Optimized Structure for Multichannel Digital Reverberation

MIROSLAV BALÍKDepartment of Telecommunications

Faculty of Electrical Engineering and Communication TechnologiesBrno University of TechnologyPurkyňova 118, 612 00 Brno,

CZECH REPUBLIC

Abstract: The paper deals with a structure for implementing of 8-channel simulation of acoustic rooms (known asartificial reverberation) as a discrete system. The paper begins with a description of essential terminology andparameters. The next chapter starts with the description of total structure block diagram. The following chapterscontinue with structures for early reflection simulation and for subsequent reverberation simulation. The main partsof structures are described by their transfer function. In the end the impulse responses for different parameter settingsare shown.

Key-Words: artificial reverberation, multichannel, simulation, digital audio signal processing

1 IntroductionSimulating an acoustic room means finding a

system that simulates the properties of sound beampropagation from a source of sound towards the listenerin an open or closed room. The following physicalparameters are of fundamental effect on the propertiesof this system – the size of the closed room, itsgeometrical lay-out, the material that the walls and theobjects in the closed room are made of, the position ofthe source of sound and the listener’s position. Thereverberation structures are based on the physical or theperceptual approach or the approximation approach ofthe acoustic room properties (see [3]). For the digitalmusical effect it is best to use the perceptual approach.The perceptual approach starts from the sensoryperception of a sound that is being played back for thelistener in the respective acoustic room. The perceptualapproach aims at reducing the computationrequirements of the simulation algorithm. The structuredoes not simulate the IR of an acoustic room directly, itonly simulates the specific features of an acoustic room.These specific features are defined as humanperceptions. Let the space of all the perceptions causedby sound beam propagation in an acoustic room bedistributed into D dimensions that would correspond toall independent perceptions caused by sound raypropagation. If each of the perceptions can be describedby a definite physical property of the acoustic room, adigital filter can be designed with D parameters thataccurately simulate each of the D independentperceptions. In the structure based on perceptualapproach two basic structures can in most cases be

distinguished, which are based on completely differentalgorithms. These are the structure that simulates earlyreflections of the sound beam and the structuresimulating subsequent reverberation.

dB

directsound

subsequentreverberation

time

earlyreflections

Fig. 1. Echogram of a general reverberation structurebased on perceptual approach

In Fig. 1 the main parts of echogram can bedistinguished, which correspond to different types ofstructure output.

To specify all D parameters it is necessary to specifythe number of outputs for early reflection simulationand for subsequent reverberation. For speculativepanning it is necessary to produce two outputs for earlyreflection simulation and eight uncorrelated outputs forsubsequent reverberation to have a structure compatiblewith all present-day multichanel standards (fromconventional two channel strereophonics 2.0 tomultichannel 7.1).

For each of these types of output next D parametersshould be controlled by the user:• the ratio of the original and the processed signal• the ratio of the output signal of the early reflections

structure and the subsequent reverberation structure

• the mutual delay between the output signals ofindividual structures and the input signal

• the size of acoustic room described by the decaytime

• the shape of acoustic room defined by anotherreflection distribution

• the colour – it describes the amount of absorption ofa certain part of the frequency spectrum of soundbeams that originates on material with frequency-dependent absorptiveness during beam propagationin a room• for early reflections the absorption is modelled by

frequency filters placed in the direct path ofeffected signal

• for subsequent reverberation the absorption ismodelled by a frequency-dependent decay time,i.e. frequency filters placed in the system’sfeedback or in the direct path with unit feedback

• the density of reflections in the acoustic room, thenumber of non-zero values in the discrete IR

This paper is based on my extensive research firsttime published in my PhD thesis [1]. The contributionof this paper is a presentation of new reverberationstructures, which will be prepared for real-timeprocessing and which will meet the requirement for 8-channel output prepared for the speculative pannigmethod (see [2]).

2 Total structure block diagramThe block diagram of total structure is shown in Fig.

2. The part of structure for early reflection simulation isbased on a series of three identical delay lines, but withdifferent parameters. The delay line outputs represent 18most significant reflections. These outputs lead to linearscaling matrix 18x2 in size (18 inputs and 2 outputs forleft and right early reflections channel). This matrixcontrols the absorption coefficients for each delay lineoutput and its presence in the left or right channel ofearly reflections.

11ySR8(n)

10ySR7(n)

9ySR6(n)

8ySR5(n)

7ySR4(n)

6ySR3(n)

5ySR2(n)

4ySR1(n)

3yERR(n)

2yERL(n)

1yDR(n)z-(P)

xnsaf(n) ymcf6(n)

xnsaf(n) ymcf5(n)

xnsaf(n) ymcf4(n)

xnsaf(n) ymcf3(n)

xnsaf(n) ymcf2(n)

xnsaf(n) ymcf1(n)

xdf26(n)

xmcf1(n)

xmcf2(n)

xmcf3(n)

xmcf4(n)

xmcf5(n)

xmcf6(n)

xmcf7(n)

xmcf8(n)

ySR1(n)

ySR2(n)

ySR3(n)

ySR4(n)

ySR5(n)

ySR6(n)

ySR7(n)

ySR8(n)

xP(n) ynsa(n)

xnsa(n)ynsaf(n)

xnsaf(n) ymcf8(n)

xnsaf(n) ymcf7(n)

xdf16(n)

1x(n)

ydf31(n)

ydf32(n)

ydf33(n)

ydf34(n)

ydf35(n)

ydf36(n)

ydf21(n)

ydf22(n)

ydf23(n)

ydf24(n)

ydf25(n)

ydf26(n)

x(n)

ydf11(n)

ydf12(n)

ydf13(n)

ydf14(n)

ydf15(n)

ydf16(n)

xdf11(n)

xdf12(n)

xdf13(n)

xdf14(n)

xdf15(n)

xdf16(n)

xdf21(n)

xdf22(n)

xdf23(n)

xdf24(n)

xdf25(n)

xdf26(n)

xdf31(n)

xdf32(n)

xdf33(n)

xdf34(n)

xdf35(n)

xdf36(n)

yERL(n)

yERR(n)

3 x Hdlf(z) LSM 18 x 2 LSM 8 x 88 x Hmcf(z)

Hnsa(z)

FSR(z)

AER ASR

Fig. 2: Total structure for optimized 8-channel digital reverberation

The part of structure for subsequent reverberationstarts with delay z-P of the input signal. Next blockHnsa(z) controls the proper subsequent reverberationdensity. Filter FSR(z) is used to control the initialspectrum of subsequent reverberation. Then the signalleads to eight modified comb filters Hmcf(z) to controlfrequency dependent decay time of subsequentreverberation. The eight outputs lead to a unitary linearscaling matrix. This matrix generates eight uncorrelated

surround outputs prepared for speculative panning.

3 Structure for early reflectionssimulation

The early reflections should be frequency dependentand of proper density. These two parameters are set withthe help of a structure with transfer function denotedHnaf(z) (see Fig. 3).

Fig. 3: Delay line with higher density and frequency dependent outputs

Fig. 4: Nested all-pass filter with frequency dependent output

This structure is based on a nested all-pass filterwith frequency dependent output is shown in Fig. 4.

The next few equations define transfer functions( )df p

H z of each output from one of the delay lines

used. Ha(z) is the transfer function of a common all-passfilter and Hnaf(z) is defined by substitution (4).

( )a1

a1

-1

a -11

N

NgH

g− +

=−

zzz

(1)

Transfer function G(z) describes all structures in thedirect path inside the feedback and feedforward paths.

( ) ( ) ( ) ( )( )

a2-a er

N n

d

GG H F

G= =

zz z z z

z(2)

( ) ( ) ( )

( )

a a1 2

a1

- -1 er

-1

,

1-

N Nn

Nd

G g F

G g

= −

=

z + z z z

z z(3)

( ) ( ) ( )( ) ( )

( ) ( ) ( )

( ) ( ) ( )

a aa a 1 21 2

a aa a 1 21 2

2naf

2

-- -2 1 2 1 er er

-- -er 1 er 1 2 2

d n

d n

N NN N

N NN N

g G GH

G g G

g g g g F F

F g F g g g

+

+

− += =

−

− + − +=

− + −

z zz

z z

z z z z z

z z z z z

(4)

Equation (5) defines transfer function ( )df pH z as

the series of a delay line and the nested all-pass filter.

( ) ( )d

1df naf

p

p

p

N

H H−∑

=z z z (5)

6ydf6(n)

5ydf5(n)

4ydf4(n)

3ydf3(n)

2ydf2(n)

1ydf1(n)

Hnaf(z) z-(Nd6)z-(Nd5)z-(Nd4)z-(Nd1) z-(Nd3)z-(Nd2)1x(n)

1ynaf(n)

-g1

-g2

g1

g2

Fer(z)z-(Na1) z-(Na2)1x(n)

Note that this structure with transfer function( )df p

H z is used three times in a series. The last (p = 6)

output from the current delay line structure leads to theinput of next delay line structure.

4 Structure for subsequentreverberation

The delayed input signal leads to a structure thatcontrols the distribution and density of reflections ofsubsequent reverberation. The structure is based on aseries of four all-pass filters in combination with anested all-pass filter (see Fig. 5).

1ynsa(n)

g4

-g3

g6

-g5

g5

-g6

-g7

g3

-g4

g7

z-(Na4)

z-(Na6)

z-(Na3)

z-(Na5)

z-(Na7)1xP(n)

Fig. 5: Series of four all-pass filters combined with nested all-pass filter

The transfer function denoted Hsa(z) is defined as aseries of four all pass filters (6). Structure G(z) is aseries combination of four all-pass filters and delay z-Na7

(see (7)).

( )a

a

6

sa3 1

i

i

Ni

Ni i

gHg

−

−=

− +=

−∏ zz

z(6)

( )( )

( )( )( )

a a7

a

6

36

3

1

i

i

N Ni

ni

N di

i

g GG

Gg

− −

=

−

=

− += =

−

∏

∏

z z zz

zz(7)

Transfer function Hnsa(z) is defined by this substitution:

( ) ( ) ( )( ) ( )7

7

ansa

a

d n

d n

g G GH

G g G− +

=−

z zz

z z(8)

This relatively complex structure for controllinginitial reflection density is capable of setting differentdistributions of impulses in the impulse response. It ispossible to set a very low density by using only one all-

pass filter in series with another or with a nested one.The unused all-pass filters will have feedback andfeedforward gains equal to 0. For maximum density,which is required for large hall or church simulation, all4 all-pass filters nested with the fifth all-pass filtershould be used.

In Fig. 6 the block diagram of a modified combfilter with frequency dependent decay time is shown.This structure is typical of lossless feedback (g=1). Thetransfer function of each of q outputs is denoted

( )mcfqH z and defined by equation (9).

( ) ( )( )( )

c

c

srmcf nsa

sr1

q

q

q q

q

Nq

Nq

k FH H

k F

−

−=−

z zz z

z z(9)

In the already described structures the filters withtransfer function FER(z) or FSR(z) are used. These filtershave a fundamental influence on computationalcomplexity of the total structure. For the lowest possiblecomputational complexity, one pole low-pass filters

1ymcf(n)

k Fsr(z)z-(Nc)1xnaf(n)

Fig. 6: Modified comb filter with frequency dependent decay time

should be used. I have used two parametric filters (see[8]) in series, a low shelving filter and a high shelvingfilter. The computational complexity is higher, butparametric filters allow a more precise control offrequency dependent absorption.

The calculation of all structure parameters and linearscaling matrixes is based on the analysis of desiredroom impulse response and sound directivity. Thisanalysis is based on the statistic model in the time andfrequency domain, energy-decay curve and energy-decay relief. This analysis is complex and beyond thescope of this paper (see [1]).

5 ResultsThe total structure presented is capable of

controlling these parameters:

Early reflection simulation:• the delay time of 18 reflections for the left and the

right early reflection channel by d pN vectors

• 36 absorption and panning coefficients controlledby linear scaling matrix AER

• frequency dependent absorption for the first to thethird order reflection by three filters FER(z) for eachsextuplet of delay line outputs

• early reflection density controlled by three nestedall-pass filters Hnaf(z) for each of three orders ofreflections separately

Subsequent reverberation simulation:• the delay time between direct output and

subsequent reverberation outputs controlled bydelay z-P

• subsequent reverberation density controlled by fourall-pass filters nested with the fifth all-pass filter bystructure Hnsa(z)

• initial spectrum of subsequent reverberationcontrolled at low and high frequencies by filterFSR(z)

• eight resonance frequencies controlled by cqN vector

• frequency dependent decay time controlled at lowand high frequencies by filters ( )SRq

F z and kq

• eight uncorrelated outputs controlled by linearscaling matrix ASR

Next Fig. 7 shows examples of the impulse responseof one output of three structures with transfer function

( )df pH z connected in series. The frequency dependent

absorption has been disabled for better impulse responsereadability. Fig. 7a shows impulse response, when theincreasing of density is disabled (all gains ofcorresponding all-pass filters are equal to 0). Fig. 7bshows impulse response with enabled increasing ofdensity by one (the first) nested all-pass filter. It causesto slowly increasing impulse response density. Fig. 7cshows the case when all nested all-pass filters are used.The density gets near to its maximum very fast.

0 500 1000 1500 2000 2500 3000 3500 4000

−0.5

0

0.5

a) IR, g11~12

= {0, 0}, g21~22

= {0, 0}, g31~32

= {0, 0}, Fs = 44.1 kHz

hER

(n)

n →

h →

0 500 1000 1500 2000 2500 3000 3500 4000

−0.5

0

0.5

b) IR, g11~12

= {−.5, −.5}, g21~22

= {0, 0}, g31~32

= {0, 0}, Fs = 44.1 kHz

hER

(n)

n →

h →

0 500 1000 1500 2000 2500 3000 3500 4000

−0.5

0

0.5

c) IR, g11~12

= {−.6, −.45}, g21~22

= {−.3, −.25}, g31~32

= {−.2, −.35}, Fs = 44.1 kHz

hER

(n)

n →

h →

Fig. 7: Impulse response of one early reflection structure output a) without increasing density, b) with slowlyincreasing density and c) with maximum density

Last Fig. 8 shows details of the beginning ofimpulse response of the structure for subsequentreverberation simulation. In both cases the modifiedcomb filters were set to the same parameters. Thedifference is in the structure Hnsa(z) for densityincreasing. In first case (Fig. 8a) the series of four all-pass filters is used and in the second case (Fig. 8b) isalso the nested all-pass used. The impulse distribution iscompletely different.

6 ConclusionThe total structure was tested for different input

signals – from discrete Dirac impulse to short frames ofspeech, vocals and music. Structures were simulated inthe Simulink. The parameters and matrixes werecalculated by the Matlab function and its results werepassed on to the Simulink model. The structuredesigned is perfectly suitable for multichanel digitalreverberation used as a sound effect in HDR softwarestudios. The results were compared with Sony DRE-S777 - professional audio convolution workstation. The

first results show the reverberation quality to be higherthan quality of common reverberation tools and,moreover, with acceptable digital system loading.

7 References[1] Balík, M. 2003. Structures for Auditory Space

Simulation. Ph.D. thesis. Department ofTelecomunications, Faculty of ElectricalEngineering and Communication Technologies,Brno University of Technology. (in czech)

[2] Balík, M. 2002. Principles of Designing anAlgorithm for Acoustic Room MultichannelSimulation. Proceedings of Asia-Pacific Conferenceon Circuits and Systems APCCAS´02. Singapore, pp.303 - 610, ISBN 0780376919

[3] Balík, M. 2002. Sound Source Panning Methods forMultichannel Reverberator. Proceedings of theInternational Conference Research inTelecommunication Technology RTT 2002. EDIS-Žilina University publisher, s. 34 - 72, ISBN8071009911

0 500 1000 1500 2000 2500 3000

−0.04

−0.02

0

0.02

0.04

a) IR − detail , g3~7

= {−.5, −.5, −.5, −.5, 0}, Fs = 44.1 kHz

hSR

(n)

n →

h →

0 500 1000 1500 2000 2500 3000

−0.04

−0.02

0

0.02

0.04

b) IR − detail , g3~7

= {−.55, −.55, −.65, −.85, −.21}, Fs = 44.1 kHz

hSR

(n)

n →

h →

Fig. 8: Detail of the begining of impulse response of one subsequent reverberation output a) using series of all-passfilters, b) using series of all-pass filters and nested all-pass filter

[4] Schroeder, M. R. Logan, B. F. 1961. ColorlessArtificial Reverberation. J. Audio EngineeringSociety. Vol. 9, No. 3.

[5] Schroeder, M. R. 1962. Natural Sounding ArtificialReverberation. J. Audio Engineering Society. Vol.10, No. 3.

[6] Gardner, W.G. 1992. The virtual Acoustic Room.Master Science Thesis at the MIT.

[7] Rochesso, D., and Smith, J. O. 1997. Circulant andElliptic Feedback Delay Networks for ArtificialReverberation. IEEE trans. Speech & Audio 5(1).

[8] Zolzer, U. 1997. Digital Audio Signal Processing.John Willey & Sons Ltd., ISBN 0471972266