Analysis and Fast Implementation of
Oversampled Modulated Filter Banks
StephanWeiss
Dept.Eletronis &ComputerSiene,University ofSouthampton,UK
Abstrat. Oversampledmodulatedlterbanks(OSFBs)arepopularlyemployedfor
anumberofappliationssuhasaoustiehoanellationinordertoreduethe
pro-essingomplexityofasignalproessingalgorithm. Hene,aneÆientimplementation
of OSFBsthemselvesis mandatory. Inthis paper,apolyphasedesription isusedto
removeredundaniesinthelteroperationsandtofatorisetheOSFBintolter
om-ponentsdependingonthe prototype lter, andthe modulatingtransform. Based on
astate-spaerepresentationofthisderivedpolyphasefatorisation,signal owgraphs
anbeobtainedwhihpermitaverysimpleandeÆientOSFBimplementation. The
analysisisperformedfor anumberofdierentlassesofOSFBs,andaomparisonto
existing methodsisdrawn.
1 Introdution
Oversampledlterbanks(OSFBs)ndawiderangeofappliations,where
om-putational redutions for resoure-demanding signal proessing algorithms are
soughtby means of subband approahes. Examples inlude subband adaptive
lters used in aousti ehoontrol [1;2℄, line enhanement[3℄, or
beamform-ing[4℄. AnotheruseofOSFBsare,forexample,transmultiplexersforthe
trans-mission of several users overa single hannel [5℄. Partiularly in the light of
omputational eÆieny,simplelowomplexityrealisationsforthelter banks
themselvesare thereforedesirable. Despite this motivation and in ontrast to
their ritially deimated ounterparts[6; 7℄,numerially eÆient
implementa-tionsofnon-ritiallysampled(or\oversampled")lterbankshavereeivedlittle
attention.
AsimplelterbanksystemgivingK subbandsignalsdeimated byN K
isshowninFig.1. AneÆientimplementationisbasedonmodulationofallK
analysisltersH
k
(z)andallsynthesislterG
k
(z)fromoneprototypelter[8℄.
For OSFBs with non-ritial deimation N < K, an eÆient implementation
0
Y
1
Y
Y
analysis filter bank
synthesis filter bank
N
N
N
)
(
z
1
H
)
(
z
0
H
)
(
z
1
K-H
N
N
N
)
(
z
0
G
)
(
z
G
1
)
(
z
1
K-G
)
(
z
X
)
(
z
)
(
z
)
(
z
1
K-)
(
z
X
Fig.1. AnalysisandsynthesislterbankforsubbanddeompositionofX(z).
to afatorisationoftheanalysis lterbank operationinto alteringoperation
linked to the prototype lter oeÆients, aylial shift, and the appliations
of theappropriate modulating transform (e.g. aDFT). A similar approah
re-sulting in adierentsequene ofexeutionis presentedin [10; 5℄, witha
time-varyingexitationofdierentomponentsoftheprototypelterfollowedbythe
modulatingtransform. Morereently,polyphasefatorisations inthez-domain
have been presented[11; 12℄, whih also permit a separationinto ltering
op-erations based on the prototypelter, and the modulating transform. Forall
ases [9; 10; 11; 12; 5℄, a dual implementation anbe found for the synthesis
lterbankoperation.
Thepolyphaseapproah[11;12℄anbeutilisedasastartingpointtoderivea
lterbankfatorisationyieldingaverysimpleandlowostimplementation[13℄.
Here, this fatorisation is generalised to arbitrary length prototypelters and
arbitrary modulations, for whih the analysis and fatorisation is presented in
Setion 2. Based on a state-spae representation of the OSFB operations in
Setion 3, signal ow graphsfor analysis and synthesis lter bank are derived
inSetion 4. Further,inthelattertheimplementationandomputational
om-plexityoftheresultingiruitsisdisussedandompared toexistingmethods.
In our notation, boldfae upperase variables are matrix valued, boldfae
lowerase or upperase underlined quantities refer to vetor valued variables.
AnNN identitymatrixisdenotedbyI
N
,anNM matrixwithzeroelements
by0
NM .
2 Filter Bank Analysis
2.1 Polyphase Notation
Letus onsiderthe analysislterbank ofFig.1produingK subbandsignals.
To exploit omputational redundanies arising from the deimation by N, a
polyphasedesriptionis utilized [7℄. A polyphasenotation forthekth analysis
lters,
H
k (z)=
N 1
X
z n
H
k jn (z
N
0
1
0
Y
Y
Y
1
H
N
N
N
X
(
z
)
)
(
z
X
)
(
z
X
N-
1
)
(
z
)
(
z
)
(
z
)
(
z
1
K-)
(
z
X
Fig.2. AnalysislterbankwithdemultiplexerandH(z)desribinganNK
MIMOsystem.
produesadeompositionintoN type-IpolyphaseomponentsH
k jn
(z)[7℄.
Sim-ilarly,theinputsignalX(z)isdeomposedintoN type-IIpolyphaseomponents
X
n (z),
X(z)= N 1
X
n=0 z
N+n 1
X
n (z
N
) : (2.2)
Ifthepolyphaseomponentsareorganisedinvetorform,
H
k (z) =
H
k j0 (z) H
k j1
(z) H
k jN 1 (z)
T
(2.3)
X(z) = [X
0 (z) X
1
(z) X
N 1 (z)℄
T
(2.4)
asubbandsignalY
k
(z)anbedenoted as
Y
k
(z)=H T
k
(z)X(z) : (2.5)
Forompatibility, in thefollowing weassume that lters are alwayssubjeted
to type-Iandsignalstotype-IIpolyphasedeompositions.
2.2 AnalysisFilter Bank
Fora ompat notation of theanalysis lter bank operations, the K subband
signals are olletedin avetorY(z)= [Y
0 (z) Y
1
(z) Y
K 1 (z)℄
T
. Inserting
(2.5)gives
Y(z) =
H
0 (z) H
1
(z) H
K 1 (z)
T
X(z) (2.6)
= H(z)X(z) ; (2.7)
whereH(z)2C KN
(z)isthepolyphaseanalysismatrix[11℄.With the
desrip-tion (2.7), the analysis lter bank in Fig.1 an be implemented by a
demul-tiplexer followed by a linear time-invariant multi-input multi-output (MIMO)
systemH(z) asshownin Fig.2. Weare nowinterestedin apartiular
fatori-sationof H(z).
ItisassumedthattheanalysisltersareFIRwithL
p
oeÆients,andderived
from a prototype lter P(z) by modulation. With the oeÆients of the kth
analysis lterorganisedinavetorh
k 2C
L
p
,
h =
h [0℄ h [1℄ h [L 1℄
T
k=
1
k=K-
1
k=
0
k=
1
k=K-
1
|
H
k
(
e
j
|
Ω
)
|
H
k
(
e
j
Ω
)
|
1/
2π
K
1/
4π
K
2π
2π
0
0
(a)
(b)
Ω
Ω
k=
0
Fig.3. (a)Evenstakedand(b)oddstakedK-hannellterbank.
thepolyphaseomponentsin (2.3)anbewrittenas
H k (z)= I N z 1 I N z bL p =N+1 I N z bL p =N I R 0 N RR
| {z }
L T 1 (z) h k (2.9) whereI N
isanNN identitymatrix,btheooroperator,andR=mod
N L
p
the remainder of the division of the lter length L
p
by the deimation fator
N. ThedependenyofH
k
(z)ontheunderlyingprototypelterwithoeÆients
p[i℄,i=0:::L
p
1isinorporatedas
h k = 2 6 6 6 4 p[0℄ 0 p[1℄ . . . 0 p[L p 1℄ 3 7 7 7 5
| {z }
P 2 6 6 6 4 t k [0℄ t k [1℄ . . . t k [L p 1℄ 3 7 7 7 5
| {z }
t
k
; (2.10)
basedonthegenerallyomplexmodulationsequeneontainedint
k 2C
Lp
. The
matrixP2R LpLp
isadiagonalmatrixholdingtheprototypelteroeÆients.
Depending onwhih modulationis invokedforthelter bank,dierent
pe-riodiitiesofthesequenet
k
[i℄,i=0:::L
p
1,result. Ifthelterbankiseven
staked as shown in Fig. 3(a), the periodiity is K. In this ase, a ompat
notationofthemodulationsequeneanbefound
t k = I K I K I K I S 0 K SS
| {z }
L T 2 T ~ t k ; (2.11) where ~ t k 2 C K , L 2 2 N LpK
, and S = mod
K L
p
. Odd-staked lter banks
modulation sequene anhowever be exploited by alternatingsign hanges on
bloksofK oeÆients,
t
k =[I
K I
K I
K I
K ℄
T
~
t
k
: (2.12)
InsteadofmodifyingthematrixL
2
ofoddstakedlterbanks,thissignhange
an be inorporated into P in (2.10) and hene theprototype lterP(z)| a
trikthatisalsoknownfromdisreteosinetransformimplementations[7℄. Itis
assumed that foroddstakedlterbanks suh amodiationof theprototype
lterisperformedbynegatingtheoeÆients'signsin everyseondK-blokof
oeÆients. Withthisassumptions,in thefollowingevenandoddstakedlter
banksanbetreatedalikeusing (2.11).
Themodulationsequenes ~
t
k
,k=0:::K 1,areolletedinamatrix
T=[ ~
t
0
~
t
K 1 ℄
T
2C KK
(2.13)
whih for examplefor aDFT modulated lterbank wouldbe aK-pointDFT
matrix. Applying(2.13) tothesubstitutionof(2.11)and(2.10)into (2.6)gives
H(z)=TL
2 PL
1
(z) (2.14)
asnotation forthepolyphase analysismatrix. With (2.14)a fatorisationinto
prototype lteromponentsand a rotationby atransformmatrix T hasbeen
established similar to [11; 12℄. The dierene is that the diagonal matrix P
ontainsnosparseltersbutonlytheprototypelteroeÆients,whihwillbe
exploited fortheimplementationinSetion 4.
2.3 SynthesisFilter Bank
Dual to the analysis lter bank, the synthesis lter bank as shown in Fig. 1
upsamples the subband signals by a fator N and applies interpolation lters
G
k
(z). TheonditionthatallltersG
k
(z)andH
k
(z)arederivedfromthesame
prototype lowpass lter and that the lter bank is perfetly reonstruting is
guaranteed by H(z) being paraunitary [11℄. Reonstrution is then given by
thepolyphasesynthesismatrixG(z)2C NK
(z),whihistheparahermitianof
H(z),andrelatesthesubbandsamplesbaktothepolyphaseomponentsofthe
fullband signal,
^
X(z)=G(z)Y(z) : (2.15)
Forausality, adelayhas to beintrodued suh that ^
X(z)=z Lp+1
X(z)in
Fig.1. IntheN-polyphasedomain,this isexpressedas[7℄
^
X(z)=
0
N RR z
bLp=N+1
I
N R
z bL
p =N
I
R
0
RN R
| {z }
(z)
0
1
Y
0
Y
1
Y
G
(
z
)
)
(
z
X
N
N
N
)
(
z
X
)
(
z
X
)
(
z
X
N-
1
)
(
z
1
K-)
(
z
)
(
z
Fig. 4. Synthesislterbank withG(z)desribingaKN MIMO system
fol-lowedbyamultiplexer.
Inorporatingthisdelayintotheperfetreonstrutionondition,G(z)H(z)=
(z),thepolyphasesynthesismatrixis givenby
G(z) = (z)H H
(z 1
) (2.17)
= (z)L T
1 (z
1
)PL T
2 T
H
: (2.18)
Evaluating(z)L T
1 (z
1
)in(2.18) gives
^
L T
1
(z)=(z)L T
1 (z
1
)=L T
1 (z)J
Lp
; (2.19)
where J
Lp
is an L
p L
p
reverse identity matrix left-right ipping the ausal
matrix L T
1
(z). The polyphase synthesis matrix in (2.18) leads to the signal
ow graph in Fig. 4 with the MIMO system G(z) followed by a multiplexer.
Its funtionality is idential to theoriginal synthesis lter bank in Fig. 1, but
multipliationswithexpandingzerosin theinterpolationsltersareavoided.
Paraunitarityof H(z) is equivalent to the lterbank implementing atight
framedeomposition,whihoersusefulpropertiessuh asaxedenergy
rela-tion betweenthefullband signalX(z)and the subbandsamples in Y(z). In a
more generalase, H
k
(z) and G
k
(z) anbe basedon dierent prototype
low-pass lters (with dierent oeÆients oreven lter lengths) and therefore the
lter bank system is not neessarily perfetly reonstruting. In this ase the
fatorisation of the polyphase analysis matrixremains asin Setion 2.2, while
the fatorialmatries of G(z) in (2.18) are builtaordingly, whereby the
pa-rametersof the synthesis prototypelterhave to be appliedfor L
1
(z),P, and
L
2 .
3 Canonial State-Spae Representations
Before tryingto ndsuitable implementations for thepreviouslyfatorised
l-ter bankoperations,anonialstate-spaerepresentationsforFigs.2and4are
derivedinthissetion. Theserepresentationstaketheform
zW(z)
V(z)
=
A B
C D
W(z)
U(z)
; (3.1)
A owgraphof (3.1)isgiveninFig.5. AppropriatesystemmatriesA,B, C,
A
C
D
z
)
(
U
V
(
z
)
z
)
(
W
z
W
(
z
)
B
Fig.5. Flowgraphofstate-spaerepresentationwith statesW(z).
3.1 AnalysisFilter Bank Representation
Fortheanalysislterbank,U(z)ontainsN demultiplexedsamplesoftheinput
signal(i.e.U(z)=X(z),theoutputV(z)=Y(z)holdstheK subbandsignals.
Foraanonialform,thestatevetorW(z)mustnothavemorethanL
p
N
ele-mentsandanbefoundbyreformulatingtheanalysisoperationbyintrodution
ofanintermediatevariableQ(z):
Y(z) = TL
2
PQ(z) with Q(z)=L
1
(z)X(z) (3.2)
Q(z) =
0
NLpN 0
NN
I
LpN 0
LpNN
z 1
Q(z)+
I
N
0
LpNN
X(z):(3.3)
With thereursive update in (3.3), the lower portion with L
p
N elementsof
Q(z)nowformsthestatevetorW(z),andthestatespaesystemmatriesan
beidentiedas
A=
0
NLp 2N 0
NN
I
Lp 2N 0
Lp2NN
B=
I
N
0
Lp2NN
(3.4)
C=TL
2 P
0
NLp N
I
Lp N
D=TL
2 P
I
N
0
LpNN
: (3.5)
Notethatallmemoryexhibitingmatriesin(2.14)havebeenreplaedby
mem-orylessoperationsduetothereursiveupdatingin(3.1).
3.2 SynthesisFilter Bank Representation
For the synthesis, the input in Fig. 5 now ontains the K subband samples,
U(z)=Y(z),whihareusedforthereonstrutionofN polyphaseomponents
of thefullband signal ^
X(z),held in theoutput V(z)= ^
X(z). A suitablestate
vetorW(z)issoughtfrom (2.18)with(2.19)inserted:
^
X(z) = ^
L T
1
(z)PL T
2 T
H
Y(z)=
0
NL
p N
I
N
Q(z) (3.6)
Q(z) =
0
NLpN 0
NN
I
LpN 0
Lp NN
z 1
Q(z) + PL T
2 T
H
Y(z) (3.7)
The upper L
p
thefollowingstate-spaematries:
A=
0
NL
p 2N
0
NN
I
L
p 2N
0
L
p 2NN
B=
I
L
p N
0
L
p NN
PL T
2 T
H
(3.8)
C=
0
NLp 2N I
N
D=
0
NLp N I
N
PL T
2 T
H
(3.9)
The system matries A in both (3.4) and (3.8) represent tapped delay lines
(TDL),in whihthestatevaluesareshiftedbyN statesforeveryupdate
itera-tion.
4 Filter Bank Implementation
Based onthe fatorisationsin Setion 2and thestate-spaerepresentations in
Setion3,wenowaimtondsignalowgraphsthatprovidesimpleandeÆient
OSFBimplementations.
4.1 Analysis Filter Bank Implementation
Inspeting (3.4) and (3.5), the analysis lter bank operation in (2.14) an be
exeutedintwosteps. Asmentionedabove,thestatevaluesfromaTDL,whih
is shiftedby A and updated with N fresh samples in everysubband sampling
period by B. Hene, C andD are exitedby L
p
N old and N urrentinput
samplesinW(z)andU(z),respetively. DuethesimiliarityofCandD,asingle
TDL[U T
(z) W T
(z)℄ T
holdingatotalofL
p
urrentandpastinputsamplesan
beassembled. Theresultis theowgraphin Fig.6. There,thedemultiplexing
ofthesalarOSFB inputintoN parallel samplesinU(z)asshownin Fig.2is
alreadyinorporatedintotheTDL.
Aordingto(3.5),in Fig.6theTDLvetor[U T
(z) W T
(z)℄ T
ispassedinto
theblokP,multiplyingeah valuebyaprototypelteroeÆientsp[i℄. From
theresults,L
2
reatesKsubsummations. Afterrotatedofthesesubsumsbythe
modulationmatrixT,nallyasetofK subbandsampleshasbeenalulated.
Note,thattheonlymemory-exhibitingoperationin thisanalysislterbank
realisation is the TDL holding L
p
samples of the input signal. The number
of states has inreased by N over the anonial state-spae representation in
(3.4)and(3.5)duetotheinlusionofthedemultipledofFig.2. Thereforewith
respettotheoveralliruitrunningatthefullband rate,theiruitinFig.6is
anonial.
4.2 SynthesisFilter Bank Implementation
Let us onsider the system matriesB and D dened for the synthesis OSFB
in (3.8)and (3.9),respetively,and thestate-spaerepresentationin Fig.5. It
is obviousthat the subbandsamples in U(z)=Y(z)are derotated byT H
and
dupliatedtoL
p
valuesbyL T
2
tonallyexitetheL
p
prototypelteroeÆients
in P. Ofthese L
p
produts,theupperL
p
N valuesarelathedbyBonto the
TDLimplemented by A in (3.8). The lowerN produt valuesareadded with
the lower N elements of the state vetor to form the output V(z) = ^
X(z).
2
1
demux /
(
z
)
Y
1
(
z
)
Y
0
(
z
)
Y
K-1
L -1
p
p
N
N
N
N
N
N
N
N
p
2K-1
p
2K
K+1
p
p
K-1
p
K
p
1
p
0
(
z
)
X
L
(
z
)
L
1
2
S
K
T
P
Fig.6. Analysislterbanksignalowgraph.
anberealisedasshownin Fig.7. This analsobemotivated bymergingthe
multiplexerofFig.4with ^
L T
1 (z).
Notethat similartotheanalysis OSFBin Fig.6thisiruitonlyrequiresa
single TDL,onto whih theresults ofthe operation PL T
2 T
H
^
Y(z) are
au-mulated. This operation is performed one in everysubband sampling period.
Afterwards,thevaluesintheTDLanbeshiftedNtimes,lokedatthefullband
rate, toform thefullband outputsamples. Similarargumentsasin Setion4.1
hold fortheanonialpropertyoftheowgraphin Fig.7.
4.3 Speial Filter Bank Cases
Anumberofspeiallterbankimplementationsarisefromdierenthoiesofthe
transformmatrixT,whihinthefollowingistoberenedinitsimplementation.
In aseofaDFTmodulatedlterbank, resultingin aneven stakeddesignas
shown in Fig. 3(a), T is a KK DFT matrix, T = T
DFT
. In ase of a
generalised DFT (GDFT, [14℄) lterbank, T is aGDFT matrix. This GDFT
matrixomprisesof elementst
k ;n =e
j 2
K
(k +k0)(n+n0)
, with n=0:::L
p
1and
k=0:::L
p
1. Notethatfork
0 =
1
T
1
2
T
mux /
N
N
N
N
N
N
N
N
N
1
2
S
K
T
L -1
p
p
p
2K-1
p
2K
K+1
p
p
K-1
p
K
p
1
p
0
(
z
)
Y
0
(
z
)
Y
1
(
z
)
Y
K-1
(
z
)
X
L
(
z
)
L
P
H
Fig.7. Synthesislterbanksignalowgraph.
ofFig.3(b)arises. AGDFTmatrixgenerallypermitsafatorisation
T=D
1 T
DFT D
2
: (4.1)
For an odd staked lter bank, all matries are KK and D
1
and D
2 have
diagonalformifinversionofthesignsofprototypeoeÆientsareperformedas
disussedin Setion2.2[12℄.
EÆientimplementationsmakeuseofanFFTroutineinsteadofperforming
T
DFT
asamatrixoperation. Further,iftheinputsignalX(z)isrealvalued,(i)
alloperationsexeptthetransformmatrixanbeperformedwithrealarithmeti
and (ii) redundanies arise due to approximately half of the subband signals
beingomplexonjugateopies ofothers. Hene abouthalf (dependingon the
DFT/GDFTtransformand Kbeingoddoreven)ofthesubbandsdonotneed
tobealulatednorproessed.
Singlesideband(SSB)modulatedOSFBsforrealvaluedsubbandsignalsan
beobtainedbymodiationofaGDFTmodulatedlterbankdeimatedbyonly
N=2[8℄. Anadditionalomplexmodulationisperformedonthesubbandsignals
Table1 ComputationalComplexityofFilter BanksImplementations
C
real
/[MACs℄ C
omplex
/[MACs℄
DFT
1
N (L
p
+4Klog
2 K)
1
N (2L
p
+4Klog
2 K)
GDFT 1
N (L
p
+4Klog
2
K+4K) 1
N (2L
p
+4Klog
2
K+8K)
SSB 2
N (L
p
+4Klog
2
K+5K) 2
N (2L
p
+4Klog
2
K+10K)
4.4 ComputationalComplexity
From the signal ow graphs for analysis and synthesis in Figs. 6 and 7, the
omputational omplexities for both operations an be evaluated in terms of
multiply-aumulates (MACs)persampling period. The latteristheperiodof
thefullbandsignalsprioranalysisoraftersynthesis. Theomplexitiesaregiven
inTab.1andareidentialforanalysisandsynthesis,butdierforthehoieof
T anddependonwhethertheinputsignalisrealoromplexvalued. Note,that
the multipliationof theomplexsamples withthereal valued prototypelter
oeÆients arues to 2L
p
MACs. The modulation matrix T
DFT
is assumed
to be implementedby a K-pointFFT requiring4Klog
2
K real valued MACs,
whihisinvariablyappliedforalltypesoflterbanks.
Althoughanumberofmethodsreportedintheliteraturegiveidential
om-plexitiesin termsofMACs,therealisationsin Figs.6and7donotrequireany
additionaltime-varyingirular shifts[9℄orswithing[5℄,theindexingof
time-varyinglters[10℄,orlterswithsparseoeÆients[11;12℄. Further,thesignal
ow graphs in Figs. 6 and 7 only require a single irular buer and hene a
minimumamountofpointersforaddressing,andpermitanarbitraryprototype
lter length L
p
independent of both the deimation ratio N and the hannel
numberK.
5 Conlusion
Oversampledlterbankswhereallltersarederivedfrom aprototypelterby
modulationhavebeenanalysed usingthewell-knownpolyphasedeomposition.
Similartopreviousanalysesin theliterature,thisdeompositionwasfatorised
to redue alllter operationsto operations onthe prototypelter oeÆients,
andamultipliationbythemodulationmatrix. Thisensuredaminimumamount
ofmultiply-aumulateoperations. However,thefatorisationwasexploitedvia
astate-spaerepresentationtoloateallmemory-requiringoperationsnexttothe
multiplexersanddemultiplexersoftheiruit. Thebenetisanimplementation
withaonlysingleTDLthatanbeonvenientlyupdated.
Thepresentedimplementationanbeappliedtoavarietyofmodulatedlter
banks, suh as shown for DFT, GDFT, orSSB lter banks, and with a large
exibility forthelengthof theprototypelter. Althoughnotexpliitlyderived
here, the analysis and synthesis lter bank implementations an be similarly
applied to lterbankswhere ltersH
k
(z)and G
k
(z)originatefrom morethan
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