adaptive filters

Departement Elektrotechniek ESAT-SISTA/TR 2000-9

Fullband Error Adaptation of Subband Adaptive Filters 1

Koen Eneman, Marc Moonen 2

April 2000

Published inthe Proceedings of the IEEE International Symposium

2000 on AdaptiveSystems for Signal Processing, Communications

and Control(AS{SPCC), pp. 293{298

1

This report is available by anonymous ftpfrom ftp.esat.kuleuven.ac.be in the

directorypub/SISTA/eneman/reports/00-9.ps.gz

2

ESAT (SISTA) - Katholieke Universiteit Leuven, Kardinaal Mercier-

laan 94, 3001 Leuven (Heverlee), Belgium, Tel. 32/16/321809,

Fax 32/16/321970, WWW: http://www.esat.kuleuven.ac.be/sista. E-mail:

koen.eneman@esat.kuleuven.ac.be. MarcMoonenisaResearchAssociatewith

theF.W.O.(FundforScienticResearch{Flanders). Thisresearchworkwas

carriedoutat theESAT laboratoryof theKatholiekeUniversiteit Leuven,in

theframeoftheBelgianState,PrimeMinister'sOÆce{FederalOÆceforSci-

entic,TechnicalandCulturalAairs{InteruniversityPolesofAttractionPro-

gramme{IUAPP4{02(1997{2001): Modeling,Identication,Simulationand

Control ofComplexSystems, theConcertedResearch ActionMIPS(`Model{

basedInformationProcessingSystems')andGOA{MEFISTO{666(Mathemat-

ical Engineeringfor Informationand CommunicationSystemsTechnology)of

theFlemishGovernment,ResearchProjectF.W.O.nr. G.0295.97(`Designand

implementationofadaptivedigitalsignalprocessingalgorithmsforbroadband

applications'). Thescienticresponsibilityisassumedbyitsauthors.

Koen Eneman Marc Moonen

ESAT { KatholiekeUniversiteit Leuven

KardinaalMercierlaan 94,B{3001Heverlee {Belgium

email : koen.eneman@esat.kuleuven.ac.be

marc.moonen@esat.kuleuven.ac.be

Abstract

For many years now, subband and frequency{domain

adaptive ltering techniqueshave beenproposedfor the

identicationofhigh{orderFIRsystems. Standardfull-

band algorithms are less attractiveas the implementa-

tion cost is higher and their convergence behaviour is

typically worse.

Subband processing has many desirable properties.

However,whenusedtoimplementadaptivelters,vari-

oussideeectsoccurwhichreduceperformance. Onthe

other hand, frequency{domain adaptive lters, suchas

thePBFDAF,donotsuerfromtheseproblemsdespite

being(nearly)equivalent tosubbandadaptive lters,be

itwith\poor"lterbanks.

In this paper an alternativefullband error basedadap-

tationschemeforsubbandadaptivesystemswillbepro-

posed. Itwillbeshownthatthe weightupdatingmecha-

nismoftheso{calledunconstrainedPBFDAFisclosely

related to the proposed fullband error adaptation algo-

rithm.

1 Introduction

Subbandandfrequency{domainadaptiveschemeshave

beenatopic ofinterestfor manyyearsnow. Theyare

employed to identify high{order FIR systems and are

a promising alternative for standard fullband adapta-

tion algorithms such as LMS. Still, with the available

multirate techniques, it is diÆcult to meet all the re-

quirements.

Frequency{domain techniques are well understood [6]

andhencetheirperformanceisverytractable. Onthe

other hand, subband adaptive lters |at rst sight|

mayhavealowercomplexityandabetterperformance.

Unfortunately,thispictureofthesubbandapproachis

certainlytoooptimistic.

In this paper we will focus on an alternative fullband

error adaptation algorithm for subband adaptive sys-

tems and show that the unconstrainedPBFDAFuses

ter weights. This is an attempt to generalise and ex-

tendthefrequency{domainaliasingcompensationtech-

niquesto subbandadaptivesystems.

2 Subband Adaptive Filtering

2.1 General setup

The general setup for a subband adaptive system is

showningure1. Theloudspeakerandmicrophoneare

...

...

...

... ...

+ -

-

+ +

- near-end signal

far-end signal

error signal

adaptive filters

analysis filter bank synthesis filter bank

+ +

+ +

e

0

1

M

1 d0

d1

dM1 y0

y1

yM1 L

L

L

L

L L

L L L

d=s+w?x s

x

f

f

f i=0

i=1

i=M 1

H

0 H0

H1 H1

HM

1 HM

1

G

0

G1

GM

1 F0

F1

FM

1 w[k]

Figure 1: Subband adaptive lter with ideal lter

banks: echocancellation setup

addedforconvenience,indicatinghowtheadaptivel-

terstructuremaybeemployedinanacousticechocan-

cellation setup. While acoustic echo cancellation has

beenadrivingapplicationformanyresearchersinthis

eld, all results obviously apply to other applications

too.

TheinputsignalsxanddarefedintoidenticalM{band

analysis lter banks. After subsamplingwith afactor

+

... ... ...

+

... ...

... ...

+

F0 H (z)

H (z)

JG T

(z) L

L

L

L

L

L

L L L

z 1

z 1

z 1 z

1

z 1

z 1 j=0

x d

e j=L 1

j=L 1

F1

FM

1

Figure 2: Subband adaptive lter : polyphase imple-

mentation

each subband. The outputs of the subband adaptive

lters are recombined in the synthesislter bank and

fedtotheoutput. Theidealcharacteristicsoftheana-

lysis bank lters H

i

and synthesisbank lters G

i are

shown (idealbandpasslters). Dueto aliasing eects,

thissetupwillonlyworkforM>L.

2.2 Polyphase implementation

Theoutputsoftheanalysisbankltersareimmediately

downsampled. Henceitischeapertodoalllteropera-

tionsatthedownsampledrate. Byre{arranginggure

1weobtain gure2. H(z) and G (z) are respectively

called theanalysis andsynthesispolyphase matrix. J

istheanti{diagonalmatrix. Element(i;j)ofH(z)is

[H(z)]

ij

=H

ij:L (z)



i=0!M 1

j=0!L 1

(1)

H

i

j:L

(z)is the j{th outof L polyphase componentof

the i{th subband lter h

i

[k], in other words the z{

transformofh

i

[j+Lk]. Similarly,

[G(z)]

ij

=G

i

j:L (z)



i=0!M 1

j =0!L 1

(2)

2.3 DFT modulatedsubband adaptive lters

Subband adaptive systems are often based on DFT

modulated lterbanks. M subbandltersare derived

fromasingleprototypelterh

0 [k] :

h

i [k]=h

0 [k]e

j 2 k i

M

; i=0!M 1

,

H

i

(z)=H

0 (e

j 2 i

M

z) (3)

Itappearsthattheanalysisltersarefrequencyshifted

versions of each other and each ltercoversa partof

thefrequencyspectrum.

+

... ...

...

... ...

... ...

+ +

F

F

F 1 F0

B(z)

B(z)

C(z) L

L

L

L

L

L

L L L

z 1 z

1

z 1

z 1

z 1

z 1 j=0

j=0

x d

e j=L 1

j=L 1

F1

FM

1

Figure3: DFTmodulatedsubbandadaptivelter

DFTmodulatedlterbanksmaybe implementedeÆ-

cientlyusing polyphase decomposition and fast signal

transforms. In [1] ageneralframework forDFT mod-

ulatedsubbandsystemswasproposed. ADFTmodu-

latedlterbankwithL{folddownsamplingcanbeim-

plementedasatappeddelay lineof sizeL followedby

astructuredMLmatrixB(z),containingpolyphase

componentsoftheprototypeh

0

,and anMM DFT

matrixF. ForDFTmodulatedlterbanksgure2can

beredrawnresultingingure3. Thesynthesisbankis

constructedinasimilarfashionwithanLM matrix

C(z).

Bysplittingsignalsintosubbandsandsubsequentsub-

sampling faster (initial)convergenceand bettertrack-

ingpropertiesarehopedfor. Astheadaptivecomputa-

tionsaswellasthelterbankconvolutionscanbedone

at a reduced sampling rate, the subband approach is

supposed togiveabetterperformanceat alowercost.

Unfortunately,thispictureofthesubbandapproachis

certainlytoooptimistic[2].

3 Frequency{domain Adaptive Filters

AsacheaperalternativetoLMS,thefrequency{domain

adaptive lter(FDAF)wasintroduced, which is adi-

recttranslationofBlockLMSinfrequencydomain[6].

TheimplementationcostfortheFDAFislow,but the

input{outputdelayintroducedbythealgorithmistyp-

icallytoohigh.

The FDAF can be extended by splitting the acoustic

impulse response in equal parts. In this way a kind

of mixed time{ and frequency{domain adaptive lter

is obtained, called the Partitioned Block Frequency{

Domain AdaptiveFilter (PBFDAF)[5][7]. Here block

lengthscanbeadjusted, resultingin acheapadaptive

lterwithacceptableprocessingdelay.

TheL

FB

{tapsfullband adaptivelterw (n)

[k]atblock

indexnispartitionedin LFB

equalpartsw

p

n

oflength

P each:

w

pn 8p

= 2

6

4 w

(n)

[pP]

.

.

.

w (n)

[(p+1)P 1]

3

7

5

; p=0: L

FB

P 1

W

p

n 8p

= F



w

p

n

0



lP

lL 1

(4)

TheequationsdeningthePBFDAFare 2

:

X

pn 8p

= diag 8

<

: F

2

6

4

x[(n+1)L pP M+1]

.

.

.

x[(n+1)L pP]

3

7

5 9

=

; (5)

y =



0

P 1 0

0 I

L



F 1

L

FB

P 1

X

p=0 X

pn W

pn (6)

d =



0

d

n



lP 1

lL

; d

n

= 2

6

4

d[nL+1]

.

.

.

d[(n+1)L]

3

7

5 (7)

e = d y (8)

W

pn+1 8p

= W

pn +F



I

P 0

0 0

L 1



F 1

X H

p

n Fe(9)

IneachiterationLnewx{samplesaretakenin,andL

newlteroutputsamplesareproduced. Liscalledthe

blocklength, the corresponding input/output delayis

2L 1. Fis anMM DFTmatrix,
=2diag(

m )

contains the subband dependent step sizes and M =

P+L 1. IfP isdivisiblebyL(whichistypicallythe

case), X

pn

=X

0

n pP=L

and henceequation 5requires

only1DFToperation,whichcorrespondstop=0.The

other X

pn

can be recoveredfrom previous iterations.

VeryoftenMischosenequaltoM=P+L 1+s=2 r

withs>0andraninteger. Inmostpracticalapplica-

tionsP=L.

There exists twovariantsof this algorithm, called the

constrained and the unconstrained PBFDAF. For the

unconstrainedversiontheweightupdatecompensation

isleftoutresultinginthefollowingweightupdateequa-

tion:

W

p

n+1 8p

=W

p

n +
X

H

pn

Fe (10)

The unconstrained updating requires 3 DFTs per

iteration whereas the constrained PBFDAF is more

expensive, having an extra 2LFB

P

DFTs to compute.

The latter on the other hand has better convergence

properties.

The PBFDAF turns out to be a special subband

adaptive lter having interesting convergence prop-

erties at a low implementation cost [2]. Despite the

low frequency{selective lter banks upon which the

1

Weassumethat L

FB

P

isinteger.

2

PBFDAF is based the algorithm doesn't seem to

suer from aliasing eects and hence the convergence

propertiesare comparabletothoseof LMS.

4 Alternative adaptation scheme

Inanattempttogeneraliseandextendthefrequency{

domain errorcorrection to subbandadaptive systems,

wenowfocuson analternativeadaptation schemefor

subbandadaptivesystems, which adjuststhe subband

lters F

m

(see gure4) based on thefullband errore

insteadof using thesubband errorsignals

m

=d

m

y

m

directlyasisnormallydonein astandardsubband

scheme(seegure1).

- - -

... ...

... ...

... ...

+ +

+

+

e z

1

z 1

L L L

e0 eL

2 eL

1 d0

d1

dM

1

x0

x1

xM

1

y

0

y

1

yM

1 F

0

F

1

F

M 1

JG T

(z)

JG T

(z)

Figure 4: Oversampled subband adaptivelter : syn-

thesispart

4.1 Fullbanderror adaptation scheme

Dene:

2

6

4 Y

k

0

.

.

.

Y k

M 1 3

7

5

| {z }

Y

k

= 2

6

4 X

k

0

::: 0

.

.

. .

.

. .

.

.

0 ::: X k

M 1 3

7

5

| {z }

X

k

2

6

4 F

k

0

.

.

.

F k

M 1 3

7

5

| {z }

wk

;

(11)

in which(seegure4)

X k

m

= 2

6

4 x

m [k L

f

+1] ::: x

m [k L

f L

SB +2]

.

.

.

.

.

.

.

.

.

x

m

[k] ::: x

m [k L

SB +1]

3

7

5

(12)

F k

m

= 2

6

4 f

k

m [0]

.

.

.

f k

m [L

SB 1]

3

7

5

; Y k

m

= 2

6

4 y

m [k L

f +1]

.

.

.

y

m [k]

3

7

5

w

k

is theadaptiveltervector. L

SB

isthelength of

the subband adaptive lters, L

f

is the length of the

synthesisbankpolyphaselters. Furtherdene

2

6

4 e

0 [k]

.

.

.

e

L 1 [k]

3

7

5

| {z }

E FB

k

= 2

6

4 S

00:L

::: S

M 10:L

.

.

. .

.

.

.

.

.

S

0

L 1:L

::: S

M 1

L 1:L 3

7

5

| {z }

S T

2

6

4

 k

0

.

.

.

 k

M 1 3

7

5

| {z }

E SB

k

(14)

inwhich

2

6

4

 k

0

.

.

.

 k

M 1 3

7

5

| {z }

E SB

k

= 2

6

4 D

k

0

.

.

.

D k

M 1 3

7

5

| {z }

Dk

2

6

4 Y

k

0

.

.

.

Y k

M 1 3

7

5

| {z }

Yk

; (15)

D k

m

= 2

6

4 d

m [k L

f +1]

.

.

.

d

m [k]

3

7

5

;  k

m

= 2

6

4



m [k L

f +1]

.

.

.



m [k]

3

7

5

(16)

S

ml:L

=



g

ml:L [L

f

1] ::: g

ml:L [0]



(17)

E FB

k

isthefullbanderrorvector,E SB

k

containsthesub-

band errors,Srepresentsthesynthesislterbankand

g

ml:L [k] = g

m

[kL+l] is the l{th out of L polyphase

componentsofthem{thsynthesislterg

m [k].

Optimal errorsuppression isobtained whenEq. 14 is

assmallas possible:

min

w

k E

n

jjE FB

k jj

2

2 o

(18)

Theoptimalw is

w

opt

=argmin

w

k jjS

T

D

k S

T

X

k w

k jj

2

2

: (19)

In practice the optimal subband lters are estimated

adaptivelyusingasteepestdescentalgorithm. Thegra-

dientwithrespecttow

k is

rw

k

=2X H

k S



S T

(X

k w

k D

k

): (20)

Therefore,w

opt

canbefoundinaniterativewayusing

afullband errorblockadaptationalgorithm:

w

k +1

= w

k

+2
X H

k S



S T

(D

k X

k w

k ) (21)

= w

k

+2
X H

k S



S T

E SB

k

(22)

with
= diag(

m

) a diagonal matrix with subband

dependent stepsizes. The subband errors are passed

throughthesynthesisbanktwice before theyarecom-

binedwiththefar{endsignalx.

3

Inastandardsubbandadaptationschemethesubband

errorsareminimised,leadingto

min

w

k E

n

jjE SB

k jj

2

2 o

: (23)

Thegradientwithrespecttow

k is

rw

k

=2X H

k (X

k w

k D

k

): (24)

Therefore, theoptimal w canbefound in aniterative

waybyoptimisingthesubbanderrors:

w

k +1

=w

k

+2
X H

k E

SB

k

: (25)

Equation25correspondstoastandardsubbandweights

updating algorithm using a block{LMS algorithm in

each band. The compensation matrix S



S T

is absent

here. It is hoped that employing the fullband error

adaptation scheme i.o. subband error adaptationwill

leadtoimprovedperformance.

4.2 PBFDAFweightupdating

The alternative adaptation scheme studied in para-

graph4.1,leadingtoaweightupdatingequationbased

on fullband errors (Eq. 22) is now applied to the

PBFDAF.

For DFT modulated lter banks the synthesis bank

polyphasematrixG(z)canbewrittenas

F T

C T

(z)J: (26)

ForthePBFDAFitcanbeproven[2]thatwhenP isa

multiple ofL

C(z)=



I

L 0

L

::: 0

L 0



| {z }

M

=C (27)

isindependentofzandhencealsoG(z)isindependent

of z. This means the synthesis polyphase lters have

length L

f

= 1. S as dened in Eq. 14 then equals

G(z)= S=F T

C T

J. For thePBFDAFthe update

equation22thenbecomes

w

k +1

= w

k +2

M X

H

k F C

T

CF 1

(D

k X

k w

k ) (28)

= w

k +2

M X

H

k F



I

L 0

0 0

M L



F 1

E SB

k (29)

E SB

k

arethesubbanderrors,FC T

CF 1

doestheerror

correction. It can be proven (see [4]) that the weight

updateequationofaso{calledunconstrainedPBFDAF

for which P is a multiple of L (Eq. 10) is closely re-

lated to equation 29, derived here based on fullband

errorfeedback.