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Unitary Matrix Design for Diffuse Jot Reverberators

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This paper presents different methods for designing unitary mixing matrices for Jot reverberators with a particular emphasis on cases where no early reflections are to be modeled. Possible applications include diffuse sound reverberators and decorrelators. The trade-off between effective mixing between channels and the number of multiply operations per channel and output sample is investigated as well as the relationship between the sparseness of powers of the mixing matrix and the sparseness of the impulse response.
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Journal of the Audio Engineering Society.
Unitary Matrix Design for Diffuse Jot Reverberators
Fritz Menzer1and Christof Faller1
1Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
Correspondence should be addressed to Fritz Menzer (fritz.menzer@epfl.ch)
ABSTRACT
This paper presents different methods for designing unitary mixing matrices for Jot reverberators with a
particular emphasis on cases where no early reflections are to be modeled. Possible applications include
diffuse sound reverberators and decorrelators. The trade-off between effective mixing between channels and
the number of multiply operations per channel and output sample is investigated as well as the relationship
between the sparseness of powers of the mixing matrix and the sparseness of the impulse response.
1.INTRODUCTION
In 1991, Jot and Chaigne [1] presented a reverber-
ator based on the feedback delay network structure
introduced by [2] and proposed a systematic method
for calculating the parameters of the reverberator.
Figure 1 shows the feedback loop of a four-channel
Jot reverberator, containing a delay element and a
filter in each channel and amplification and summing
elements assuring the mixing between channels. To
simplify the analysis, the amplification factors are
normally represented as the so-called mixing matrix:
A=
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
.
The mixing matrix Ais crucial for the stability of the
feedback loop and it was proposed by [3] to use uni-
tary feedback matrices, which is a sufficient condi-
tion for keeping the total power of the signals in the
feedback loop constant when no filters are present.
The frequency-dependent reverberation times can
therefore be easily controlled by the filters in the
loop.
In practice however, not only the power conserva-
tion matters, but also mixing capability of the ma-
trix is important, i.e. the capability to spread power
from one channel to all the other channels. While
for example an N-by-Nidentity matrix would be a
perfectly valid mixing matrix, it does not have any
mixing capability and will reduce the Jot reverbera-
tor to the first stage of a Schroeder reverberator [4],
Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
z-m1
z-m2
z-m3
z-m4
h1
h2
h3
h (z)
(z)
(z)
(z)
4
Fig. 1: Feedback loop of a 4-channel Jot reverber-
ator.
i.e. Nparallel comb filters. A matrix with high
mixing capability has a low crest factor, as defined
in (1). The minimum achievable crest factor is 1, in
which case all elements must be either 1 or 1.
The requirements for mixing matrices vary depend-
ing on the application of the reverberator. A Jot
reverberator designed to model the increase in echo
density found in measured room impulse responses
may not require a very efficient mixing matrix, but
rather a mixing matrix that leads to the desired in-
crease in echo density.
The aim of this study was to find mixing matrices
for decorrelators and diffuse sound reverberators. A
decorrelator is a reverberator that implements two
or more short and statistically independent reverb
tails while a diffuse sound reverberator simulates a
room impulse response from which the direct sound
and the early reflections have been removed. Diffuse
sound reverberators and decorrelators both require
a high mode density, in order not to introduce col-
oration to the signal, and a high echo density to
make the reverberation sound smooth. For decor-
relators it is also crucial that a high echo density
is reached quickly because the reverberation tail is
typically very short.
Because the mode density is directly related to the
total delay length [1] and a rapid increase in echo
density implies short average delays, a high number
of channels is required for a decorrelator or a diffuse
sound reverberator. In practice it may be desirable
to have 20 to 40 channels to make the reverbera-
tor sound good. For such high numbers of channels,
random N-by-Nunitary mixing matrices are com-
putationally very expensive and should be avoided.
To reduce the computational complexity, the use
of Hadamard matrices has been proposed before
[5], which allows to implement mixing matrices
with a crest factor of 1 using only Nlog2Noper-
ations. However, for a 32-by-32 matrix, log2N=
log232 = 5, and therefore the implementation of the
Hadamard matrix needs 5N= 160 operations. The
goal of this research is to study mixing matrices that
can be implemented using even less operations, and
matrix structures have been proposed that can be
implemented with 4
3Nto 5Noperations, regardless
of N.
It must be mentioned that, besides the already men-
tioned Hadamard matrix, several other special cases
of mixing matrices are known to have highly effi-
cient implementations [5]. Contrary to most of these
cases, which rely on elements of the matrix having
the same magnitude, the approach chosen here is dif-
ferent (and to some extent orthogonal to the same-
magnitude approach) and imposes that the majority
of elements in the mixing matrix is zero, i.e. that the
matrix is sparse. This may seem contradictory to the
goal of achieving efficient mixing between channels,
but it needs to be considered that an impulse fed to
one of the channels will go many times through the
mixing matrix before its amplitude becomes negligi-
ble. It is possible to design a sparse unitary matrix U
such that Unhas only nonzero elements for a small
n, meaning that after passing ntimes through the
mixing matrix, an impulse in an arbitrary channel
will have spread to all other channels.
Studying the sparsity of Ungives only an approx-
imative indication on the behavior of the feedback
loop. On one side, because the delays in the feed-
back loop are all different, it is impossible to define a
single time instant when all impulses have passed n
times through the feedback loop, meaning that Un
does not represent the real spreading of energy from
one channel to the other, especially for large n. On
the other side, a matrix with only nonzero elements
can still behave like a sparse matrix if the magni-
tudes of the elements are very different (e.g. some
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
elements “stick out”). To gain more detailed infor-
mation, the crest factor of Uncan be studied. For
a matrix Awith elements ai,j (1 i, j N), the
crest factor is defined as
C(A) =
max
i,j |ai,j |
v
u
u
u
t
N
P
i=1
N
P
j=1
a2
i,j
N2
.(1)
However, despite the shortcomings of studying only
the sparseness of Un, this method turned out to give
a simple yet useful indication and was therefore used
throughout this research.
It should be mentioned also that there is extensive
mathematical literature on unitary matrices. How-
ever, while it is known how to factorize any unitary
matrix into a series of sparse unitary matrices [6,
Section 2.B], little seems to be known about which
non-sparse unitary matrices can be expressed as a
power of a single sparse unitary matrix. The ap-
proach for designing suitable sparse unitary matrices
presented here is a bottom-up approach, combining
small and simple unitary matrices to generate a big
unitary matrix with the desired properties.
This paper is structured as follows: Section describes
2 the method of evaluating the different sparse ma-
trix types proposed in Section 3 while Section 4
presents the results and Section 5 discusses them.
Conclusions are drawn in Section 6.
2.MATRIX EVALUATION
In this research different structures for sparse uni-
tary matrices (denoted Uin the following) are pro-
posed and evaluated with respect to different aspects
and under different conditions. The aspects are the
sparsity of Unas a function of n, as well as the
time- and frequency-domain density of the impulse
responses produced by Jot reverberators using Uas
the mixing matrix.
Different application conditions were simulated by
three different scenarios. In the first scenario, the
number of channels (and therefore the matrix size)
is constant. Wherever possible, 24 channels were
used and 25 was used in the cases where 24 was
not possible due to the matrix design. The second
and third scenario simulate complexity constraints
as they could arise when implementing a diffuse re-
verberator or a decorrelator in an environment with
limited computational resources.
As the measure of computational complexity, the
number of multiplications per output sample was
chosen. This measure is expected to be roughly pro-
portional to the number of clock cycles per output
sample needed for the implementation of the rever-
berator on a CPU in the case where the multiply op-
eration is much more costly than the add operation
(which may be the case with older or low-end CPUs)
and also in the case where a multiply-accumulate
(MAC) operation exists, which is the case for DSPs
and many multimedia-oriented CPUs. Each element
in the mixing matrix that is neither 0 nor 1 is sup-
posed to require one multiplication per output sam-
ple, as long as Ucontains at most one element equal
to 1 per column. This condition is necessary in or-
der to have a realistic complexity estimate for CPUs
with multiply-accumulate and is fulfilled for all ma-
trix types proposed in this study.
Counting the number of elements different from 0 or
1 does not take into account the fact that many ma-
trix types exist that can be implemented in a more
efficient way because many nonzero elements have
the same magnitude (different from 1). However,
such simplifications are not of primary concern for
this study since the main focus here is on the struc-
ture of the sparse matrices, not on the actual ele-
ment values, and the nonzero elements are in prac-
tice calculated from random parameters, therefore
not allowing simplifications based on equal element
values. In practice, it is of course possible to design
matrices that take advantage of both complexity re-
ductions, due to sparseness and due to equal magni-
tudes. This study does not include the equal magni-
tude approach because it imposes many constraints
on the matrices, as often for a given matrix size only
few possible matrices are known, which would be
contradictory with the approach used here, evaluat-
ing a large number of matrices of the same type and
taking the mean over the results.
In the reverberator scenario with constrained com-
plexity, the total number of multiplications for the
recursive loop was required to be less than or
equal to 200, including 4 additional multiplications
per channel for the filters modeling the frequency-
dependent reverberation times. For testing the ma-
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
trices in the “fixed size” and the “fixed cost reverber-
ator” scenarios, the reverberation time (RT60) was
fixed to 1 s for all frequencies and the delays were
mutually prime numbers randomly generated from
a Gaussian distribution with a mean of 400 samples
and a standard deviation of 300 samples.
In the constrained complexity decorrelator scenario,
the total number of multiplications is limited to 100,
including 1 additional multiplication per channel
(since a decorrelator should have a decaying white
noise tail as an impulse response, only one attenua-
tion factor per channel is needed inside the recursive
loop). The attenuation factors were calculated to
achieve a reverberation time (RT60) of 250ms and
the delays were mutually prime numbers randomly
generated from a Gaussian distribution with a mean
of 300 samples and a standard deviation of 200 sam-
ples.
3.MATRIX TYPES
In the following, the different matrix types studied in
this study are presented. The first two types of uni-
tary matrices were introduced just as a reference and
are the two most extreme cases of all possible ways
of designing mixing matrices for a Jot reverberator:
an identity matrix and a random (non-sparse) uni-
tary matrix. As mentioned before, the goal of this
study is to design mixing matrices that have many
zero elements and still produce a temporally dense
reverb. These conditions are not fulfilled by the two
mentioned matrices: the identity matrix does not
provide any mixing and the random unitary matrix
is not sparse.
Since in an identity matrix only the elements on the
diagonal are non-zero, using a N-by-Nidentity ma-
trix as a mixing matrix will reduce the resulting Jot
reverberator to the first stage of a Schroeder rever-
berator [4], i.e. Ncomb filters in parallel. The
relationship between the Jot reverberator and the
Schroeder reverberator is discussed in detail by [1].
The non-sparse random unitary matrices can be eas-
ily obtained using the singular value decomposition
(SVD) of a random matrix. Using a random unitary
matrix as the mixing matrix assures a very good
mixing because the signal from each channel imme-
diately propagates to all the other channels. How-
ever, from the implementation point of view it is
the worst possible choice because all elements are
nonzero and N2multiplications are needed.
The first attempt made to make a mixing matrix
with the desired properties was a matrix composed
of Bblocks of 2–by–2 unitary matrices, arranged to
the following structure:
U2(B) =
0 0 G20 0 0 0
0 0 0 0 · · · 0 0
0000 G30 0
0000 00
.
.
....
000000 GB
000000
G10000 00
0000· · · 0 0
where Bis the number of blocks and Giare Givens
rotations
Gi=cos αisin αi
sin αicos αi
and the αiare randomly chosen using a uniform dis-
tribution on the interval [0,2π]. Implementing an
N-by-Nmatrix of this type requires 2Nmultiplica-
tions.
An attempt was also made to design a computation-
ally very efficient matrix requiring only 4
3Nmulti-
plications to implement a N-by-Nmatrix. These
matrices are composed of B3–by–3 unitary matrices
that are sparse by themselves. The general structure
then looks like this:
U3(B) =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
000 b
U2
000 000
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 ··· 0 0 0
0 0 0 0 0 0 b
U3
0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
.
.
....
0000 00000 b
UB
0000 00000
0000 00000
b
U1
0 00000 000
0 0 0 0 0 0 ··· 0 0 0
0 00000 000
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
where b
Uiare 3x3 unitary matrices of one of the
following forms:
b
Ui8
<
:
2
4
0Gi
0
10 0
3
5,2
4
10 0
0Gi
0
3
5,2
4
0 0 1
Gi
0
0
3
5,2
4Gi
0
0
0 0 1
3
59
=
;
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
and Giare random Givens rotations. Even though
it would have been possible to use special values for
the rotation angle (e.g. π
4), allowing a reduction of
computational complexity, this was not done in or-
der to have a fair comparison between matrix struc-
tures and also to stay with the most general case,
avoiding possible unwanted effects due to one spe-
cific set of values.
While U2(B) and U3(B) can be considered as valid
candidates for good mixing matrices (see discus-
sion), they never become non-sparse, and therefore
do not fulfill the goal set above. However, a sim-
ple way was found to modify U2such that the new
matrix U21 fulfills the constraint that Un
21 should be
non-sparse for some finite nby using the following
structure:
U21(B) =
2
6
6
6
4
0U2(B)
.
.
.
0
10··· 0
3
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
000G20 0 0 0
000 00··· 0 0
00000G30 0
00000 00
.
.
....
0000000 GB
0000000
0G10 0 0 0 0 0
0 0 0 0 0 ··· 0 0
1000000 00
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
.
The same method can be used also on U3:
U31(B) =
2
6
6
6
4
0U3(B)
.
.
.
0
10··· 0
3
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 0 0 0 b
U2
0 0 0 0 0 0
0000 000 000
0000 000··· 000
00000 00 b
U3
000
00000 00 000
00000 00 000
.
.
....
00000 00000 b
UB
00000 00000
00000 00000
0b
U1
0 00000 000
0 0 0 0 0 0 0 ··· 000
0 0 0 0 0 0 0 0 0 0
10000 00000 000
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
,
where b
Uiare the same sparse 3x3 unitary matrices
as in U3(B).
A systematic way of generating sparse unitary ma-
trices Usuch that Unbecomes non-sparse for small
values of nwas found by using the coefficients of B
random unitary m×mmatrices and arrange them
in such a way on a Bm ×Bm matrix that in the
resulting Jot reverberator the signal from channel i
is fed to channels ((i1)m+1 mod Bm)+ 1 to (im
mod Bm) + 1. For m= 2 this means that the out-
put of channel 1 goes to channels 2 and 3, channel 2
to channels 4 and 5, channel 3 to channels 6 and 7,
etc.
For m= 2 and B= 3, the resulting matrix U2f(3)
(the subscript fstanding for “fast”) looks like this:
U2f(3) =
0 0 c30 0 c4
a10 0 a20 0
a30 0 a40 0
0b10 0 b20
0b30 0 b40
0 0 c10 0 c2
where
a1a2
a3a4,b1b2
b3b4,c1c2
c3c4
are random 2 ×2 unitary matrices, e.g. random
Givens rotations.
For this study, matrices designed in the same way
but with m= 3, m= 4 and m= 5 are used and are
denoted U3f(B), U4f(B) and U5f(B).
For all matrices except the first two types, versions
with randomized column orders have been gener-
ated. They are denoted f
Uxinstead of Ux. An
overview of matrix sizes and numbers of multipli-
cations for the different scenarios and matrix types
are shown in Table 1.
4.RESULTS
The matrices and the impulse responses generated
by using them in a reverberator were examined un-
der four aspects. First the evolution of the matrices
(i.e. their different powers) was studied graphically
in order to see how they converge to a non-sparse
matrix. Then, the number of iterations of the ma-
trix needed to become non-sparse was computed and
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
fixed size reverberator decorrelator
channels channels multiplications channels multiplications
I24 50 200 100 100
Ufull 24 12 192 9 90
U224 32 192 32 96
U324 36 192 42 98
U21 25 33 196 33 97
U31 25 37 196 43 99
U2f24 32 192 32 96
U3f24 27 189 24 96
U4f24 24 192 20 100
U5f25 20 180 15 90
Table 1: Channel numbers and multiplications per output sample as a function of matrix design and
scenario (the randomized versions have been omitted from this table because they have the same size as
their non-randomized counterparts)
the time needed for the impulse response to achieve
100% echo density was calculated as well as the stan-
dard deviation of the spectrum of the late impulse
response (in 1-ERB bands). Because the matrices
(except the identity matrix) depend on random val-
ues and the (random) delays used in the recursive
loop also have an influence on the performance, the
measures described above may change as a function
of the random numbers used to generate the matri-
ces and the delays. Each case was repeated 100 times
for different random number generator seeds and the
mean and standard deviations were calculated.
In the following, all illustrations of matrices show
their absolute values on a scale from 0 (white) to
1 (black). Using absolute values and white for the
value 0 makes it easy to estimate the sparseness of
the matrices. Furthermore, the signs of the values
do not carry relevant information in this context.
4.1.Fixed matrix size
As shown in Table 1, in the “fixed matrix size” sce-
nario, all reverberators have either 24 or 25 channels.
The difference is due to the fact that no single matrix
size could be generated by all the design methods,
so in the following one should keep in mind that a
difference in the results may be due to a difference
in the number of channels of 5%.
Figure 2 shows the evolution (i.e. different pow-
ers) of the identity matrix and a non-sparse random
unitary matrix. Both matrix types do not show any
qualitative change when taken to higher powers: the
identity matrix always stays the same, and a random
unitary matrix always stays a random unitary ma-
trix.
I I 2I5I10 I20
Ufull Ufull
2Ufull
5Ufull
10 Ufull
20
Fig. 2: Evolution of the two most extreme cases
of mixing matrices for “fixed size” scenario. Top:
24 ×24 identity matrix. Bottom: 24 ×24 random
unitary matrix.
Figure 3 shows the evolution of matrices generated
directly by the different design methods. It can be
seen that U2and U3do not converge, but rather
a diagonal “chain” of small (2 ×2 or 3 ×3) uni-
tary matrices moves across the matrix. The cases
U21 and U31 both converge, but very slowly (taken
into account the approximately logarithmic display
of matrix powers). In U20
21 , a diagonal band of higher
values can be distinguished. This is much less the
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
U2U2
2U2
5U2
10 U2
20
U3U3
2U3
5U3
10 U3
20
U21 U21
2U21
5U21
10 U21
20
U31 U31
2U31
5U31
10 U31
20
U2fU2f
2U2f
5U2f
10 U2f
20
U3fU3f
2U3f
5U3f
10 U3f
20
U4fU4f
2U4f
5U4f
10 U4f
20
U5fU5f
2U5f
5U5f
10 U5f
20
Fig. 3: Evolution of studied sparse mixing matrices
for “fixed size” scenario.
case in U20
31 , which in turn shows some single high
values that “stick out”. A value close to 1 would
mean that – if all delays were equal – after passing
20 times through the recursive loop, the signal from
one channel would predominantly show up in one
single (different) channel.
U2
U2
2
U5
2
U10
2
U20
2
U3
U2
3
U5
3
U10
3
U20
3
U21
U2
21
U5
21
U10
21
U20
21
U31
U2
31
U5
31
U10
31
U20
31
U2f
U2
2f
U5
2f
U10
2f
U20
2f
U3f
U2
3f
U5
3f
U10
3f
U20
3f
U4f
U2
4f
U5
4f
U10
4f
U20
4f
U5f
U2
5f
U5
5f
U10
5f
U20
5f
Fig. 4: Evolution of studied sparse mixing matrices
with randomized column ordering for “fixed size”
scenario.
The cases U2fto U5fdo not show any such behavior
and also converge much more quickly: already after
5 or 10 iterations, these matrices look line a random
unitary matrix generated using an SVD. The more
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
nonzero elements the original matrix has, the quicker
is the convergence.
Figure 4 shows instances of the same matrix types,
but with randomized column order. It can be no-
ticed that the differences in convergence between U2
and U21 completely disappeared after the random-
ization. The same holds also for the randomized
versions of U3and U31 in general. However, in this
instance of e
U31 one can see a drawback of random-
ization: randomizing can actually impair the con-
vergence behavior. Because e
U31 has two elements
equal to 1 on the diagonal, it never converges to a
non-sparse matrix.
On the “fast” matrices U2fto U5f, the effect of the
column randomization seems to be rather adverse in
the short term: the number of iterations needed for
achieving complete non-sparsity increases (which is
also confirmed by the data in Figure 5), but in the
long term, no significant change can be seen: U20
nf
and e
U20
nf ,n∈ {2,3,4,5}all look like random unitary
matrices.
Figure 5 shows the number of iterations kmin that
were needed to obtain a non-sparse matrix as a func-
tion of the matrix type, separately with and without
column randomization. The observations made on
Figure 3 are confirmed by the averages: the matri-
ces I,U2, and U3never converge; the “fast” matrices
UNf converge more rapidly than all the others (ex-
cept for Ufull of course); randomization makes U2
and U3converge faster, while it slows down the con-
vergence for the those matrices that are “fast” by
design. The reason why U31 converges much slower
than U21 is that U31 is much more sparse.
Figure 6 shows the time needed to reach 100% echo
density (i.e. non-sparsity of the impulse response).
It may be observed that this time is very closely
related to the value kmin shown in Figure 5, with
one notable exception: for the time needed to reach
100% echo density, U2and U3behave like their ran-
domized versions and also like U21 and U31. Fur-
thermore, only an insignificant difference between
U2and U2fcan be observed.
Figure 7 shows the spectral deviation of the late
tail of the reverberators impulse responses. For all
matrices, except for the identity matrix, no signif-
icant difference between spectral deviations can be
kmin
minimal k such that Uk is nonsparse
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
5
10
15
20
25
infinity original column order
random column order
Fig. 5: Number of iterations needed to obtain non-
sparse matrix for “fixed size” scenario.
time [msec]
time to 100% echo density
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
20
40
60
80
100
120
140
never original column order
random column order
Fig. 6: Time needed to achieve full echo density for
“fixed size” scenario.
observed. This is in line with the finding that the
mode density (which is related to the spectral devi-
ation) of a feedback delay network only depends on
the total length of delays [1]. Since here the number
of channels is always 24 or 25, i.e. varies only by 5%,
the average total length of the delays also varies by
5%. That the reverberator using an identity matrix
performs significantly worse even though it has the
same total delay length as all the other cases may
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
be explained by the fact that it never reaches 100%
echo density.
It is interesting to note that even the case U3which
has no complete mixing and a very sparse matrix
performs as well as the other cases with respect to
the spectral deviation.
standard deviation [dB]
spectral deviation
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
1
2
3
4original column order
random column order
Fig. 7: Spectrum standard deviation in late tail for
“fixed size” scenario.
4.2.Reverberator scenario
Figure 8 shows the evolution of the identity matrix
and a random unitary matrix for the “reverberator”
scenario, where the number of multiplications is lim-
ited to 200 and each channel contains a 4-tap FIR
filter (thus consuming 4 multiplications per chan-
nel, independently of the mixing matrix). It can be
observed that this set of constraints leads to large
differences in matrix size.
Figures 10 and 11 show the evolution of instances
of the other matrix types, with original column or-
der and with random column order, respectively. In
general, the same observations can be made as in
the “fixed size” scenario. Due to the bigger size of
the matrices, which makes the convergence of U31
and U21 very slow, it can be observed well how the
random column order improves the convergence be-
havior in these two cases.
Figure 9 shows the number of iterations needed for
I I 2I5I10 I20
Ufull Ufull
2Ufull
5Ufull
10 Ufull
20
Fig. 8: Evolution of the two most extreme cases of
mixing matrices for “reverberator” scenario. Top:
50 ×50 identity matrix. Bottom: 12 ×12 random
unitary matrix.
kmin
minimal k such that Uk is nonsparse
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
5
10
15
20
25
infinity original column order
random column order
Fig. 9: Number of iterations needed to obtain non-
sparse matrix for “reverberator” scenario.
convergence to a non-sparse matrix and confirms the
improvement of convergence due to randomized col-
umn ordering of U31 and U21. The same figure also
confirms the degradation of the convergence for the
“fast” matrix types U2fto U5f.
Figure 12 shows the time to reach 100% echo density.
The same observations as in the “fixed size” scenario
can be made. In particular this figure shows that
the improvement in convergence for U2fdoes not
translate in any significant improvement of the time
to 100% echo density.
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
U2U2
2U2
5U2
10 U2
20
U3U3
2U3
5U3
10 U3
20
U21 U21
2U21
5U21
10 U21
20
U31 U31
2U31
5U31
10 U31
20
U2fU2f
2U2f
5U2f
10 U2f
20
U3fU3f
2U3f
5U3f
10 U3f
20
U4fU4f
2U4f
5U4f
10 U4f
20
U5fU5f
2U5f
5U5f
10 U5f
20
Fig. 10: Evolution of studied sparse mixing matri-
ces for “reverberator” scenario.
Figure 13 shows the spectral deviation of the late
tail of the reverberators impulse responses. Knowing
that the mode density of a reverberator depends on
the total delay length, it can be expected that the
lowest spectral deviations occur for the reverberators
with the highest number of channels. This is true
U2
U2
2
U5
2
U10
2
U20
2
U3
U2
3
U5
3
U10
3
U20
3
U21
U2
21
U5
21
U10
21
U20
21
U31
U2
31
U5
31
U10
31
U20
31
U2f
U2
2f
U5
2f
U10
2f
U20
2f
U3f
U2
3f
U5
3f
U10
3f
U20
3f
U4f
U2
4f
U5
4f
U10
4f
U20
4f
U5f
U2
5f
U5
5f
U10
5f
U20
5f
Fig. 11: Evolution of studied sparse mixing matri-
ces with randomized column ordering for “reverber-
ator” scenario.
indeed, as the reverberators based on U2,U3,U21,
U31, and U2fperform best.
AES 128th Convention, London, UK, 2010 May 22–25
Page 10 of 16
Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
time [msec]
time to 100% echo density
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
20
40
60
80
100
120
140
never original column order
random column order
Fig. 12: Time needed to achieve full echo density
for “reverberator” scenario.
standard deviation [dB]
spectral deviation
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
1
2
3
4original column order
random column order
Fig. 13: Spectrum standard deviation in late tail
for “reverberator” scenario.
4.3.Decorrelator scenario
Figure 14 shows the evolution of the identity ma-
trix and a random unitary matrix for the “decorre-
lator” scenario, where the number of multiplications
is limited to 100 and each channel contains a sin-
gle amplifier (thus consuming 1 multiplications per
channel, independently of the mixing matrix). It
can be observed that this constraint leads to very
big differences in matrix size.
Figures 15 and 16 show the evolution of instances
of the other matrix types, with original column or-
der and with random column order, respectively. In
general, the same observations can be made as in
the “fixed size” scenario.
I I 2I5I10 I20
Ufull Ufull
2Ufull
5Ufull
10 Ufull
20
Fig. 14: Evolution of the two most extreme cases
of mixing matrices for “decorrelator” scenario. Top:
100 ×100 identity matrix. Bottom: 9×9 random
unitary matrix.
Figure 17 shows the number of iterations needed for
convergence to a non-sparse matrix and generally
confirms the observations made in the “reverbera-
tor” case.
Figure 18 shows the time to reach 100% echo density.
The same observations as in the other two scenarios
can be made.
Figure 19 shows the spectral deviation of the whole
impulse responses of the decorrelators. The triangles
below each error bar are the minimum values found
while testing 100 instances of each matrix type. This
figure shows that spectral standard deviations nearly
as low as 1 dB can be reached with such a decorre-
lator. It is interesting that the minimum value was
reached with U2f. This indicates that fast conver-
gence plays an important role in the design of decor-
relators.
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
U2U2
2U2
5U2
10 U2
20
U3U3
2U3
5U3
10 U3
20
U21 U21
2U21
5U21
10 U21
20
U31 U31
2U31
5U31
10 U31
20
U2fU2f
2U2f
5U2f
10 U2f
20
U3fU3f
2U3f
5U3f
10 U3f
20
U4fU4f
2U4f
5U4f
10 U4f
20
U5fU5f
2U5f
5U5f
10 U5f
20
Fig. 15: Evolution of studied sparse mixing matri-
ces for “decorrelator” scenario.
U2
U2
2
U5
2
U10
2
U20
2
U3
U2
3
U5
3
U10
3
U20
3
U21
U2
21
U5
21
U10
21
U20
21
U31
U2
31
U5
31
U10
31
U20
31
U2f
U2
2f
U5
2f
U10
2f
U20
2f
U3f
U2
3f
U5
3f
U10
3f
U20
3f
U4f
U2
4f
U5
4f
U10
4f
U20
4f
U5f
U2
5f
U5
5f
U10
5f
U20
5f
Fig. 16: Evolution of studied sparse mixing matri-
ces with randomized column ordering for “decorre-
lator” scenario.
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
kmin
minimal k such that Uk is nonsparse
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
5
10
15
20
25
infinity original column order
random column order
Fig. 17: Number of iterations needed to obtain non-
sparse matrix for “decorrelator” scenario.
time [msec]
time to 100% echo density
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
20
40
60
80
100
120
140
never original column order
random column order
Fig. 18: Time needed to achieve full echo density
for “decorrelator” scenario.
standard deviation [dB]
spectral deviation
I Ufull U2 U3 U21 U31 U2f U3f U4f U5f
0
1
2
3
4original column order
random column order
Fig. 19: Spectrum standard deviation of entire im-
pulse response for “decorrelator” scenario. The tri-
angles below the error bars show the minimum val-
ues found in 100 random instances.
5.DISCUSSION
It was found that the minimum power of the ma-
trix that is non-sparse allows to predict after which
time the echo density in the impulse response reaches
100%. However, there is a notable exception be-
cause the matrix types U2and U3never converge,
but still produce impulse responses that reach 100%
echo density relatively fast. The explanation lies in
the fact that the powers of the mixing matrix only
indicate how a signal spreads among channels if all
delays in the recursive loop are equal. In a real re-
verberator the delays are normally chosen to be mu-
tually prime and are therefore different.
Figure 20 shows the signals in a 4-channel rever-
berator, where channel 1 was excited with a dirac
impulse at time 0. The mixing matrix is of type U2
AES 128th Convention, London, UK, 2010 May 22–25
Page 13 of 16
Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
0 20 40 60 80 100 120 140 160 180 200
sum
4
3
2
1
time [samples]
channel
0 20 40 60 80 100 120 140 160 180 200
sum
4
3
2
1
time [samples]
channel
Fig. 20: Response of a four-channel reverberator to
a single impulse in the first channel. Matrix type:
U2.Top: Equal delays. Bottom: Mutually prime
delays.
and has the structure
U2(2) =
0 0 b1b2
0 0 b3b4
a1a20 0
a3a40 0
.
This matrix does not converge to a non-sparse ma-
trix. In the top plot the delays are all equal to 20
samples and it can be seen that at each iteration
of the recursive loop only two channels are nonzero.
This is what the powers of the mixing matrix pre-
dict.
The top plot of Figure 21 shows the same signals
for a reverberator with a mixing matrix of type U2f
0 20 40 60 80 100 120 140 160 180 200
sum
4
3
2
1
time [samples]
channel
0 20 40 60 80 100 120 140 160 180 200
sum
4
3
2
1
time [samples]
channel
Fig. 21: Response of a four-channel reverberator to
a single impulse in the first channel. Matrix type:
U2f.Top: Equal delays. Bottom: Mutually prime
delays.
which has the structure
U2f(2) =
0b30b4
a10a20
a30a40
0b10b2
,
and which has, contrary to U2, the property that
U2
2fis non-sparse. From 40 samples after the ini-
tial impulse – corresponding to two iterations in the
recursive loop – all channels are nonzero at each it-
eration.
So far the behavior of the reverberator follows closely
the behavior of the powers of the matrix. However,
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
if one looks at reverberators with mutually prime de-
lays, the behavior of the reverberator is more compli-
cated. In the bottom plots of Figures 20 and 21 the
signals for reverberators with mutually prime delays
are shown. It can be observed that the total number
of impulses in the channels as a function of time is
roughly the same for both reverberators. The dif-
ference is that for the reverberator using U2they
appear in bursts where at one iteration channels 1
and 2 are active and at the next iterations channels
3 and 4 are active, while for the reverberator using
U2fall channels are equally active over time. In fact,
both matrices have the effect that each impulse will
generate two impulses at the next iteration. The
only difference is how the impulses are distributed
to the channels, as well as their exact timing. As a
result, the evolution of the echo density is similar for
both matrices.
While this study focused on the structure of sparse
unitary matrices and on composing such matrices
from random unitary matrices, in a practical imple-
mentation it is desirable to take advantage also of the
reduction of computational complexity due to choos-
ing particular values for the elements of the mixing
matrix. As an example, a highly efficient mixing
matrix can be obtained by composing U4f(4) from
4-by-4 Hadamard matrices with randomized column
order. Such a matrix is shown below.
U4fh =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0001000 10001000 1
1000100010001000
10001000 10001000
1000 100010001000
1000 1000 10001000
010001000100010 0
010001000 100010 0
01000100010 0 0 10 0
010001000 1000 10 0
0010001000100010
0 0 100010001000 10
0010001000 1000 10
0 0 10001000 100010
0001000100010001
00010001000 1000 1
0001000 1000 10001
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
/2.
This matrix has the properties that U2
4fh is non-
sparse and the crest factor C(U2
4fh) is equal to 1.
Due to the efficient implementation of Hadamard
matrices, the computational cost of this matrix is
32 operations per output sample, i.e. half the cost
of a U4fmatrix of the same size with random values
or a 16-by-16 Hadamard matrix.
6.CONCLUSIONS
In this study it was shown that using sparse unitary
matrices as mixing matrices in Jot reverberators is
an efficient way to reduce the computational com-
plexity. Matrices of different types were analyzed
visually and by studying how fast the powers of the
matrices become non-sparse, i.e. after how many
iterations in the recursive loop a signal fed to an
arbitrary channel will spread to all other channels.
The resulting impulse responses were analyzed with
respect to echo and spectral density.
The mixing matrix of a Jot reverberator has a strong
influence on how fast the echo density in the impulse
response increases. When the goal of a reverbera-
tor is to model only diffuse reverberation or to be
used as a decorrelator, it is desirable to reach full
echo density as soon as possible. Different methods
for designing sparse unitary matrices were proposed
and evaluated in 3 different scenarios with differ-
ent constraints. In the first scenario the matrix size
was fixed, while in the second and third scenario the
number of multiplications was limited. The differ-
ence between the second and the third scenario was
that the former used parameters suitable for a re-
verberator and the latter aimed at implementing a
decorrelator.
The exact choice of a mixing matrix can be made
only in the context of the other design parameters
of the reverberators (e.g. memory constraints, con-
straints on the time when 100% echo density should
be reached, etc.) and should also depend on the
choice of delays. However, it was found that in the
reverberator scenario it is favorable to use a mix-
ing matrix that is sparse and big rather than small
and converging fast, while in the decorrelator case
fast convergence plays an important role and par-
ticularly good results were obtained with the fast
converging matrix types U2f,U3f.
While several types of mixing matrices have been
proposed before that can be implemented with a low
computational complexity because they contain ele-
ments with equal magnitude, this study focused on
the complexity reduction in reverberators due to the
sparse structure of the mixing matrix, i.e. the dis-
tribution of zero and nonzero elements. It could be
AES 128th Convention, London, UK, 2010 May 22–25
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Menzer AND Faller Unitary Matrix Design for Diffuse Jot Reverberators
shown that the advantages of both methods can be
combined by choosing the values of the nonzero ele-
ments from the previously proposed efficient mixing
matrices.
7.REFERENCES
[1] Jean-Marc Jot and Antoine Chaigne, “Digital
delay networks for designing artificial reverbera-
tors,” in Proc. 90th AES Convention, 1991.
[2] John Stautner and Miller Puckette, “Designing
multi-channel reverberators,” Computer Music
Journal, vol. 6, no. 1, pp. 52–65, 1982.
[3] Jean-Marc Jot, “An analysis/synthesis approach
to real-time artificial reverberation,” in Proc.
ICASSP-92, 1992, vol. 2, pp. 221–224.
[4] M. R. Schroeder, “Natural sounding artificial
reverberation,” J. Aud. Eng. Soc., vol. 10, no. 3,
pp. 219–223, 1962.
[5] Jean-Marc Jot, “Efficient models for reverber-
ation and distance rendering in computer mu-
sic and virtual audio reality,” in Proc. Interna-
tional Computer Music Conference, September
1997, pp. 236–243.
[6] Martin Vetterli and Jelena Kovcevi´c, Wavelets
and Subband Coding, Prentice Hall, Englewood
Cliffs, New Jersey, USA, 1995.
AES 128th Convention, London, UK, 2010 May 22–25
Page 16 of 16
... The choice of the feedback matrix is crucial for the FDN algorithm to work correctly. The popular matrix types used in FDN implementations that fulfill the requirement of being unilossless are Hadamard [27], Householder [27], random orthogonal, and identity matrices [28]. Where the first three are chosen to enhance specific properties of the algorithm, e.g., density of the impulse response, the identity matrix, however, reduces the FDN to a Schroeder reverberator, or a parallel set of comb filters [6,28]. ...
... The popular matrix types used in FDN implementations that fulfill the requirement of being unilossless are Hadamard [27], Householder [27], random orthogonal, and identity matrices [28]. Where the first three are chosen to enhance specific properties of the algorithm, e.g., density of the impulse response, the identity matrix, however, reduces the FDN to a Schroeder reverberator, or a parallel set of comb filters [6,28]. The plugin presented in this study allows the user to choose between these four matrices through a drop-down menu and to learn about the differences in the sound obtained by changing this part of the FDN reverberator. ...
Conference Paper
Full-text available
Reverberation is one of the most important effects used in audio production. Although nowadays numerous real-time implementations of artificial reverberation algorithms are available, many of them depend on a database of recorded or pre-synthesized room impulse responses, which are convolved with the input signal. Implementations that use an algorithmic approach are more flexible but do not let the users have full control over the produced sound, allowing only a few selected parameters to be altered. The real-time implementation of an artificial reverberation synthesizer presented in this study introduces an audio plugin based on a feedback delay network (FDN), which lets the user have full and detailed insight into the produced reverb. It allows for control of reverberation time in ten octave bands, simultaneously allowing adjusting the feedback matrix type and delay-line lengths. The proposed plugin explores various FDN setups, showing that the lowest useful order for high-quality sound is 16, and that in the case of a Householder matrix the implementation strongly affects the resulting reverberation. Experimenting with delay lengths and distribution demonstrates that choosing too wide or too narrow a length range is disadvantageous to the synthesized sound quality. The study also discusses CPU usage for different FDN orders and plugin states.
... The dense set of late reflections are added by a feedback network. Matrix theory has been invoked for the design of "good reverberation" [5][6][7][8]. ...
... To preserve the stability of the system, the matrix U d has to be a unitary matrix, which restricts the design of U in the s-domain to skew-hermitian matrices. For practical applications it is suitable to design the matrix U d directly [5][6][7]. ...
Conference Paper
Full-text available
The attachment of feedback loops to physical or musical systems enables a large variety of possibilities for the modification of the system behavior. Feedback loops may enrich the echo density of feedback delay networks (FDN), or enable the realization of complex boundary conditions in physical simulation models for sound synthesis. Inspired by control theory, a general feedback loop is attached to a model of a vibrating membrane. The membrane model is based on the modal expansion of an initial-boundary value problem formulated in a state-space description. The possibilities of the attached feedback loop are shown by three examples, namely by the introduction of additional mode wise damping; modulation and damping inspired by FDN feedback loops; time-varying modification of the system behavior.
... Various unitary feedback matrices have been proposed to enhance specific properties of the FDN: computational efficiency (circulant [1], sparse [171], Householder [161]), dense impulse response (Hadamard [161], circulant based on Gallois sequences [172]), fitness to measured impulse response with a genetic algorithm [95], spectral flatness [173] and approximating a geometric model [94,174]. • block triangular concatenation: By Theorem 8, two unilossless matrices E and F may be concatenated into a block triangular matrix E G 0 F where G is an arbitrary matrix. ...
... m N are the lengths of the N delays in samples. The echo density of the parallel comb-filter section is constant over time and it was soon realized that the echo density can be increased by interconnecting the feedback of the comb-filters resulting in the Feedback Delay Network (FDN)[161,171,172].In this chapter, we determine the echo density of an FDN over time and by this provide insight into the temporal evolution and the mixing time of an FDN's impulse response.Opposed to counting echoes in the impulse response, the presented prediction method gives means to determine the analytic properties of the echo density. It is shown that, under certain assumptions, the echo density follows a polynomial function. ...
Thesis
Full-text available
In today's audio production and reproduction as well as in music performance practices it has become common practice to alter reverberation artificially through electronics or electro-acoustics. For music productions, radio plays, and movie soundtracks, the sound is often captured in small studio spaces with little to no reverberation to save real estate and to ensure a controlled environment such that the artistically intended spatial impression can be added during post-production. Spatial sound reproduction systems require flexible adjustment of artificial reverberation to the diffuse sound portion to help the reconstruction of the spatial impression. Many modern performance spaces are multi-purpose, and the reverberation needs to be adjustable to the desired performance style. Employing electro-acoustic feedback, also known as Reverberation Enhancement Systems (RESs), it is possible to extend the physical to the desired reverberation. These examples demonstrate a wide range of applications where reverberation is created and enhanced artificially employing signal processing techniques. A major challenge of designing artificial reverberators is the high complexity of the physical reverberation process. Even small office spaces of 40 m^3 exhibit more than 10^7 acoustic modes, in concert halls the number of acoustic modes can surpass 10^9 in the audible range. The room geometry, as well as the interaction with the boundary materials, can be as well fairly complex. Whereas these complex considerations are mandatory for simulations of specific spaces, used for example for the acoustic and architectural planning of a concert venue, they are somewhat misleading in the realm of artistic applications. The focus on perceptually convincing artificial reverberation algorithms provides the freedom to make some simplifications to the generation process, leading to the recursive systems, which play a central role in this dissertation. Two specific formulations of recursive systems for artificial reverberation are considered: Firstly, Feedback Delay Networks (FDNs) which are built around multiple delays which are fed back to their inputs and by this mimic the recursive process of sound waves bouncing back and forth in an acoustic space. And secondly, RESs, which are installed in rooms to extend the physical reverberation via electro-acoustic feedback between microphones and loudspeakers. The main objective of artificial reverberators is to recreate and enhance room impulse responses while considering three aspects: i) accurate recreation of physical spaces; ii) delivering perceptually convincing spaces; and iii) efficiency of processing and parameterization. The primary goal of this dissertation is to achieve better control over the evolution of the artificial reverberation over time, namely the evolution of normal modes and reflections over time. The decay rate of normal modes most importantly determines the stability of the system, but also the perceptual quality of the artificial reverberation. For this purpose, existing network topologies for artificial reverberation are unified in the general FDN framework. For the FDN, an analytic formulation of the polynomial governing the recursive behavior is presented from which analytic constraints on the angular distribution of the decaying modes are derived. Lossless FDNs are commonly used as a design prototype for artificial reverberation algorithms for which all normal modes neither decay nor rise. The lossless property is dependent on the feedback matrix, which connects the output of a set of delays to their inputs, and the lengths of the delays. This work presents the most general class of feedback matrices which constitutes lossless FDNs regardless the lengths of the delays. As a secondary goal, the temporal features of impulse responses produced by FDNs, i.e., the number of echoes per time interval and its evolution over time, are analyzed. This so-called echo density is related to known measures of mixing time and their psychoacoustic correlates such as perception of the room size. It is shown that the echo density of FDNs follows a polynomial function, whereby the polynomial coefficients can be derived from the lengths of the delays for which an explicit method is given. The mixing time of impulse responses can be predicted from the echo density, and conversely, the desired mixing time can be achieved by a derived mean delay length. In the last part of this dissertation, a novel time-variant reverberation algorithm is introduced. By modulating the feedback matrix nearly continuously over time, an intricate pattern of concurrent amplitude modulations of the feedback paths evolves. It is demonstrated that the perceived quality of the decaying normal modes can be enhanced by the feedback matrix modulation. The same technique of time-varying feedback matrices is applied in multichannel sound systems to improve the system's stability. It is shown with a statistical approach that time-varying mixing matrices can achieve optimal stability improvement for a higher number of channels. A listening test demonstrates the improved quality of time-varying mixing matrices over comparable existing techniques.
... There are also works which proposed novel feedback delay structures [Dahl and Jot, 2000] or feedback matrices [Menzer and Faller, 2010;Rocchesso, 1997]. [Frenette, 2000] [Välimäki et al., 2012]. ...
Thesis
There is an increasing interest in creating interactive virtual worlds due to the wide variety of potential applications in entertainment and education. The 3D acoustic scene can be synthesized from two perspectives : the physical approaches and the perceptual approaches. Acoustic radiance transfer method is an efficient ray-based method to model the diffuse reflections and the late reverberation. An extension of the Radiance Transfer Method (RTM) is proposed in this thesis, which allows modeling the early part of specular reflections while keeping the advantage of the original model for the late reverberation simulation. Feedback delay networks are widely used structures to generate the late reverberation. A new method is presented in this thesis, which inherits the efficiency of the Feedback Delay Network (FDN) structure, but aims at linking the parameters of the FDN directly to the geometries of the modeled environment. The relation is achieved by assigning a physical meaning to each delay line and studying the sound energy exchange between them. Then the physical approachand the perceptual approach are combined. The simplified acoustic Radiance Transfer Method, with extension for both specular and diffuse reflections, is incorporated with the Feedback Delay Networks. The new reverberator, despite of modeling the diffuse and late reverberation, is also capable of simulating the early and specular reflections with accuracy.
... Notice the absorption filters in the feedforward path that allow controlling the decay rate at different frequencies, and a tonal correction filter that is used to equalize the reverberator frequency response so that the generated reverberation sounds more natural. The original design uses a Householder matrix for the feedback path, however other unitary matrices can also be used [90]. These matrices can also be time-varying, resulting in improved perceptual characteristics [91]. ...
Article
Developments in immersive audio technologies have been evolving in two directions: physically motivated systems and perceptually motivated systems. Physically motivated techniques aim to reproduce a physically accurate approximation of desired sound fields by employing a very high equipment load and sophisticated, computationally intensive algorithms. Perceptually motivated techniques, however, aim to render only the perceptually relevant aspects of the sound scene by means of modest computational and equipment load. This article presents an overview of perceptually motivated techniques, with a focus on multichannel audio recording and reproduction, audio source and reflection culling, and artificial reverberators.
... This can be accounted to the FDN's flexibility in design, computational efficiency, and independent control over energy decay, amount of diffusion and total energy in each frequency band [13]. Further work has been done to maximize the echo density of FDNs by using specific feedback matrices [18]- [20]. ...
Article
Feedback delay networks (FDNs) are frequently used to generate artificial reverberation. This contribution discusses the temporal features of impulse responses produced by FDNs, i.e., the number of echoes per time unit and its evolution over time. This so-called echo density is related to known measures of mixing time and their psychoacoustic correlates such as auditive perception of the room size. It is shown that the echo density of FDNs follows a polynomial function, whereby the polynomial coefficients can be derived from the lengths of the delays for which an explicit method is given. The mixing time of impulse responses can be predicted from the echo density, and conversely, a desired mixing time can be achieved by a derived mean delay length. A Monte Carlo simulation confirms the accuracy of the derived relation of mixing time and delay lengths.
... Various unitary feedback matrices have been proposed to enhance specific properties of the FDN: computational efficiency (circulant [10], sparse [31], Householder [14]), dense impulse response (Hadamard [14], circulant based on Gallois sequences [32]), fitness to measured impulse response with a genetic algorithm [33], spectral flatness [34] and approximating a geometric model [35], [36]. All those examples of feedback matrices are unitary and therefore by Lemma 1 unilossless. ...
Article
Lossless Feedback Delay Networks (FDNs) are commonly used as a design prototype for artificial reverberation algorithms. The lossless property is dependent on the feedback matrix, which connects the output of a set of delays to their inputs, and the lengths of the delays. Both, unitary and triangular feedback matrices are known to constitute lossless FDNs, however, the most general class of lossless feedback matrices has not been identified. In this contribution, it is shown that the FDN is lossless for any set of delays, if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix. The necessity of the generalized class of feedback matrices is demonstrated by examples of FDN designs proposed in literature.
... To achieve a high-quality late reverberation, the feedback loop should be lossless, i.e. energypreserving, hence typically the feedback matrix A is designed to be unitary. Each particular choice of the feedback matrix has corresponding implications on subjective or objective qualities of the reverberator [32]; e.g. in the particular case of the identity matrix, A = I, the FDN structure reduces to N comb filters connected in parallel and acts as the Schroeder reverberator [3]. Note, however that unitary matrices are only a subset of possible lossless feedback matrices [10], [33]; we elaborate on this point in the next subsection. ...
Article
Full-text available
An acoustic reverberator consisting of a network of delay lines connected via scattering junctions is proposed. All parameters of the reverberator are derived from physical properties of the enclosure it simulates. It allows for simulation of unequal and frequency-dependent wall absorption, as well as directional sources and microphones. The reverberator renders the first-order reflections exactly, while making progressively coarser approximations of higher-order reflections. The rate of energy decay is close to that obtained with the image method (IM) and consistent with the predictions of Sabine and Eyring equations. The time evolution of the normalized echo density, which was previously shown to be correlated with the perceived texture of reverberation, is also close to that of IM. However, its computational complexity is one to two orders of magnitude lower, comparable to the computational complexity of a feedback delay network (FDN), and its memory requirements are negligible.
... Algorithms based on delay networks are comprised of delays, filters, and feedback paths and they are characterized by low memory requirements, low computational complexity, and good parameterization but often lacking realism. The original algorithms by Schroeder and Logan, which were subsequently improved by Moorer [4], and the feedback delay networks (FDN) originally proposed by Jot and Chaigne are popular examples of delay network algorithms [5,6,7,8]. ...
Conference Paper
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An efficient algorithm approximating the late part of room reverberation is proposed. The algorithm partitions the impulse response tail into variable-length segments and replaces them with a set of sparse FIR filters and lowpass filters, cascaded with several Schroeder allpass filters. The sparse FIR filter coefficients are selected from a velvet noise sequence, which consists of ones, minus ones, and zeros only. In this application, it is sufficient perceptually to use very sparse velvet noise sequences having only about 0.1 to 0.2% non-zero elements, with increasing sparsity along the impulse response. The algorithm yields a parametric approximation of the late part of the impulse response, which is more than 100 times more efficient computationally than the direct convolution. The computational load of the proposed algorithm is comparable to that of FFT-based partitioned convolution techniques, but with nearly half the memory usage. The main advantage of the new reverberator is the flexible parameterization.
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Modelling, simulation and auralisation of room acoustics plays an important role in computer games and virtual reality applications by increasing the level of realism. Accurate simulation of room acoustics is a computationally costly process which is often substituted with artificial reverberators that provide a computationally simpler alternative. However, such systems lack the accuracy and are not in general able to accurately simulate important aspects of room acoustics such as early reflections, source/microphone directivity, and frequency-dependent absorption. A new type of interactive and scalable room simulator named the scattering delay network (SDN) was recently proposed by the authors. A frequency-domain analysis and implementation of that simulator is presented in this paper. Numerical simulation examples which demonstrate the utility of the proposed system are provided.
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This paper discusses efficient digital signal processing algorithms for real-time synthesis of dynamically controllable, natural-sounding artificial reverberation. A general modular framework is proposed for configuring a spatial sound processing and mixing system according to the reproduction format or setup and the listening conditions, over loudspeakers or headphones. In conclusion, the implementation and applications of a spatial sound processing software are described, and approaches to control interface design and effective distance effects are reviewed.
Article
A general approach is proposed to the problem of realizing a recursive digital display network capable of simulating in real time the perceptively relevant characteristics of the reverberation decay in a room. The analysis/synthesis method presented makes it possible to imitate the late reverberation of a given room by optimizing some of the reverberant filter's parameters. The analysis phase is based on a time-frequency representation of the energy decay, computed from an impulse response measured in the room. The energy decay relief is proposed as a spectral development of the integrated energy decay curve introduced by Schroeder. Its three-dimensional representation allows perceptively relevant visual comparison of two room responses (measured or artificial) and accurate calculation of some widely used objective criteria of room acoustic quality.