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Audio Engineering Society

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Presented at the 128th Convention

2010 May 22–25 London, UK

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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

Page 2 of 16

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

Page 3 of 16

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 αi−sin 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

Ui∈8

<

:

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

=

;

AES 128th Convention, London, UK, 2010 May 22–25

Page 4 of 16

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 ((i−1)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

Page 5 of 16

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

Page 6 of 16

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

Page 7 of 16

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 non−sparse

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 non−sparse

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 non−sparse

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

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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 Kovaˇcevi´c, Wavelets

and Subband Coding, Prentice Hall, Englewood

Cliﬀs, New Jersey, USA, 1995.

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