design was that the recirculating
signal also produced a comb-filtering effect that impacted the magnitude
spectrum of the reproduced signal.
As the production cost of these delay units became more accessible,
more complex delay-network reverberators were created. An important
milestone came with the publication of the first reverberation algorithm
offering colorless reverberation [
172
,
16
], where Schroeder introduced the
AP filter (see Section 3.3 and Figure 3.3) that adds a feed-forward path to a
feed-back comb filter to produce a spectrally-flat delayed signal. However,
from Equation 3.7, we see that an AP filter produces an uneven distribution
of phase delays across frequencies. Thus, several AP filters in cascade
with different delay lengths are recommended to ensure a good modal
distribution suitable for reverberation [16].
The combination of AP filters in series provides a good method to cre-
ate increasingly dense copies of a signal without directly impacting its
spectrum, thus creating colorless reverberation. Having a colorless rever-
beration prototype is a fundamental building block to build an artificial
51
Artificial Reverberation
x(n)AP1(z) AP2(z)z−m1y1(n)
g1
z−m2y2(n)
g2
z−m3y3(n)
g3
z−m4y4(n)
g4
Figure 4.5.
Schroeder’s multichannel reverberator [
17
]. In his original formulation, each
output
yi
is first copied and its phase inverted to double the number of signals
going through a mixing matrix, which would further combine the signals to
produce a total of 16 dense and lowly correlated output channels.
reverberator where the frequency-dependent attenuation is controlled sep-
arately. In his original design, Schroeder added feed-forward comb filters
in parallel to introduce frequency-dependent attenuation. The comb filters
can be placed either at the beginning or the end of this system due to its
linear and time-invariant property. To obtain multi-channel outputs, a
mixing matrix is placed at the end of the signal flow to combine different
signal paths, which creates output signals with low correlation [
17
] (Fig-
ure 4.5). This reverberation model was the starting point of increasingly
complex delay-network reverberation methods that are still being used
and studied today.
An important characteristic in artificial reverberation is mixing time. As
discussed in Section 2.4, it highlights the moment where the reverberant
sound field transitions into a more diffuse state of reverberation. For
artificial reverberation, it refers to the moment of transitions between
distinct reflections and more temporally dense echoes [144, 128, 118, 59].
Many reverberation methods separate the early reflections from the
late reverberation to allow each part to be processed separately. In [
120
],
Schroeder added a multi-tap delay at the beginning of the system to ensure
that the perceptually critical early reflections were reproduced.
Through the various combinations of gains, delays, and filters, delay
networks offer a wide range of tuning capabilities while being efficient.
This makes them convenient to use in creative design. However, the vast
realm of possible arrangements posed a design challenge for the early
delay networks, which had to be tuned perceptually [
116
,
173
]. In [
174
],
Moorer further studied these structures and proposed replacing the gains
with frequency-dependent absorbent filters, inspired by the filter design
52
Artificial Reverberation
x1(n)z−m1y1(n)
x2(n)z−m2y2(n)
.
.
..
.
..
.
.
xN(n)z−mNyN(n)
g g ... g
A
Figure 4.6. Flow diagram of Gerzon’s delay network [180].
proposed in [
175
] to account for the attenuation of high frequencies due to
air absorption. Moorer also provided a specific set of practical values for
the delays and gains yielding pleasing perceptual qualities, which made
delay networks more accessible.
The commercialization of good-sounding reverberation units means that
the practical design of many delay networks are not publicly available for
researchers to explore, but a few exceptions exist [
176
,
177
]. In [
178
], the
design of a popular commercial reverberator is presented and it demon-
strates how complex they can get.
Through a series of articles [
179
,
180
,
181
], Gerzon introduced the idea
of organizing sets of delay lines in a recirculating network to create a
reverberator capable of producing high density of echoes from a small set
of interconnected delay lines. In this design, the delay lengths were chosen
to be mutually prime, which was more challenging at the time because
digital delay units were sold with 5-ms size increments.
The recirculating matrix
A
(Figure 4.6) controls the feedback gain for
each input-output combination of delays. The matrix is also constrained
to be orthogonal to ensure lossless recirculation. A practical value of
0.7
is given for the recirculating gain
g
to obtain a pleasant sounding
reverberation.
Stautner proposed a delay-network structure for quadraphonic sound
reproduction [
182
]. In this design, each delay path is connected to an
output channel spatially distributed around the listener.
The signal from the direct sound is inserted into the system through the
input
xi
, which corresponds to its incident direction (Figure 4.7). Early
reflections are simulated using the ISM [
13
] to calculate the pre-delay, at-
tenuation, and incident direction for each reflection. These early reflections
53
Artificial Reverberation
x1(n)z−m1F1(z)y1(n)
x2(n)z−m2F2(z)y2(n)
x3(n)z−m3F3(z)y3(n)
x4(n)z−m4F4(z)y4(n)
g g g g
A
Figure 4.7.
Flow diagram for Stautner’s quadraphonic delay network [
182
]. The input
xicontains both panned direct sound and a set of early reflections generated
using geometrical acoustics positioned at the nearest input channel. The
filters
Gi
combine the low-pass, band-pass, and high-pass filters from the
original formulation.
are also distributed in the system through the xi(n)inputs.
In this method, each channel feeds the input of its two neighbouring
delay path resulting in the orthogonal recirculating matrix
A=⎡
⎢
⎢
⎢
⎢
⎢
⎣
011 0
−100−1
100−1
01−10
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,(4.3)
where some of the recirculating paths invert the phase of the signal. The
attenuation gain used in the recirculation is set to
g< 1
√2.(4.4)
Absorbent filters
Fi(z)
are used to attenuate each delay line. In the article,
this correspond to a set of low-pass, band-pass, and high-pass filters tuned
with the absorbent coefficient of wall materials, and a low-pass filter that
accounts for air absorption. The article also suggests continuously varying
the lengths of the delay lines slightly (
≤1ms
) to vary the peaks in the
frequency response and minimize flutter echoes.
In [
8
], Jot made a great leap forward in the formalism of feedback-
delay networks (FDNs), allowing them to reach a level of popularity still
maintained today. The FDN adds input and output gains
bi
and
ci
as well
as a path for the direct sound with attenuation
d
(see Figure 4.8). The
article provides a formal definition for the design of absorbent filters, and
an in-depth analysis of the frequency response of the system is also given.
54
Artificial Reverberation
d
x(n)
b1
z−m1G1(z)
c1
y(n)
b2
z−m2G2(z)
c2
.
.
..
.
..
.
.....
.
..
.
.
bN
z−mNGN(z)
cN
A
Figure 4.8. Flow diagram of a generalized feedback delay network (FDN) [8].
The exponential decay of energy is produced from a set of attenuation
parameters defining frequency-dependent decay rates as
gdB(ω)=−60 1
T60(ω)fs
,(4.5)
where
T60(ω)
is the specified frequency-dependent reverberation time. The
per-sample linear attenuation gain is obtained from
glin(ω)=10
gdB(ω)
20 ,(4.6)
which is used to calculate the attenuation required at the output of each
delay line using
gi(ω)=(glin(ω))mi,(4.7)
where miis the length of the delay line.
In turn, these gains are used to design absorbent filters for their corre-
sponding frequencies. For instance, the transfer function of a first-order
low-shelf filter is specified with [183]
G(z)=gtan( ω
2)+√g+[gtan( ω
2)−√g]z−1
tan(ω
2)+√g+[tan(ω
2)−√g]z−1,(4.8)
for an attenuation coefficient
g
and a center frequency
ω
. Finally, in the
z-domain, the system is described as
y(z)=cTG(z)s(z)+dx(z),(4.9)
s(z)=Z(z)[AG(z)s(z)+bx(z)],(4.10)
55
Artificial Reverberation
where s(z),b, and care
s(z)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
s1(z)
s2(z)
.
.
.
sN(z)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,b=⎛
⎜
⎜
⎜
⎜
⎜
⎝
b1
b2
.
.
.
bN
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,c=⎛
⎜
⎜
⎜
⎜
⎜
⎝
c1
c2
.
.
.
cN
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(4.11)
and Z(z), and G(z)are
Z(z)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
z−m10··· 0
0z−m2··· 0
.
.
..
.
.....
.
.
00··· z−mN
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(4.12)
G(z)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
G1(z)0··· 0
0G2(z)··· 0
.
.
..
.
.....
.
.
00··· GN(z)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
.(4.13)
This formulation from [
8
] has a single input and a single output, but this
design is easily expanded to multiple inputs and outputs by considering
the formulation in [182].
The lossless properties of the recirculating matrix was further developed
in [
184
] with the introduction of circulant matrices, and [
185
] presented a
general definition for the requirements of lossless matrices. Since these
networks are designed as complex feedback filters, they do not have AP
properties. More complex network designs with AP properties were pro-
posed in [180, 186] and further formalized in [187].
To achieve natural sounding reverberation it is important that the system
quickly reaches a sufficiently high echo density, which can be done by using
shorter delays or adding more delay lines. To avoid multiple impulses
landing on the same sample, thus losing density, it is important to ensure
that delay lengths are mutually prime [16].
However, this is not sufficient to prevent different recirculation paths
from landing on the same sample. For example, an impulse going through
the first and second delay line will land on the same sample as one going
through the second delay followed by the first. To minimize this occur-
rence, [
188
] proposed to quantifying this overlap to help select an optimal
set of lengths. Recently, [
189
] proposed adding small delays in the re-
circulating matrix to ensure that every recirculation path has a unique
combined length. However, these small delays have an impact on the
modal distribution, which was improved in [190].
The modal density is another key characteristics of reverberation [
191
].
A popular method to increase the modal density in delay networks is
56
Artificial Reverberation
d
x(n)B1(z)z−m1G1(z)C1(z)y(z)
B2(z)z−m2G2(z)C2(z)
.
.
..
.
..
.
.....
.
..
.
.
BN(z)z−mNGN(z)CN(z)
A
Figure 4.9.
Flow diagram of the velvet-noise FDN proposed in Publication IV.
Bi(z)
and
Ci(z)contain short sequences of velvet noise.
to vary slightly the length of delay lines over time thus producing more
modes [
182
,
192
,
116
,
178
], which requires the use of fractional delay lines
[
193
]. In [
194
], a time-varying recirculating matrix is used to modulate
the recirculation coefficients that changes the amplitude of modes over
time. To reproduce the frequency-response of real rooms, the absorbent
filters require precise design, usually through the use of a set of high-order
equalization filters [195, 183, 196, 197, 198].
4.5.1 Velvet-Noise Feedback-Delay Network
As delay networks often suffer from a slow echo density build-up, the use
of more delay lines is necessary to ensure that a sufficient echo density is
reached in a short period of time. On the other hand, the echo density is
a key strength of noise-based reverberators that are very dense from the
very beginning.
In Publication IV (p.139), short sequences of velvet noise are added to
the input and output of an FDN to increase its echo density. Using this
method, it was demonstrated that the number of required delay lines
could be reduced without impacting the perceptual characteristics of the
reverberator. In the proposed method, the input and output gains from
a traditional FDN (Figure 4.8) are replaced with velvet-noise sequences
Bi(z)and Ci(z)(Figure 4.9).
With this design, a 32-delay-line FDN was found to be comparable to
a 16-delay-line reverberator in the proposed model. The time-domain
convolution with the velvet-noise sequences required extra computation,
but due to the reduced number of delay lines in the system, up to a 52%
reduction of floating-point operations (FLOPS) is be obtained.
57
5. Hybrid Reverberation Methods
The methods detailed in the previous chapter offer perceptually motivated
and tuneable reverberation. Their design was substantially simplified
when Jot introduced the formulation for absorbent filters, providing a
comprehensive set of parameters for the system using
T60(ω)
as the target
[
8
]. However, some applications of artificial reverberation require a more
physically-informed reproduction of a reverberant sound field. This means
that certain physical attributes of a room, such as its dimensions, are used
to specify some of the properties of the reverberator.
This chapter describes hybrid reverberation algorithms combining delay
networks with geometrical acoustics as well as methods capable of produc-
ing more complex multichannel outputs, such as Ambisonics and binaural
signals. Reverberation algorithms are grouped into sub-categories, al-
though many of the methods described here overlap in various aspects
used in the categorization.
The chapter concludes with a section on directional reverberation and
contains a detailed description of the contributions made to this topic in
Publication V, VI, and VII. In these publications, an analysis framework is
proposed to assess the perceptual importance of directional characteristics
in late reverberation, along with reverberation algorithms leveraging the
use of delay networks to reproduce these characteristics.
5.1 Geometrically-Informed Reverberators
In [
120
], Schroeder suggested the use of geometrical acoustics to specify
the multi-tap output of a delay line. Each tap accounts for an early reflec-
tion in a simulated room. The delayed signals serve as input for a late
reverberation block (Figure 4.5). Because early reflections are a percep-
tually critical aspect of reverberation [
1
,
64
], this design was proposed to
offer simulated reverberation in the context of architectural design.
The reproduction of moving sources in reverberant rooms was studied by
Chowning in [
199
]. Based on the location of the sound source relative to
59
Hybrid Reverberation Methods
the listener, frequency modulation is applied to produce the Doppler shift of
a moving source. The source signal is then panned between loudspeakers
on a four-channel system using pairwise panning defined as [199]
ya=
1−1
2'1+tan'θ−θmax
2((,
yb=
1
2'1+tan'θ−θmax
2((,
(5.1)
where
θ
is the source’s incidence angle in the horizontal plane,
θmax
is the
maximum angle between two adjacent loudspeakers, and
ya,y
b
are two
adjacent loudspeakers encompassing
θ
. The amplitude of the direct sound
is also modulated based on the distance
D
between the source and the
listener (1/D).
Each output channel has an independent reverberator. The input of each
of the reverberators consists of a mix between the panned signal, which is
distributed to two channels as determined by Equation 5.1, and attenuated
using
1−1
D b
√D,(5.2)
where
b
is a parameter to adjust the overall level of the reverberant signal.
Similarly, the dry signal is also mixed into all channels using
1
D b
√D,(5.3)
thus controlling the mix between the dry and wet signal. Here, the spe-
cific design of the late reverberator is not documented, and based on the
reference in the article a Schroeder reverberator is assumed.
5.2 Binaural Reverberation
In a series of articles, Kendall developed a binaural reverberator using
the ISM to specify a delay network [
200
,
9
,
201
,
202
]. In this design, a
set of delay lines are set from the distance of the first and second-order
reflections, calculated using the ISM. The incident direction of individual
reflections also determines their directions in the output of the system. A
binaural output is produced by panning the direct sound and first-order
reflections using HRTFs (Figure 5.1).
Together, the first-and second-order delays form a network of recirculat-
ing delays that produces decorrelated late reverberation. In the network,
the second-order delays are split into two recirculating delay lines, each
60
Hybrid Reverberation Methods
x(n)Direct sound
Early reflections
Panner
y(n)
Delay network
/
M
\
M
\
N
/
K
Figure 5.1. General building blocks of an hybrid binaural reverberator.
representing the propagation to and from a wall. Having two recircu-
lating paths yield more delayed signals and thus a higher echo density.
Frequency-dependent filters are placed before and after each delay path to
account for wall and air absorption, respectively.
The early 1990s was a prolific time for binaural reverberators using
geometrical and statistical methods [
203
,
204
,
205
,
206
]. In [
203
,
150
],
Martin divided the reverberation into three stages: an early reflection
stage producing a fixed amount of geometrically-informed reflections, a
pseudo-statistical stage using linear prediction to estimate higher-order
reflections, and finally, a late reverberation stage using Gaussian white-
noise to produce two decorrelated decaying signals y1, and y2.
The two noise sequences
y1
, and
y2
were mixed together to reintroduce
some correlation and produce binaural signals using
yL(n)=cos1
2arcsin(γIACC)y1(n)+sin1
2arcsin(γIACC)y2(n),
yR(n)=sin1
2arcsin(γIACC)y1(n)+cos1
2arcsin(γIACC)y2(n),
(5.4)
where
γIACC
is a measure of the IACC, such as the one defined in Equation
2.33. In this system, the IACC is estimated from the binaural signal
produced in the second stage of reverberation.
In [
204
], Gardner calculated image sources up to the fifth order and
pruned by merging taps occurring within 1 ms of each other and keeping
only the 40 loudest taps. These reflections are panned across a multichan-
nel loudspeaker system using intensity panning [
207
], and the signal is
fed to a delay network, tuned to follow the appropriate decay rate.
In [
208
,
209
], Jot et al. cascaded early reflections and late reverberation
in three stages. Starting from early reflections, which are panned either
left or right, each reflection also serves as the input to the individual delay
lines of a delay network. The output of the first delay network is used as
the input to a second late reverberation unit. The final output is encoded
61
Hybrid Reverberation Methods
to either binaural, multichannel, or Ambisonics format. For a binaural
output, the late reverberation stage is filtered in order to adjust the IACC
between the left and right output channels. The reverberator was also
built into a larger system, producing moving sources, source directivity,
distance attenuation, and room equalization.
A reverberator reproducing the characteristics of a captured binaural
IR is proposed by Väänänen et al. in [
210
]. In this method, a set of early
reflections are produced at each ear by analyzing the individual peaks
contained in the first 100 ms of the binaural IR. The direct sound and
the early reflections are then fed to a delay network that produces the
late reverberation. In this delay network, each recirculating delay path
contains an AP and a low-pass filter, and the latter is specified by frequency-
dependent decay rates measured in the captured IR. Here, instead of using
a recirculating matrix, the output of all the delay paths are summed
together and fed back to the input of each delay.
In [
211
,
212
], Christensen et al. proposed a multichannel reverberator
with distinct signal paths for early and late reverberation. The last stage
of the reverberator has a directional encoding stage that supports different
output formats such as Ambisonics, HRTF, and channel-based formats
through panning [
105
]. A key distinction with other models is that each
channel has a separate reverberation block. Hence, this reverberator
is capable of producing anisotropic decaying sound fields. However, the
design outlined in the paper only specifies the intended goal as obtaining
fully decorrelated outputs by using independent delay networks.
Recently, the emergence of applications for binaural sound reproduction
has motivated the development of several binaural reverberation methods.
As with previous binaural reverberators [
200
,
208
], a key characteristic of
these reverberators is the use of individually-panned early reflections and
decorrelated late reverberation.
Recently, Hacıhabibo˘
glu et al. introduced an algorithm to cluster and
reduce a set of simulated early reflections resulting in computational sav-
ings [
213
]. Leveraging on the precedence effect, the simulated reflections
reaching the listener within a small time frame, and thus perceptually
indistinguishable, are collapsed into the same sample before reaching a
late reverberation stage.
In [
214
] by Menzer et al., the decorrelated stereo output of an FDN is
mixed using a frequency-dependent IACC control to match the spectral
envelope of an existing binaural response. In [
215
], the method is expanded
to use two cascaded delay networks. The first network is specified by the
directions and lengths of first-order reflections, and the second network
is used to produce the diffuse reverberation similar to the one in [
214
].
Finally, in [
216
], the method is modified to take multiple sound sources as
the input.
An extended FDN, informed by a set of simulated early reflections, is pro-
62
Hybrid Reverberation Methods
posed by Wendt etal. in [
217
]. Early reflections are individually processed
using HRTF filters for binaural auralization and the hybrid reverberator
expands on previous systems [
9
,
204
] by incorporating spectral filtering
based on a room’s material.
In [
218
], Borß et al. introduced a two-stage reverberator which uses a
filter bank and a set of white-noise sequences to reproduce the frequency-
dependent energy decay and IACC of a recorded binaural IR. A multichan-
nel reverberator was proposed by Oksanen et al. in [
219
], that uses velvet
noise to produce a set of direction-dependent decay characteristics from
an existing SIR. The use of decorrelated noise sequences in a parametric
binaural reverberation is further explored in [220].
In [
221
], the transfer of energy between the early reflection and the late
reverberation stage is formalized to control the amount of energy entering
the late reverberation stage based on the location of the listener in a sound
scene. In [
154
], a simplified one-stage binaural delay network, using a set
of delay lines determined by reflection paths, is perceptually compared to
the ISM and captured binaural responses.
5.3 Digital Waveguide
Another approach to specify delay networks using room properties is called
digital waveguide (DWG) networks [
192
]. DWG networks, originally a mod-
ified form of DWG filters, was first proposed by Smith as a reverberation
algorithm [
192
]. It consists of a network of interconnected bi-directional
delay lines representing physical propagation paths. The delay lengths
represent propagation time, and the connections between delay lines are
controlled through scattering junctions defining the sound pressure using
the following equation [222]:
pj(n)= 2
N
i=1 Γi
N
i=1
Γip+
i(n),(5.5)
where
Γi
and
p+
i(n)
are the admittance and sound pressure, respectively,
of individual incoming waveguides iat junction j.
The bi-directional delay lines of a DWG network are suitable to simulate
a sound source propagating outwards from a specific point in the delay
network. In its simplest form, a one-dimension DWG network is capable of
producing the behavior of a guitar string, which shares some similarities
with the Karplus-Strong algorithm [
223
,
224
]. This method was later gen-
eralized through the discretization of a two-dimensional grid space, called
DWG mesh, suitable for the physical modeling of musical instruments
[
225
]. In three-dimensional space, a DWG mesh may be used to simulate
sound propagating in a room [226, 227].
Since FDNs are the general building blocks to define a network of delays,
63
Hybrid Reverberation Methods
a DWG network may be converted to an FDN design where the scattering
junctions are equivalent to the filters and the matrix controlling the recircu-
lation of delays [
184
,
228
]. More complex DWG meshes are mathematically
comparable to FDTD under certain conditions [229].
A DWG network is formalized in [
222
] by using the location of geo-
metrically-specified first-order reflections from a shoe box. The location
of the reflection on each wall is used to create paths from the sound
source to the walls, from the walls to the listener, and between walls. The
distance of each path informs the delay-lengths in a recirculating delay
network. Another DWG network was later proposed, which simulates
sound propagation in a forest by incorporating the properties of wave
scattering around cylindrical objects in the physical model, which form
scattering junctions simulating trees [230].
The reverberation model proposed by [
222
] was later improved by De
Sena et al. in [
231
,
232
], removing some propagation paths to reduce the
computational cost and using a more intuitive approach to define the
filters at the scattering junctions where wall absorption coefficients are
used. A DWG network design was proposed in [
233
] using higher-order
propagation paths defined within for complex three-dimensional scenes.
To reduce the computational cost of three-dimensional meshes in the
context of room acoustics simulation, the use of two-dimensional meshes
has been studied in [
234
]. Various hybrid approaches combining two-
dimensional approximations, Gaussian noise, and ray tracing to replace
segments of propagation were proposed by Murphy et al. in [235].
In [
236
], a method to design unilossless and spatially-aware recirculating
matrices is proposed. Using this method, the recirculating matrix of
an FDN may be designed to simulate two interconnected rooms with
different decay rates and a controlled amount of recirculation between
them. This matrix design approach also has applications to obtain angle-
aware scattering junctions for a DWG network.
5.4 Ambisonics Delay Networks
For immersive applications, an Ambisonics reverberator offers the benefit
of supporting a wider range of output configurations, including head-
tracked binaural reproduction. In [
237
], an Ambisonics input signal with
Q
channels is first converted into discrete
K
plane waves (Figure 5.2). The
early part of the reverberation is produced by using multiple cascaded
delay stages to increase the density gradually. These early reflections
are encoded individually to Ambisonics. Multiple decorrelated signals are
obtained from a late-reverberation delay network and a gain parameter
controls the amount of the reverberant signal in different directions.
A first-order Ambisonics (FOA) reverberation is obtained using a set of
64
Hybrid Reverberation Methods
x(n)
Ambisonics
decoder
Delay network
(early reflection)
Panner
y(n)
Delay network
(late reverberation)
/
Q/
K/
K/
Q
Figure 5.2.
Flow diagram of the Ambisonics reverberator proposed in [
237
]. Thick lines
represent signals containing multiple channels.
Schroeder-type reverberators in [
238
] by distributing the output of each
reverberator spatially. A scattering matrix controls the redistribution of
reverberated signals amongst the directions to compensate for the direc-
tivity of the input signal. Three steering parameters control the output
gain between the left-right, front-back, and top-down directional axes. In
this formulation, contrary to what was proposed in [
8
], every direction
has an independent reverberation unit and the recirculating gains in the
reverberator are set across all directions and independently of the delay
length. For this reason, a longer delay line in the reverberator has a slower
decay, resulting in different reverberation times in different directions.
After encoding the final output of the reverberator in FOA, a spread pa-
rameter controls the proportion of the reverberated signal taken from the
omni-directional channel.
5.5 Directional Reverberation
The physical properties of a room determine its ability to diffuse sound
energy as waves propagate through it. Poor diffusion in a room due, for
instance, to flat rigid walls, leads to interference patterns increasing the
potential sound pressure at certain locations further away from the center
(Section 2). As such, achieving good diffusion is one of the challenging
aspects of building reverberant rooms suitable for scientific measurements
[
46
]. Whereas concert halls are built to favor diffusion, since it gives a
better musical experience, other types of rooms are well varied.
In the context of immersive sound reproduction, such as augmented real-
ity, the accuracy of reproduction is imperative to ensure that a reproduced
sound blends with the real world. Hence, it is important to assess the char-
acteristics of reverberant sound fields and understand the extent in which
we perceive their characteristics. A direction-dependent reverberant sound
field, or directional reverberation for short, is also sometimes referred to
as a non-diffuse field [54], while others use anisotropic sound fields [46].
65
Hybrid Reverberation Methods
012345
0
50
100
150
200
250
300
350 -60
-50
-40
-30
-20
-10
0
Figure 5.3.
Directional decays (
EDCdB(φ, θ)
) of the example SIR on the horizontal plane.
This section contains an overview of key contributions made to the objec-
tive and subjective analyses of sound fields as well as their reproduction
using artificial reverberation.
5.5.1 Analysis
Publication V (p.149), proposes an analysis framework to measure the
anisotropy in a captured reverberant sound field as well as a perceptual
study methodology to evaluate our capacity to hear anisotropic character-
istics. This method is also useful to analyze the output of a multichannel
reverberator and to evaluate its capacity to preserve direction-dependent
characteristics in the decay.
The proposed objective analysis method is based on energy-decay mea-
sures such as EDC and EDR (Equation 2.22 and Figure 2.23). It compares
directional decay properties to a mean in all directions to measure how
much each direction deviates from an isotropic sound field similarly to the
method proposed in [239].
From a SIR encoded in Ambisonics, a set of directional IRs are extracted
using a beamformer, as expressed in Equation 2.31. The directional IRs
are used to produce a set of directional decay curves in dB from
EDCdB(n, φ, θ) = 10 log10 'N
i=ny(i, φ, θ)2(,(5.6)
where
N
is the total number of samples in the signal. The
EDCdB(n, φ, 0)
analysis of the captured shoe-box room is illustrated in Figure 5.3. We
can see that the distribution of energy, between the analyzed directions,
diverges increasingly with time, which illustrates the difference in the rate
of decay based on the direction on the horizontal plane. The energy decays
66
Hybrid Reverberation Methods
Figure 5.4.
Energy decay deviation (
EDD(φ, θ)
), on the horizontal plane, of the example
SIR.
faster at 0◦and 180◦compared to 90◦and 270◦.
From the properties of a diffuse field (Section 2.3), the energy of an
isotropic sound field is uniformly distributed in all directions. As such, a
diffuse field’s directional decay characteristics should remain close to an
average decay curve
EDRdB(n)
. To quantify the anisotropy of a reverberant
sound field, we look at the deviation of each of the directional decay curves
from the mean. The energy decay deviation (EDD) is calculated as
EDD(n, φ, θ)=EDR
dB(n, φ, θ)−EDRdB (n),(5.7)
which yields a time- and direction-dependent measure containing nega-
tive and positive values highlighting the anisotropic characteristics of a
reverberant sound field, as illustrated in Figure 5.4. One benefit of using
an analysis based on the EDC is that it does not require any windowing
of the signal in the time domain, as opposed to other methods using the
mean of energy over a short time window. For these reasons, this method
is not characterized by a windowing function and its length. However, this
method is impacted by the Ambisonics order and the beamformer used,
which may smooth out the results over multiple directions.
In the context of human perception, taking into account a specific head-
orientation and binaural hearing is useful. Using a binaural IR, which may
be obtained from a SIR in a specific orientation, we calculate the interaural
energy-decay deviation (IEDD) from
IEDD(t, ω)=EDR
L
dB(t, ω)−EDRR
dB(t, ω),(5.8)
67
Hybrid Reverberation Methods
where
ω
is a frequency band and
EDRL
dB
is computed from the left channel
of the binaural IR and the
EDRR
dB
from the right channel. In order to get a
general idea of the energy deviation per frequency, the root-mean-square
in time is calculated as
IEDD(ω)=
1
T
T−1
t=0
IEDD(t, ω)2.(5.9)
The EDD and IEDD measures are useful to analyze a sound field objec-
tively. To assess the same characteristics of late reverberation perceptively,
a few important aspects must be considered. For instance, one challenging
aspect of the perceptual evaluation of late reverberation is the effect of
early reflections on perception. For this reason, in Publication V, we only
assess the perception of the sound field after the estimated mixing time
(Section 2.4). In the perceptual study, a reverberant sound field is repro-
duced in an anechoic room equipped with a spherical loudspeaker array,
and an ABX test [
240
] is used to assess the perception of small differences
occurring when the late reverberant sound field is rotated. This assesses
the perception of the directional characteristics of the reverberant sound
field.
Another aspect to consider is the presence of noise in a captured SIR,
which may be perceivable during reproduction. To ensure that noise is not
a factor in the perceptual evaluation, a spatial denoising algorithm [
38
]
was used to remove it.
From the rooms analyzed in the study, the results suggest that a sound
field with an
IEDD(t, ω)
value as small as 1 dB is perceivable, implying
that these characteristics are an important factor to consider for spatial
sound reproduction when accuracy is critical.
By proposing objective analysis measures as well as a formal perceptual
study methodology for reverberant sound fields, Publication V offers a
complete framework to evaluate anisotropic reverberant sound field.
5.5.2 Reproduction
The direction-dependent characteristics detailed in the previous section
are analyzed in captured SIRs, which can also be used for their repro-
duction. However, capturing every possible combination of sound source
and listener locations is not always practical. The computational cost of
convolution with long impulse responses may also be prohibitive in some
applications. For these reasons, the use of artificial reverberation methods
in the reproduction may be preferable.
Using existing methods (Section 5.1) it is possible to simulate the early
reflections of an interactive scene using geometric acoustics and to use a
multichannel artificial reverberator to produce decorrelated late reverber-
68
Hybrid Reverberation Methods
012345
0
50
100
150
200
250
300
350 -60
-50
-40
-30
-20
-10
0
Figure 5.5.
Directional decays (
EDCdB(φ, θ)
) using direction-dependent output gains,
while the decay rate remains constant for all directions.
ation. However, this type of reverberator does not inherently produce the
directional decay characteristics illustrated in the previous section.
As an illustrative example, Figure 5.5 shows the output of an Ambisonics
reverberator using direction-dependent output gains to control the rever-
beration. However, unlike the captured SIR in Figure 5.3, Figure 5.5 has a
constant decay rate in all directions.
5.5.3 Directional Feedback Delay Network
Publication VI (p.163), proposes to expand the design of delay network
reverberators to produce direction-dependent decay characteristics from
a set of decay parameters (Figure 5.6). This multichannel reverberation
method is called the directional feedback delay network (DFDN).
In this formulation, the delay lines of an FDN (Figure 4.8) become delay-
groups containing the
Q
channels of an Ambisonics signal (Figure 5.6). An
input signal is weighted in each delay group using
bi
, an input-weighing
vector.
The directional decay characteristics of the reverberator are controlled
through a set of direction-dependent decay parameters
T60(φ, θ)
, which are
converted to per-sample directional attenuation gains using Equations 4.5
to 4.7. The resulting direction-dependent gain vector
gi
for each delay-
group is used to define a directional weighting transform
Ti
using a method
similar to the one presented in Equation 2.32, although a more thorough
formulation of the transform is given in Publication VI.
Due to the Ambisonics processing, the individual delay-lines contained
in a delay group are constrained to share the same length. To produce
direction-dependent decay characteristics, the recirculation matrix
A
pre-
69
Hybrid Reverberation Methods
d
x(n)
b1
z−m1
g1c1
y(n)
b2
z−m2
g2c2
.
.
..
.
..
.
.....
.
..
.
.
bN
z−mN
gNcN
T1T2... TN
A
/
Q
/
Q
/
Q
/
Q
Figure 5.6.
Flow diagram of the DFDN from Publication VI. The thick lines represent
multiple connections, which are Ambisonics channels in this formulation.
vents the individual channels of a delay group from mixing with other
channels. This is done to preserve the spatial distribution of energy and to
ensure that the directional decay follows the ones specified by the
T60(φ, θ)
values. The Hadamard product is used to multiply individual channels
with an output gain vector ciin Figure 5.6.
5.5.4 Frequency-Dependent Directional Feedback Delay
Network
In Publication VII (p.177), the delay groups of the DFDN are simplified to a
set of
K
spatially distributed channels, each represented as a thick line in
Figure 5.7. These channels replace the Ambisonics signals in Publication
VI. While this formulation requires more channels to retain an equivalent
spatial resolution, the simplifications in the processing ultimately allow
for a more efficient direction- and frequency-dependent control of the decay.
Similarly to the methods presented in Section 5.1, a series of geometrically-
based early reflections are used as the input x(n)to the reverberator.
The decay in the reverberator is controlled through a set of direction- and
frequency-dependent decay parameters
T60(ω, φ, θ)
that are used to specify
absorbent filters at the output of each of the delay groups. The individual
delay lines of each of the delay groups have their own set of absorbent
filters, and a multi-band equalizer is used for precise tuning.
One benefit of using channels encoded for a set of directions instead of
Ambisonics is that the delay groups are no longer required to share the
same delay lengths for each individual channel. Removing this constraint
allows all channels of each delay group to contribute independently to the
reverberation, which significantly increases the echo density at the output.
The increase in echo density allows a reduction of the number of delay
70
Hybrid Reverberation Methods
d
x(n)
b1
z−m1G1(z)
c1
y(n)
b2
z−m2G2(z)
c2
.
.
..
.
..
.
.....
.
..
.
.
bN
z−mNGN(z)
cN
A
/
K/
K
Figure 5.7.
Flow diagram of the DFDN formulated in Publication VII. The thick lines
represent multiple connections, which are signals distributed around a sphere
in this formulation. The input
x(n)
contains the direct sound as well as a set
of early reflections generated using geometrical acoustics. The output
y(n)
,
containing a set of directional signals, is converted to an output configuration
(not illustrated here).
groups.
Further processing of the directional reverberation is possible using the
directional input and output gains
bi
and
ci
, respectively. Finally, the
output
y(n)
of the reverberator is encoded to a specified output format.
which is either channel-based, Ambisonics, or binaural.
y(z)=
N
i=1
cT
iGi(z)si(z)+dx(z),(5.10)
si(z)=Zi(z)bix(z)+
N
j=1
Aij Gj(z)sj(z),(5.11)
where
denotes element-wise multiplication (Hadamard product) and
vectors s(z),b, and care
si(z)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
si,1(z)
si,2(z)
.
.
.
si,K (z)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(5.12)
bi=⎛
⎜
⎜
⎜
⎜
⎜
⎝
bi,1
bi,2
.
.
.
bi,K (z)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(5.13)
71
Hybrid Reverberation Methods
012345
0
50
100
150
200
250
300
350 -60
-50
-40
-30
-20
-10
0
Figure 5.8.
Directional energy-decay curves (
EDCdB(φ, 0)
on the horizontal plane from
the output of a DFDN.
ci=⎛
⎜
⎜
⎜
⎜
⎜
⎝
ci,1
ci,2
.
.
.
ci,K
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(5.14)
and the matrices Zi(z), and Gi(z)are
Zi(z)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
z−mi,10··· 0
0z−mi,2··· 0
.
.
..
.
.....
.
.
00··· z−mi,K
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,(5.15)
Gi(z)=⎛
⎜
⎜
⎜
⎜
⎜
⎝
Gi,1(z)0··· 0
0Gi,2(z)··· 0
.
.
..
.
.....
.
.
00··· Gi,K(z)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
.(5.16)
Figure 5.8 visualizes the
EDCdB(φ, θ)
taken from the output of a DFDN.
This specific DFDN was specified by a set of directional decay times
T60(φ, θ)
measured in the captured shoe-box room (Figure 5.3). In Figure 5.9,
the EDD (Section 5.5.1) is analyzed at the output of the same DFDN
configuration. Therefore, the DFDN is suitable for reproduction of decay
characteristics contained in an existing SIR. Furthermore, since these
decay characteristics are controlled through a set of gains and filters, an
interactive scene may be reproduced by simply modulating these gains and
filters, which allows for six-degrees-of-freedom in the reverberation [10].
72
Hybrid Reverberation Methods
Figure 5.9.
Energy-decay deviation
EDD(φ, θ)
, on the horizontal plane of the output of
the DFDN.
In the article, the echo density of different configurations is compared,
along with the reproduction accuracy of distinct uniformly distributed
spatial grids. Ultimately, this formulation allows flexibility to increase
or decrease the spatial and frequency accuracy to accommodate various
computational-cost requirements.
73
6. Summary of Main Results
This section presents the main results from the featured publications that
are related to the author’s work.
Publication I - “Velvet-Noise Decorrelator”
In Publication I, the use of velvet noise as a decorrelation filter is intro-
duced. Velvet-noise sequences have a perceptually flat spectrum and a
random phase, which makes them suitable to be used as broadband time-
domain decorrelation filters. Since a velvet-noise sequence usually consists
of the values
−1
, 0, and 1, the article investigates the use of decaying
envelopes together with the sequences. A decaying envelope is necessary
to preserve the reproduction of transients in the decorrelated signals. The
proposed method is compared with white noise, which has well-known
decorrelation properties, and the results show a similar decorrelating
potential.
Publication II - “Optimized Velvet-Noise Decorrelator”
Publication II improves on the velvet-noise decorrelator introduced in Pub-
lication I and proposes an optimization process to minimize the coloration
and find ideal pairs of filters for stereo decorrelation. The optimization
process slightly alters the location of individual impulses in a velvet-noise
sequence to minimize its spectral coloration. A set of 500 optimized se-
quences is produced, and the best low-correlated pairs are selected from
this set. A weighing factor is used to determine how to prioritize be-
tween spectral flatness and low cross-correlation. The use of optimized
velvet-noise sequences using lower echo density is also investigated. Two
perceptual studies are conducted to evaluate the method. The first eval-
uates the spectral differences between the optimized and non-optimised
sequences. The second perceptual study evaluates the decorrelating prop-
75
Summary of Main Results
erties of the filters for up-mixing a mono audio signal to stereo. Both
studies were conducted over headphones, and white-noise decorrelation
was used as the reference method. The results showed that the optimized
sequences using less impulses were appropriate for use in stereo decorrela-
tion, which yields 88 % fewer floating-point operations when compared to
the FFT-based convolution required for comparable white-noise filters.
Publication III - “Late Reverberation Synthesis Using Filtered Velvet
Noise”
Publication III extends the late-reverberation method introduced in [
165
]
using filtered velvet noise to reproduce the characteristics of an existing
impulse response. The method uses velvet noise to reproduce segments of
an existing impulse response. Velvet-noise sequences are used to produce a
perceptually flat spectrum that is then altered using a coloration filter. This
article improves on the previous formulation by using cascaded differential
coloration filters, which reduces the required impulse density of the velvet
noise used for later segments, thus lowering the overall computational
cost. A new segmentation method is also proposed to find the optimal
segment boundaries. A perceptual study evaluates the similarities between
a recorded IR and the output of the reverberator. The results showed that
the differences were noticeable in most cases, but these differences were
rated as small by the test subjects. The computational cost is comparable
to FFT-based convolution methods, but using much less memory.
Publication IV - “Velvet-Noise Feedback-Delay Network”
In Publication IV, a delay-network reverberator is modified to increase
its echo density using velvet noise. Delay network reverberators, such as
the FDN, require many delay lines to ensure a sufficiently high density of
echoes early in their response. In this article, the velvet-noise sequences
proposed for decorrelation in Publication I and II are used to increase the
echo density in an FDN. Objective measures demonstrate that filtering
both the input and output of each of the delay lines with velvet-noise
sequences produces an optimal increase in echo density and with low
spectral deviations. One of the key benefits of this method is the reduction
in the number of delay lines required to achieve a sufficient echo density.
The propose method requires between 35% and 52% less floating-point
operation in practical configurations.
76
Summary of Main Results
Publication V - “Perceptual Analysis of Directional Late
Reverberation”
In Publication V, an analysis framework is proposed to assess the per-
ception of directional characteristics of a reverberant sound field. Based
on existing methods used to analyze impulse responses, the publication
introduces two objective measures to analyze the directional properties of
SIRs. The proposed method measures the deviation of energy in the decay
between a mean and a set of directions, or between the two channels of
a binaural signal. This process is also extended to the frequency domain
for further analysis. These objective measures are used to analyze SIRs
from four different locations, and a perceptual study is conducted to assess
whether these directional characteristics are subjectively significant. A
testing methodology is proposed where a rotation is applied to the SIRs
after their estimated mixing time. This provides a set of stimuli containing
a rotated late reverberant sound field without altering the early part of the
reverberation. In the proposed methodology, existing denoising techniques
are also used to ensure that the background noise from the captured SIR
does not influence the results of the perceptual study. Using the stimuli,
an ABX perceptual study was conducted using eighteen test subjects, and
the results indicate that a deviation of approximately 1 dB in the proposed
binaural objective measure is sufficient to differentiate between the origi-
nal and the rotated stimuli. These results demonstrate that the directional
characteristics of individual SIR should be evaluated to ensure the appro-
priate level of details in applications where the accuracy of reproduction is
important.
Publication VI - “Directional Feedback Delay Network”
Publication VI introduces a delay-network reverberation method capa-
ble of reproducing directional reverberation characteristics found to be
perceptually important in Publication V. This method is based on previ-
ous formulations of an FDN. For multichannel sound reproduction, FDNs
may provide sets of mutually decorrelated signals. However, the proposed
method improves the multichannel capabilities of the FDN to allow the
reproduction of directional decay characteristics, something that was not
previously possible. Delay lines are modified as multichannel delay groups,
where the signal is encoded in the spherical harmonics domain and the
reverberator is characterized using a set of direction-dependent reverbera-
tion times. A spatial transform in the Ambisonics domain is placed after
each delay group to alter the gains of different directions and to produce
the specified reverberation times.
77
Summary of Main Results
Publication VII - “Frequency-Dependent Directional Feedback Delay
Network”
Publication VII improves the method introduced in Publication VI to ef-
ficiently produce frequency- and direction-dependent reverberation. In
this formulation, the delay groups are encoded as individual plane waves
distributed on a spatial grid. While this spatial representation of sig-
nals requires more channels, when compared to the previous formulation,
this ultimately reduces the overall number of computations necessary
to produce frequency- and direction-dependent reverberation. A set of
geometry-based early reflections distribute the energy spatially at the
input of the reverberator. The article demonstrates that by removing the
Ambisonics processing from the recirculation loop, we are able to increase
the perceived echo density since the delay groups are no longer constrained
to share delay length. Another benefit is to simplify the spectral process-
ing, allowing for more precise tuning. The error between the target decay
properties and the output of the reverberator is also evaluated for various
spatial grid resolutions.
78
7. Conclusion
In this overview, we have discussed the characteristics of reverberant
sound fields, the decorrelation of audio signals, and artificial reverberation
methods. The propagation of sound in a room was described, and some
of the statistical properties used in room acoustics were detailed. The
physical properties of reverberant and diffuse sound fields were reviewed
along with the reproduction.
The decorrelation of audio signals and the use of coherence measures was
summarized. The use of sparse sequences of random noise, called velvet
noise, was proposed as a decorrelation method in Publication I, and this
approach was further improved in Publication II by optimizing the location
of individual impulses in a sequence. While the method provides very
efficient broadband stereo decorrelation, more work remains to generalize
the method to more output channels.
We examined the various methods used to reproduce reverberation arti-
ficially, and more specifically, methods using random noise to mimic late
reverberation and the ones using delay networks for efficient reproduc-
tion. An improved reverberation algorithm using velvet noise along with
coloration filters was introduced in Publication III, and its ability to repro-
duce segments of an existing IR was evaluated through a perceptual study
which demonstrated that this method has good applications to reduce the
storage required by large data sets of IRs. Velvet noise was also used in
Publication IV to improve the echo density of an existing delay-network
reverberation method, thus reducing the required number of delay lines in
the system and the overall computational cost of the system.
Hybrid reverberation methods, which, for instance, may combine geo-
metrical acoustics with delay networks, were also explored as also were
binaural and Ambisonics reverberators. To better understand the re-
quirement of spatial audio reverberation methods, we proposed a novel
framework to analyze the directional characteristics of reverberant sound
fields (Publication V). The proposed objective measurements method shows
that the decay of energy in SIRs may not be uniform in all directions, and
a perceptual study methodology was developed to demonstrate that once a
79
Conclusion
certain perceptual threshold has been met this anisotropic distribution of
energy is perceivable. As such, the proposed framework is useful in assess-
ing the necessary spatial resolution for the reproduction of reverberant
sound fields.
A directional reverberation algorithm, based on the feedback-delay net-
work and capable of producing direction-dependent decay characteristics,
was proposed in Publication VI. This method was further improved in
Publication VII to allow for a more efficient processing of frequency- and
direction-dependent decay characteristics. Using this method, late rever-
beration is designed using a set of direction-dependent decay properties,
which is suitable for the reproduction of the decay characteristics contained
in existing SIRs. In this method, the late reverberation is combined with
early reflections that may be determined through the use of a geometric
virtual acoustics method. The decay characteristics of the reverberator
may be modulated in real-time for applications requiring six-degrees-of-
freedom, such as in virtual and augmented reality [10].
Although this dissertation offers new means to analyze and reproduce
reverberant sound fields containing directional characteristics, more work
remains to be done to fully understand the perceptual limits of direc-
tional reverberation, especially when other characteristics, such as early
reflections or the direct-to-reverberant ratio, may have precedence on the
perception. Nonetheless, we hope that by contributing to the understand-
ing of directional reverberation, the work contained in this dissertation
may inspire better reproduction methods in the future.
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Errata
Publication I
Equation 11 should be
Tdl(m)= TT
100 102m
M−Tdl(m−1),
where
Tdl(0) = TT
100 102m
M.
Publication II
In Equation 7,
δ
is the discrete Dirac function. Also, the right term in Equation
18 should be 0.5T, where T is the sampling period.
97
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