FDNTB: The Feedback Delay Network Toolbox

Abstract

Feedback delay networks (FDNs) are recursive filters, which are widely used for artificial reverberation and decorrelation. While there exists a vast literature on a wide variety of reverb topologies, this work aims to provide a unifying framework to design and analyze delay-based reverberators. To this end, we present the Feedback Delay Network Toolbox (FDNTB), a collection of the MAT-LAB functions and example scripts. The FDNTB includes various representations of FDNs and corresponding translation functions. Further, it provides a selection of special feedback matrices, topologies, and attenuation filters. In particular, more advanced algorithms such as modal decomposition, time-varying matrices, and filter feedback matrices are readily accessible. Furthermore, our toolbox contains several additional FDN designs. Providing MATLAB code under a GNU-GPL 3.0 license and including illustrative examples, we aim to foster research and education in the field of audio processing.

Full text

Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx-20), Vienna, Austria, September 8–12, 2020
FDNTB: THE FEEDBACK DELAY NETWORK TOOLBOX
Sebastian J. Schlecht
Acoustics Lab, Dept. of Signal Processing and Acoustics
Media Lab, Dept. of Media
Aalto University, Espoo, Finland
sebastian.schlecht@aalto.fi
ABSTRACT
Feedback delay networks (FDNs) are recursive filters, which are
widely used for artificial reverberation and decorrelation. While
there exists a vast literature on a wide variety of reverb topologies,
this work aims to provide a unifying framework to design and an-
alyze delay-based reverberators. To this end, we present the Feed-
back Delay Network Toolbox (FDNTB), a collection of the MAT-
LAB functions and example scripts. The FDNTB includes vari-
ous representations of FDNs and corresponding translation func-
tions. Further, it provides a selection of special feedback matrices,
topologies, and attenuation filters. In particular, more advanced
algorithms such as modal decomposition, time-varying matrices,
and filter feedback matrices are readily accessible. Furthermore,
our toolbox contains several additional FDN designs. Providing
MATLAB code under a GNU-GPL 3.0 license and including il-
lustrative examples, we aim to foster research and education in the
field of audio processing.
1. INTRODUCTION
If a sound is emitted in a room, the sound waves travel through
space and are repeatedly reflected at the room boundaries resulting
in acoustic reverberation [1]. Many artificial reverberators have
been developed in recent years [2, 3], among which the feedback
delay network (FDN), initially proposed by Gerzon [4] and further
developed in [5, 6], is one of the most popular. The FDN consists
of Ndelay lines combined with attenuation filters, which are fed
back via a scalar feedback matrix A. Thus, any filter topology of
interconnected delays may be represented as an FDN in a delay
state space (DSS) representation, similar to the general state space
(SS) representations, which is an interconnection of unit delays.
Therefore, FDN framework provide the means for a systematic
investigation of a wide variety of filter topologies such as Moorer-
Schroeder [7], nested allpasses [8], allpass and delay combinations
[9], and many more. Equivalent structures are digital waveguides
[10], waveguide webs [11], scattering delay networks [12] and di-
rectional FDNs [13].
Artificial reverberation can be alternatively applied by directly
convolving the source signal with a room impulse response (RIR)
[2]. Whereas the general representation as a finite impulse re-
sponse (FIR) tends to imply higher computational costs, recent
developments yielded partitioned fast convolution schemes with
highly optimized implementation [14]. As any RIR, including the
FDN impulse responses, can be applied by convolution, it might
Copyright: © 2020 Sebastian J. Schlecht. This is an open-access article distributed
under the terms of the Creative Commons Attribution 3.0 Unported License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
appear as a more general method. However, there are a few im-
portant advantages of FDNs. Where in principle, any combina-
tion of acoustic features can be applied by convolution as long as
they are linear and time-invariant, in practice numerical generation
or acoustic measurements are complex and involved topics [2].
Further, FDNs allow time-variation by modulating the input and
output gains and delays for early reflections of a moving source
[15, 12], and modulation of the feedback delays [16] and feed-
back matrix [17]. Also, synthesizing auditory scenes with mul-
tiple sources and multiple outputs for spatial reproduction scales
computationally well with FDNs compared to individual source-
to-receiver RIR convolution [18].
There are several central challenges in the design of FDNs,
which are only partly addressed in the research literature. A sig-
nificant challenge of FDN design is the inherent trade-off between
three aspects: computational complexity, mode density, and echo
density. Reduced modal density can lead to metallic sounding arti-
facts [19, 20], while reduced echo density can cause rough rattling
sounds. A higher number of delays increases both modal and echo
density, but also the computational complexity. Although attempts
have been made [21, 22], it remains open how to achieve spectrally
and temporally smooth FDNs with a minimal number of delays. A
closely connected topic is the choice of delay length. While the
co-prime criterium introduced by Schroeder [23] remains popular
and extensions exists [24], actual delay lines choices are still open
[25]. Recently, attempts have been made to quantify the spectral
quality of FDNs by statistical measures [26] and based on modal
decomposition [27], but perceptual verification and application ex-
amples need to be provided.
This work presents a Feedback Delay Network Toolbox (FD-
NTB) to support future research in this area. The toolbox contains
a wide variety of conversion functions between different FDN rep-
resentations such as delay state space, state space, modal, and ra-
tional transfer function. Further, matrix generation functions for
feedback matrices such as Hadamard, Circulant, and, random or-
thogonal are provided. Some well-known structures such as the
Moorer-Schroeder or nested allpasses are provided as well. Also,
the toolbox provides additional code for various example appli-
cations. Such applications include time-varying feedback matri-
ces, filter feedback matrices, proportional attenuation filters and
reverberation time estimation. We believe that supplying an entire
collection of different FDN approaches along with example appli-
cations within a unifying framework can be highly beneficial for
both researchers as well as educators in the field of audio process-
ing. The FDNTB can be found online1and is provided under the
GPL-3.0 license.
The remainder of this work is organized as follows. In Sec-
1https://github.com/SebastianJiroSchlecht/
fdnToolbox
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A(z)
x(z)b1(z)z−m1c1(z)y(z)
b2(z)z−m2c2(z)
b3(z)z−m3c3(z)
d(z)
Figure 1: Filter feedback delay network (FFDN) with three delays, i.e., N“3and single input and output, i.e., Nin “1and Nout “1,
respectively. All input, output and direct gains are potentially filters as well as the filter feedback matrix (FFM) Apzq.
tion 2, we review the general FDN structure, lossless and lossy
systems as well as topological variations. In Section 3, we review
a range of feedback matrix types and generation algorithms. Cor-
responding FDNTB function names are indicated as function.
2. FEEDBACK DELAY NETWORKS
In this section, we introduce FDN formally and review various
representations of FDNs. Further, we consider lossless and lossy
FDNs and the corresponding matrices and filters. This section is
concluded with an overview of topology variations of the standard
FDN.
2.1. Feedback Delay Networks (FDNs)
The single-input-single-output (SISO) FDN is given in time do-
main by the difference relation [28]
ypnq “ cJspnq ` d xpnq
spn`mq “ A spnq ` bxpnq,(1)
where xpnqand ypnqare the input and output values at time sam-
ple n, respectively, and ¨Jdenotes the transpose operation. The
FDN dimension Nis the number of delay lines and we occasion-
ally write N-FDN. The NˆNmatrix Ais the feedback matrix,
Nˆ1vector bof input gains, Nˆ1vector cof output gains
and scalar dis the direct signal gain. The lengths of the Nde-
lay lines in samples are given by the vector m“ rm1,...,mNs.
The Nˆ1vector spnqdenotes the delay-line outputs at time n.
The vector argument notation spn`mqabbreviates the vector
rs1pn`m1q,...,sNpn`mNqs. The system order of a standard
FDN in (1) is
N“
N
ÿ
j“1
mj.(2)
In Section 3, we discuss the time-varying feedback matrix Apnq
as effective manipulation of the resulting reverberation. Any gain
in Eq. (1) may consist of finite and infinite impulse response (FIR
and IIR) filters, for instance, a filter feedback matrix (FFM) Apzq
instead of a scalar feedback matrix A(see Fig. 1). The transfer
function of the filtered FDN (FFDN) in the z-domain, correspond-
ing to the difference relation in (1), is
Hpzq “ Ypzq
Xpzq“cpzqJrDm`z´1˘´Apzqs´1bpzq` dpzq,(3)
where Xpzqand Ypzqare the z-domain representations of the in-
put and output signals xpnqand ypnq, respectively, and Dmpzq “
diag`rz´m1, z´m2,...,z´mNs˘is the diagonal NˆNdelay
matrix. We abbreviate the loop transfer function with Ppzq “
Dm`z´1˘´Apzq. Alternatively, every gain and delay in (1) can
be time-varying to adjust to changing acoustic scenes.
2.2. Representations
There are multiple useful representations of FDNs. The represen-
tation (1) is called a delay state-space (DSS) representation and (3)
is the corresponding transfer function. Rocchesso [28] derived an
equivalent standard state-space (SS) representation, i.e., with all
delays equal to 1, see dss2ss. The matrix size of the equivalent
SS is then equal to Nin (2). The transfer function is equivalently
given by Hpzq “ qpzq
ppzq, where
qpzq “ dpzqdetpPpzqq ` cpzqJadjpPpzqqbpzq(4)
ppzq “ detpPpzqq,(5)
where adj denotes the matrix adjungate. The transfer function
form holds equally for the SS representation, see dss2tf and
dss2tfSym. Please note that in the multi-input-multi-output
(MIMO) case, ppzqis a scalar function, where qpzqis a ma-
trix which describes the input-output relation of size Nout ˆNin.
The polynomial coefficients are computed from the principal mi-
nors as given in [29, Lemma 1], see generalCharPoly and
generalCharPolySym.
The roots of the polynomial ppzqare the system poles,
whereas the roots of each entry of qpzqare the system zeros of
the corresponding input-output combination. The modal decom-
position of an FDN computes the partial fraction decomposition
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0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
Mode Frequency [rad/sample]
Mode RT60 [s]
(a) Reverberation time of poles
0 0.5 1 1.5 2 2.5 3
−120
−110
−100
−90
−80
−70
Mode Frequency [rad/sample]
Mode Energy [dB]
(b) Magnitude of residues
Figure 2: Poles and residues for an 8-FDN with a frequency-dependent decay. Each dot indicates one mode in (a) frequency and reverber-
ation time and (b) frequency and residues magnitude. Delays are m“ r2300,499,1255,866,729,964,1363,1491sand Ais a random
orthogonal matrix.
of the transfer function in (3), i.e.,
Hpzq “
N
ÿ
i“1
ρi
1´λiz´1,(6)
where ρiis the residue corresponding to the pole λi. Fig-
ure 2 shows the poles and residues of an FDN with frequency-
dependent reverberation time. The FDNTB provides the polyno-
mial Ehrlich-Aberth Method to compute the modal decomposi-
tion in (6), see dss2pr. Alternatively, the modal decomposition
can be computed from the SS or transfer function representation,
see dss2pr_direct. The residues in (6) can be computed in
two ways, directly from the transfer function [27, Section II-D],
dss2res or from the impulse response with a least-squares fit
[30], see impz2res.
Typically, the efficient DSS representation (1) is used to im-
plement the FDN, see dss2impz, but the impulse response can
be produced based on each of the representations. Therefore, we
provide also impulse responses from poles and residues as well
as the matrix transfer function representation, see pr2impz and
mtf2impz, respectively.
2.3. Lossless and Lossy Feedback
As a first step when designing, FDNs are commonly constructed
as lossless systems, i.e., all system poles lie on the unit circle [31].
The lossless property of general unitary-networks, which in par-
ticular applies to the FDN with a filter feedback matrix Apzq, was
described by Gerzon [31]. An FDN is lossless if Apzqis parauni-
tary, i.e., Apz-1qHApzq “ I, where Iis the identity matrix and ¨H
denotes the complex conjugate transpose [31]. For real scalar ma-
trices A, the FDN is lossless if Ais orthogonal, i.e., AJA“I.
However also non-orthogonal feedback matrices may yield loss-
less FDNs [32, 33], and we give some examples in Section 3.
Homogeneous loss is introduced into a lossless FFDN by re-
placing each delay element z-1with a lossy delay filter γpzqz-1,
where γpzqis ideally zero-phase with a positive frequency-
response. The frequency-dependent gain-per-sample γpeıω qre-
lates to the resulting reverberation time T60pωqby
γpeıωq “ ´60
fsT60pωq,(7)
where fsis the sampling frequency and ωis the angular frequency
[5]. However, as substitution with lossy delays is impractical, the
attenuation filters are lumped into a single filter per delay line. In
standard FDNs with Apzq “ UΓpzq, where Uis lossless, the
delay-proportional attenuation filters should satisfy [5]
|Γpeıωq|“diagpγpeıωqmqm(8)
where |¨|denotes the absolute value. There are various absorp-
tion filters ranging from the computationally efficient one-pole
shelving filter [5], see onePoleAbsorption, to highly accu-
rate graphical equalizers [34], see absorptionGEQ, and FIR
filters in absorptionFilters. With the modal decomposi-
tion (dss2pr), it is possible to demonstrate the lossless and lossy
behavior of FDNs. In Fig 2, a pole-residue decomposition is de-
picted for a 8-FDN with one-pole shelving filters designed with
T60 “2s for the low and T60 “0.4s for the high frequencies. Al-
though, the reverberation time follows the specified values, due to
the inaccurate magnitude and phase component, the reverberation
time can deviate from the specification.
2.4. Topology Variations
The absorption filters can be placed also directly after the delay
line such that the first pass through the delays are filtered as well.
In a similar manner, the feedback matrix is occasionally placed on
the forward path to increase the density on the first pass through.
Another related variation are extra allpass filters in the feedback
loop [35, 36]. Further, extra tap in and out points in the main delay
lines were proposed to increase the echo density and reduce the
initial time gap. More geometrically informed FDNs [12, 37, 13]
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may introduce extra filters and delays to account for source and
receiver positions.
Please note that any of the mentioned topology variations can
be represented and analyzed in formulation (3) by additional fil-
tering of the input and output gains. Although, the computational
complexity may differ from the optimal arrangements, in this work
we prioritize the comparability and generalizability and leave the
efficient implementation for the application scenario.
3. FEEDBACK MATRICES
In this section, we present a broad collection of feedback matrices
which are useful in the context of FDN design.
3.1. Lossless Feedback Matrices
The most important class of feedback matrices in FDNs are the
lossless matrices such that all system poles λiare on the unit cir-
cle. For the real matrices, we review special designs below, see
fdnMatrixGallery. To test statistically properties of FDNs,
we often choose random orthogonal matrices. Many random or-
thogonal matrix generators do not sample the space of possible
matrices equally, e.g., Gram-Schmidt orthogonalization. For this
reason, we provide the randomOrthogonal function, which
samples the space of all orthogonal matrices uniformly. The or-
thogonal matrices can be diagonally similar such that the lossless
property is retained (see diagonallyEquivalent) [32]. The
reverse process is less trivial, i.e., determining whether a matrix
Ais a diagonally similar to an orthogonal matrix. Such a pro-
cess might be necessary to determine whether a given matrix is
lossless. The provided algorithm isDiagonallySimilarTo-
Orthogonal is based on [41]. For instance, the allpass FDN
matrix (shown below) can be shown to be lossless, although not
orthogonal.
Alternatively, we can start with an arbitrary feedback matrix,
e.g., inspired by a physical design [12, 37], and find the nearest
orthogonal matrix. This so-called Procrustes problem solution is
provided in nearestOrthogonal. However, often the specifi-
cation does not necessary specify the phase (= sign) of the matrix
entries as it results from an energy-based derivation. For instance,
one might be interested in a feedback matrix which distributes
the energy from each delay line equally. As the conventional
Procrustes solution can give poor results, we have developed the
sign-agnostic version in [42] given nearestSignAgnostic-
Orthogonal.
Further, it is useful to interpolate between two given orthog-
onal matrices. However, the linear interpolation between matrix
entries does typically not yield orthogonal matrices. Instead, we
proposed to perform the interpolation for the matrix logarithms
[39]. The matrix exponentials map the antisymmetric matrices
to orthogonal matrices. Because the linear interpolation between
antisymmetric matrices remains antisymmetric, matrix exponen-
tial approach yields orthogonal interpolation matrices. We also
provide a special implementation of the inverse function, the ma-
trix logarithm, see realLogOfNormalMatrix based on [43].
For instance, interpolateOrthogonal allows to interpolate
between a Hadamard matrix and an identity matrix to adjust the
density of the matrix continuously and therefore the time-domain
density of the resulting impulse response [39].
3.2. Scalar Feedback Matrices
Here, we review a number of important scalar feedback matrices.
Table 1 lists many of the proposed matrices with the associated
operation counts. The implementation cost of the matrix-vector-
multiplication for a single time step vary from a conventional ma-
trix multiplication N2down to linear number of operations N.
Many of the examples are lossless matrices and loss is introduced
by additional attenuation filters. Some feedback matrices, most
notably from connections of allpass filters, do not have a lossless
prototype as the poles and zeros would cancel out at this limit case.
Some of the presented examples result from translating well-
known reverb topologies into a compact FDN representation.
Feedforward-feedback allpass filters have been introduced with the
delay lines to increase the short-term echo density [40, 7]. Alter-
natively, allpass filters may be placed after the delay lines [35, 36],
which in turn doubles the effective size of the FDN [29]. Gardner
proposed the nested allpass structure by [8], which recursively re-
places the delay in the allpass with another allpass. As a unified
representation, Fig. 4 depicts an overview of the present feedback
matrices.
3.3. Filter Feedback Matrices
If the sound is reflected at a flat, hard boundary, the reflection is co-
herent (specular), while it is incoherent (scattered) when reflected
by a rough surface. Towards a possible integration of scattering-
like effects in FDNs, we introduced in [33] the delay feedback ma-
trix (DFM), where each matrix entry is a scalar gain and a delay.
In [44], we generalized the feedback matrix of the FDN to a fil-
ter feedback matrix (FFM), which then results in a filter feedback
delay network (FFDN). As a special case of the FFM, we present
the velvet feedback matrix (VFM), which can create ultra-dense
impulse responses at a minimal computational cost [45].
FIR filter feedback matrices can be factorization as follows
Apzq “ DmKpzqUK¨ ¨ ¨ Dm1pzqU1Dm0pzq,(9)
where U1,...,UKare scalar NˆNunitary matrices and
m0,m1,...,mKare vectors of Ndelays. In this formulation,
the FFM mainly introduces Kdelay and mixing stages within
the main FDN loop. A few examples of FIR FFMs are de-
picted in Fig. 3. detPolynomial provides an efficient FFT-
based method for determining the polynomial matrix determinant
detpApzqq [46] for (5).
3.4. Time-Varying Feedback Matrix
Many FDN designs also introduce a time-varying component for
enhanced liveliness, improved time and modal domain density as
well as feedback stability in reverberation enhancement systems.
The most prominent variations are delay line modulation [16], all-
pass modulation [47] and matrix modulation [48, 39]. The all-
pass modulation can be represented equally as a matrix modulation
[39]. The matrix modulation is given by
Apn`1q “ ApnqR,(10)
where Ris an orthogonal matrix close to identity, see tiny-
RotationMatrix. A more robust and computationally efficient
version can be implemented by performing the modulation in the
eigenvalue domain, see timeVaryingMatrix.
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Name Definition Operations Counts Notes
Diagonal diagpvqNCorresponds to parallel comb filters
Orthogonal for |vi|”1[38]
Triangular matrix
Lower (or Upper)
Lij “0for iăj NpN`1q{2or
N
May correspond to series comb filters
Lossless for |Lii|”1
not orthogonal except diagonal [38]
Hadamard [6] H0“1
Hk`1“1
?2
»
–
HkHk
Hk´Hk
fi
fl
Nlog N
Fast Hadamard
transform
Orthogonal
|Hij |”1
?N, i.e., equal magnitude
exists only for N“1,2,4,8,12, . . .
Anderson [21] Sparse Block-Circulant matrix
KˆKblocks
NK Orthogonal, K“4recommended
Sparse structures allows larger sizes
Householder [35] I´2vvJfor unit vector v2NOrthogonal
Symmetric
Circulant [28]
»
—
—
—
–
v1vN. . . v2
v2v1. . . v3
.
.
..
.
.....
.
.
vNvN´1. . . v1
fi
ffi
ffi
ffi
fl
|DFTpvq|”1
2Nlog N`N
Fast Convolution
Orthogonal
Convolution is across channels not time
Random Orthogonal Uniform sampling of
orthogonal group OpNq
N2Orthogonal
useful for statistical tests
Tiny Rotation [39] A“Q-1ΛQ
|=Λii|«ϵ
|Λii|“1
N2Orthogonal
close to identity matrix Ifor small ϵ
Allows small matrix modifications
Diagonally Similar
Orthogonal [32]
D´1U D with
diagonal Dand orthogonal U
N2lossless, but not orthogonal
Allpasses in FDN
[35, 36]
Allpasses in
a FDN of size N{2
pN{2q2`NEquivalent to standard FDN of size N,
lossless, but not orthogonal
Moorer-Schroeder [7]
Reverberator
Series of N{2parallel comb and
N{2series allpasses
2NMoorer and Schroeder, Freeverb
Nested Allpass [8] Allpasses nested within allpasses NSISO allpass characteristic, not lossless
Table 1: Special matrices. The operations count are for a single matrix vector multiplication and are rough estimates as there are many
implementation details, e.g., the circulant matrix on a DFT implementation. Allpasses refer to Schroeder’s feedforward-feedback comb
filters [40, 36]. Some notation includes: |¨|denotes the absolute value; ”denotes a constant value for all parameters; and =denotes the
phase of a complex number.
4. CONCLUSIONS
In this paper, we have introduced the FDN toolbox (FDNTB), a
unifying MATLAB framework which contains several FDN al-
gorithms, various code examples for demo applications, as well
as additional measures that have already been used for evaluating
FDN algorithms. By doing so, we gave an overview on recent
studies and open topics. We hope that this toolbox not only pro-
vides a solid code basis to work in the field of FDN, but also helps
to direct attention to shortcomings of classical FDN algorithms, to
foster the development of new FDN techniques, and to ease the de-
sign of listening experiments. Finally, we would like to encourage
developers and researchers in the field of audio processing to use
the toolbox to realize their innovative FDN ideas.
5. ACKNOWLEDGMENT
The author thanks the anonymous reviewers for extensive and
helpful comments on the manuscript as well as the corresponding
software toolbox.
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Time [samples] - [1,17]
Amplitude [linear] - [-0.85,0.85]
(a) Delay feedback matrix (DFM) with K“1.
Time [samples] - [1,413]
Amplitude [linear] - [-0.13,0.13]
(b) Velvet feedback matrix (VFM) with K“2and δ“
1
30 .
Figure 3: Paraunitary filter feedback matrices Apzqwith N“4. The subplots depict the filter coefficients of the matrix entries Aijpzq
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Figure 4: Scalar feedback matrix gallery of size 8ˆ8. The color indicates a linear gain between -1 (red), 0 (white) and 1 (blue). More
details are given in Table 1.
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