In this paper, we evaluate the performance limitations of subband adaptive filters in terms of achievable final error terms. The limiting factors are the aliasing level in the subbands, which poses a distortion and thus presents a lower bound for the minimum mean squared error in each subband, and the distortion function of the overall filter bank, which in a system identification setup restricts the accuracy of the equivalent fullband model. Using a generalized DFT modulated filter bank for the subband decomposition, both errors can be stated in terms of the underlying prototype filter. If a source model for coloured input signals is available, it is also possible to calculate the power spectral densities in both subbands and reconstructed fullband. The predicted limits of error quantities compare favourably with simulations presented

A. Stenger

R. Rabenstein

R. W. Stewart

S. Wei

Wei Stewart

English

Weiss, Stephan

Stewart, Robert

Stenger, Alexander

Rabenstein, Rudolf

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryWeiss, Stephan and Stewart, Robert and Stenger, Alexander and Rabenstein, Rudolf (1998)Performance limitations of subband adaptive filters. In: 9th European Signal ProcessingConference, 1998-09-08 - 1998-09-11, Rhodes.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/ Proc. IX European Signal Processing Conference (EUSIPCO-98),Sept. 1998, Rhodes, Greece, pp. 1245-1248PERFORMANCE LIMITATIONS OF SUBBANDADAPTIVE FILTERSS. Wei, R.W. StewartSignal Processing Division, Dept. of EEEUniversity of StrathclydeGlasgow G1 1XW, Scotlandfweiss,bobg@spd.eee.strath.ac.ukA. Stenger, R. RabensteinLehrstuhl fur NachrichtentechnikUniversitat Erlangen-NurnbergErlangen, Germanyfstenger,rabeg@nt.e-technik.uni-erlangen.deABSTRACTIn this paper, we evaluate the performance limitationsof subband adaptive lters in terms of achievable nalerror terms. The limiting factors are the aliasing level inthe subbands, which poses a distortion and thus presentsa lower bound for the minimum mean squared error ineach subband, and the distortion function of the over-all lter bank, which in a system identication setuprestricts the accuracy of the equivalent fullband model.Using a generalized DFT modulated lter bank for thesubband decomposition, both errors can be stated interms of the underlying prototype lter. If a sourcemodel for coloured input signals is available, it is alsopossible to calculate the power spectral densities in bothsubbands and reconstructed fullband. The predictedlimits of error quantities compare favourably with sim-ulations presented.1 INTRODUCTIONAdaptive ltering in subbands is widely used for prob-lems where an adaptive system is required to identifyvery long impulse responses, since it enables to pro-cess in decimated subbands with decreased complexity[4, 2], which is e.g. exploited in acoustic echo cancella-e  [m]analysisfilter bankbankfilteranalysisbankfilterunknown systems[n]d[n]desired signalinputsignalx[n] e[n]errorsignalx  [m]0x  [m]1x         [m]K/2-1e  [m]01e        [m]K/2-1d  [m]0d  [m]d       [m]K/2-11w0w1wK/2-1synthesisFig. 1: Subband adaptive lter structure in a systemidentication setup.tion. Performance characteristics of subband adaptivelters (SAF) as shown in Fig. 1 due to the subbandsplitting have mainly been addressed in terms of con-vergence speed. Investigations into the achievable nalconvergence errors are mainly made in terms of trunca-tion errors and non-causality [4, 9], while there are littlehints for the inuence of distortions introduced by thelter banks [5, 8].In the following, we discuss convergence error limitsof subband adaptive ltering in dependency on a gen-eralized DFT (GDFT) modulated lter bank used forsubband decomposition, which will be briey reviewedin Sec. 2. In Sec. 3, we then introduce a method toobtain the power spectral density (PSD) of the alias-ing terms, which sets the lower limit for the adaptationerror. This limit can be approximated by a stopband at-tenuation measure of the prototype lter. A second partthen discusses the error inherent in the fullband modelof the adapted subband lters. Simulations supportingour results are presented in Sec. 4.2 GDFT MODULATED FILTER BANKS2.1 ModulationA general structure of a K band lter bank with deci-mation by a factor N  K is shown in Fig. 2. The anal-ysis lters hk[n] are derived from a real valued lowpassprototype FIR lter p[n] of length Lpby a generalizeddiscrete Fourier transform (GDFT),hk[n] = ej2K(k+k0)(n+n0) p[n]; k; n 2 N: (1)NNNNNN     x[n] x[n]analysis bankH (z)H (z)G (z)G (z)synthesis bank01K-1 G     (z)K-101H    (z)Fig. 2: Analysis and synthesis branch of a K-channellter bank with subbands decimated by N .jP(e    )Ω| |0frequency response / [dB]/Kpi pi/NΩpiFig. 3: Required frequency response of the real valuedprototype lter p[n] for aK channel oversampled GDFTlter bank with decimation by N .The term generalized DFT [1] stems from osets k0andn0introduced into the frequency and time indices. Withk0= 1=2, it is sucient for real valued input x[n] toprocess the rst K=2 subbands covering the frequencyinterval [0;], while the remaining subbands are redun-dant. Together with conditions on p[n], the time osetn0can be set appropriately to ensure useful propertiessuch as linear phase. The synthesis lters gk[n] can beobtained by time reversion and complex conjugation ofthe analysis lters, i.e. gk[n] =~hk[n] = hk[Lp n+1]. Themodulation approach allows for both low memory con-sumption for storing lter coecients and an ecientpolyphase implementation [7].2.2 Prototype DesignThrough the above modulation, the lter bank designreduces to an appropriate choice of the prototype l-ter, which has to fulll two criteria. Firstly, the lters'attenuation in the stopband ranging from [=N ;], asindicated in Fig. 3, has to be suciently large. Everyfrequency of the input signal in the interval [=N ;] willbe aliased into the baseband after ltering and decima-tion, and cause a distortion of the subband signal.A second constraint on the design is the perfect re-construction condition. If stopband attenuation of theprototype lter is high enough to suciently suppressaliasing, this condition reduces to the consideration ofinaccuracies in power complementarity [6]:K 1Xk=0jHk(ej)j2!= 1: (2)A prototype lter approximating these constraints canbe constructed by an iterative least-squares method [7].3 PERFORMANCE LIMITATIONSIn this section, we derive limitations in adaptation as-suming that the only disturbance originates from thelter banks employed for the subband decomposition.First, we look at the achievable error PSD and the meansquared error (MSE) term, Ee2[n]	, which is importantto minimize in e.g. acoustic echo cancellation. Secondly,(a)-u[n]L(ej)-x[n]n#N-v[m](b)u[n]-n#Nu0[m]---L(ej)qN 1(ej) #N...L(ej)q1(ej) #NL(ej)q0(ej) #NHHjAAAUh+-v[m]Fig. 4: The identity of the structures (a) and (b) isexploited to calculate the PSD of a decimated signalv[m]for system identication applications, we state a limitfor the error of the identied model.3.1 PSD of Adapted Error SignalLet us interpret the desired signal for the kth adaptivelter in Fig. 1 as the sum of two components,dk[m] = sk[m]  xk[m] + zk[m] : (3)The rst summand reects the un-aliased projection ofthe output of the unknown system, the desired signald[n] = s[n]x[n], into the kth subband. The second sum-mand, zk[m], represents the aliased signal componentscreated in the decimation stage, which can be viewedas a distortion of the LTI system and modelled as ad-ditive noise. Therefore, the Wiener solution of the kthsubband lter is given by sk[m], while the unidentiablepart zk[m] denes the minimumMSE (MMSE) by [3]MMSE = Ez2k[m]	: (4)To nd an analytical expression for the subband MMSE,we rst determine the PSD of the aliased signal parts inthe subband signals, making use of two facts: aliasing can be conveniently pictured as a superposi-tion of spectral intervals; after decimation, a previously white noise signal re-mains still white with identical variance.The further proceeding is depicted in Fig. 4. Assumingthe knowledge of a source modelL(ej), which is excitedby a white noise signal u[n], the decimation by N canbe swapped with L(ej). In each branch of Fig. 4(b)the source model is multipied with a window qi(ej)and then decimated by N . The windows have adjacentrectangular spectra with bandwidth 2=N each. Thisdecimated model is then excited by a decimated butotherwise unmodied white noise process, u[n].When we identify v[m] in Fig. 4 with the desired sig-nal dk[m], the source model for the kth subband Lk(ej)consists of a noise shaping lter F (ej), which repre-sents the source model of the input signal x[n] for ex-citation by white noise of unit variance, the unknownsystem S(ej), and the analysis lter Hk(ej),Lk(ej) = F (ej)  S(ej) Hk(ej) : (5)The PSD of dk[m] now consists of the squared sum overall N terms of the kth decimated source model in Fig. 4.The squared sum of N 1 alias-only terms, denedby N 1 rectangular windows q1: : : qN 1covering thestopband of the kth analysis lter Hk(ej), nally givesthe PSD of the minimum error corresponding to theMMSE,SMMSEekek(ej) =N 1Xi=1N 1Xn=0Lk(ej(+2n)=N)qi(ej(+2n)=N)2: (6)This assumes that all un-aliased signal parts in the sub-band error signal ek[m] have been cancelled by the sub-band adaptive lter. Thus, due to the Wiener-Khintchi-ne transform, the MMSE can be calculated asMMSE =12Z20SMMSEekek(ej)d : (7)By inclusion of the synthesis lters Gk(ej), it is alsopossible to derive the PSD of the reconstructed mini-mum error signal, and state the fullband MMSE analo-gous to (7).Approximations. The advantage of the outlined ap-proach is that for spectrally correlated signals, all cross-terms in the PSDs are considered. However, for weakspectral correlation, we may disregard the cross-termsbetween dierent aliased spectral intervals, and thusapproximate the PSDs by swapping summations andsquare operations in (6). To obtain a more practicallimit for the performance of SAFs, we calculate the ratiobetween the power levels of un-aliased and aliased sub-band components, creating an SNR-like measure, whichwe refer to as signal-to-alias ratio (SAR),SAR =R=N0jP (ej)j2dR=NjP (ej)j2d: (8)This approximation has been based on the further as-sumption F (ej) = S(ej) = 1, such that (8) only de-pends on the magnitude response P (ej) of the pro-totype lter p[n]. However, the SAR measure can beshown to yield valid results also for non-white input sig-nals and unknown systems. Note that the denominatorof (8) is a measure of the stopband attenuation discussedin 2.2.equivalent fullband model[n]--analysisbankunknownsystems[n]---````--adaptedltersanalysisbank--````----````--synthesisbanksynthesisbank--w[n]s[n]t[n]Fig. 5: Separation of system identication structure forreconstruction of equivalent fullband model.3.2 Error of Equivalent Fullband ModelDisregarding any other limiting inuences and assumingadaptation e[n] ! 0 in Fig. 1, an equivalent fullbandmodel can be reconstructed from the adapted subbandimpulse responses wk[n] by sending an impulse throughanalysis bank, adapted lters and synthesis bank. Ajustication is demonstrated in Fig. 5 by swapping sum-mers for the subband errors with the (linear) synthesisoperation. Ideally, the fullband equivalent model w[n]will match the cross-correlation function between inputand desired signal, which for white noise excitation givesthe unknown system s[n], convolved with the distortionfunction t[n] of the lter banks [8]. This distortion func-tion characterizes the serial connection of the decimatedlter banks in Fig. 2, x^[n] = x[n] t[n]. Thus, any devi-ation from perfect reconstruction will result in an errorin the equivalent fullband model, where the accuracycan be shown to be limited by the reconstruction error(RE),RE = kt[n]  [n Lp+1]k22: (9)4 SIMULATIONS AND RESULTSWe perform adaptive system identication in a set-upas shown in Fig. 1 of a recursive system s[n] with twodominant poles at  = 0:1 and 0:45 using an SAFwith K=2 = 8 complex subbands decimated by N = 14.For simulation with an NLMS algorithm and stronglycoloured input signal, Fig. 6 shows the PSDs of desiredsignal d[n] and nal error e[n] after almost completeadaptation. In contrast, the analytically calculated PSDfor the error signal at the Wiener-Hopf solution is givenin Fig. 7, overlaid with the nal error PSD of Fig. 6.Apart from deviations due to insucient convergenceat the band edges and residual peaks and a raised errorpower spectrum around the positions of the dominantpoles, clearly the predicted PSD is enveloped by the sim-ulated result, and therefore can be regarded as a lowerlimit of the error PSD.Tab. 1 compares the error limits derived in Sec. 3with simulated results for three dierently designed pro-totypes, PA, PB, and PC. The design method is a leastsquares minimization of stopband energy and the errorin power complementarity, which can be traded o byintroducing a weighting between the two measures [7].0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100−80−60−40−20020normalized frequency Ω/piPSDs  See(ejΩ), Sdd(ejΩ)  / [dB]PSD of desired signalPSD of final error signalFig. 6: PSD of desired signal and nal error signal;dashed vertical lines indicate band edges.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100−90−80−70−60−50−40normalized frequency Ω/piPSD  See(ejΩ) / [dB]predicted PSD of residual error    measured error PSD after adaptationFig. 7: PSD of nal error signal and predicted PSDbased on alias components in the reconstructed signal.Tab. 1 states design results in terms of the measures REas dened in (9) and the SAR of (8) reecting the stop-band attenuation.For simulations, the set-up in Fig. 1 was employed toidentify a delay using an RLS algorithm with whiteGaussian input. The error norm of the equivalent full-band model kw  sk22, where w is the reconstructed full-band model according to Sec. 3.2, is given in Tab. 1,which together with the reduction in error variance,2dd=2eets very closely the predicted values. For theexample in Fig. 6 using coloured input and a rathercomplex unknown system, the MSE reduction 2dd=2eeof 56.73dB closely agrees with an SAR value of -57.01dBfor the employed prototype lter in this case.Analytical Prediction Simulation ResultsRE SAR kw sk222dd=2eePA-54.0821 54.9 -54.0153 54.0PB-34.6191 65.2 -34.6143 66.2PC-18.0016 77.8 -18.0010 78.5Tab. 1: Predicted fullband model error and nal MSEcompared to simulation results (all quantities in [dB]).5 CONCLUSIONSWe have introduced measures to predict the adaptationlimits for subband adaptive lters in terms of the nalMSE and the error of the identied model. In case ofGDFT lter banks, they can be expressed in terms ofthe prototype lter, and closely agree with simulationresults. For subband adaptive lter applications thesemeasures provide convenient tools to design lter banksfullling pre-specied performance requirements. Forapplications like acoustic echo control, where the adap-tation error is the most important issue, the banks canbe designed to be just good (and short) enough to satisfyrelaxed constraints on the model error.6 REFERENCES[1] R. Crochiere and L. Rabiner. Multirate Digital Sig-nal Processing. Prentice Hall, 1983.[2] A. Gilloire and M. Vetterli. \Adaptive Filteringin Subbands with Critical Sampling: Analysis, Ex-periments and Applications to Acoustic Echo Can-celation". IEEE Trans. Signal Processing., SP-40(8):1862{1875, 1992.[3] S. Haykin. Adaptive Filter Theory. Prentice Hall,2nd ed., 1991.[4] W. Kellermann. \Analysis and Design of MultirateSystems for Cancellation of Acoustical Echoes".Proc. ICASSP, 5:2570{2573, New York, 1988.[5] D. Slock. \Fractionally-Spaced Subband and Mul-tiresolution Adaptive Filters". Proc. ICASSP,5:3693{3696, Toronto, 1991.[6] P. Vaidyanathan. Multirate Systems and FilterBanks. Prentice Hall, 1993.[7] S. Wei, M. Harteneck, and R. Stewart. \On Im-plementation and Design of Filter Banks for Sub-band Adaptive Filter Systems". In IEE Colloq. Dig-ital Filters: an Enabling Technology, London, April1998.[8] S. Wei, U. Sorgel, and R. Stewart. \Computa-tionally Ecient Adaptive System Identication inSubbands with Intersubband Tap Assignment forUndermodelled Problems". Proc. Asilomar Con-ference, 2:818{822, Monterey, CA, 1996.[9] R. J. Wilson, P. A. Naylor, and D. Brookes. \Per-formance Limitations of Subband Acoustic EchoControllers". In Proc. IWAENC, pp.176{179, Im-perial College, London, UK, September 1997.

Adaptive Filter Theory.

Adaptive Filtering in Subbands with Critical Sampling: Analysis, Experiments and Applications to Acoustic Echo Cancelation&quot;.

Analysis and Design of Multirate Systems for Cancellation of Acoustical Echoes&quot;.

Computationally Ecient Adaptive System Identi in Subbands with Intersubband Tap Assignment for Undermodelled Problems&quot;.

Fractionally-Spaced Subband and Multiresolution Adaptive Filters&quot;.

Multirate Systems and Filter Banks.

On Implementation and Design of Filter Banks for Subband Adaptive Filter Systems&quot;.

Performance Limitations of Subband Acoustic Echo Controllers&quot;.

Performance limitations of subband adaptive filters

Name not available

In this paper, we evaluate the performance limitations of subband adaptive filters in terms of achievable final error terms. The limiting factors are the aliasing level in the subbands, which poses a distortion and thus presents a lower bound for the minimum mean squared error in each subband, and the distortion function of the overall filter bank, which in a system identification setup restricts the accuracy of the equivalent fullband model. Using a generalized DFT modulated filter bank for the subband decomposition, both errors can be stated in terms of the underlying prototype filter. If a source model for coloured input signals is available, it is also possible to calculate the power spectral densities in both subbands and reconstructed fullband. The predicted limits of error quantities compare favourably with simulations presented.  1 INTRODUCTION  Adaptive filtering in subbands is widely used for problems where an adaptive system is required to identify very long impulse response..

Performance limitations of subband adaptive filters

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