Step-Size Bounds Analysis of the Generalized Multidelay Adaptive Filter

Transcription

1 WCE 007 Jly London UK Step-Size onds Analysis of the Generalized Mltidelay Adaptive Filter Jnghsi Lee and Hs Chang Hang Abstract In this paper we analyze the bonds of the fixed common step-size parameter for the generalized mltidelay adaptive filter () Freqency domain adaptive filters are attractive in applications reqiring a large nmber of coefficients sch as acostic echo cancellation (AEC) However the very restrictive convergence bond for bloc LMS has limited its seflness Derivations on step-size bonds for the partitioned freqency-domain bloc LMS have been reported recently bt are not consistent with each other Contrary to other researchers wor this paper derives a not-restrictive step-size bond that spports a stable We also derive relations of step-size parameters for NLMS and to have similar convergence properties he reslts of extensive simlation experiments are inclded in the paper hese reslts show the bonds and the convergence behavior predicted by the analysis is in very good agreement with the experimental reslts Index erms Acostic Echo Cancellation loc LMS Convergence Analysis Freqency Domain LMS I INRODUCION he normalized least-mean-sqare (NLMS) scheme has been the most poplar adaptive filtering algorithm in many applications here are qite a nmber of variations of the NLMS algorithm being developed for certain specific applications For example freqency-domain fast bloc LMS (FLMS) adaptive filters are attractive alternatives for acostic echo cancellation (AEC) which may need thosands of filter coefficients to reach the desired level of performance [1]-[6] he great redction in comptational complexity associated with FLMS is de to the sage of fast Forier transform (FF) In the literatre the FLMS is also referred to as the bloc freqency-domain adaptive filters (FDAF) It is well nown that the normalized bloc LMS (NLMS) with bloc length N and the NLMS algorithms converge at the same rate and achieve the same mis-adjstment if the step-size parameter of NLMS is N times as large as that of the NLMS [7] However both algorithms have the same Manscript received March his wor was spported in part by the AIWAN-National Science Concil nder Grant NSC95-1-E and by the Yan-Ze University Commnications Research Center J Lee is with the Department of Electrical Engineering Yan-Ze University Chng-Li aoyan 306 AIWAN ( Phone: Ext 419; fax: ; eejlee@satrnyzedtw) H C Hang is with Department of Electrical Engineering Yan-Ze University Chng-Li aoyan 306 AIWAN (Fax: ; s958501@mailyzedtw) convergence bonds for the step-size parameter herefore even for a moderate bloc length N the NLMS has to employ a fairly small step-size parameter to meet the very restrictive convergence bond his limitation greatly redces the seflness of the NLMS and its freqency-domain variations FLMS A major problem of FLMS in AEC application is the long delay associated with the large filter size Soo proposed a variation of FLMS which was referred to as the mltidelay bloc freqency domain adaptive filter (MDF) to alleviate this delay troble [4] MDF segments the filter into several partitions and employ as many sb-filters as well he MDF belongs to the class of partitioned FLMS (PFLMS) algorithms he PFLMS is most commonly implemented by normalizing its inpt in freqency domain (nown as freqency bins) and is referred to here as the NPFLMS Some researchers thoght that the freqency-bin normalization procedre resolves the problem of slow modes of the NLMS algorithm and the reslting algorithm converges faster than the NLMS [10] However some researchers reasoned that becase of the restriction on the step-size bonds that spports a stable filter the freqency domain algorithms actally do not perform better than the NLMS in convergence and tracing properties [5] And they have introdced a hybrid coefficients pdate scheme that performs comparable to that of the NLMS with a comptational complexity comparable to that of the standard freqency domain algorithms [5] Molines [1] proposed the generalized MDF () that allows one to select FF size and the bloc delay separately his advantage is owing to the controlling of the overlap between the sccessive inpt blocs ecase of this flexibility the is more general than the NPFLMS However lie we jst pointed ot researchers presented different views on the convergence performance of NPFLMS and the derivations of step-size bonds in the literatre are not consistent [1] [6] [9] In this paper we mae a frther stdy on the step-size bonds of the generalized mltidelay adaptive filter he step-size bonds we derived in this paper is N times larger than that of the NLMS and is mch bigger than the one reported in [1] for and a recent paper [6] as well ecase of this new analysis we can now choose proper step-size so that the well-designed maintains good tracing and convergence performance and has great saving in comptations as well he rest of the paper is organized as follows Section smmarizes the algorithm Section 3 investigates the range of step-size that spports a stable filter We then compare or derivations with other wor

2 WCE 007 Jly London UK crrently available in the literatre Extensive simlation reslts confirming or performance analysis are presented in Section 4 he conclsions are made in the last section of the paper II SUMMARY OF HE Let n ( ) and dn ( ) represent the reference inpt and desired otpt signal respectively to the adaptive filter with order M he NLMS is formlated by en ( ) = dn ( ) w( n) ( n) (1) w( n+ 1) = w( n) + ( n) e( n) Mr0 () where ( n) and w ( n) are inpt and coefficient vectors respectively and r 0 is an estimate of the variance of n ( ) he step-size is bonded in (0 ) for convergence he NLMS with bloc size N is formlated by e( N + i) = d( N + i) w ( N + i) i = 01 N 1 (3) N 1 w+1 = w + ( N + i) e( N + i) (4) Mr0 N i= 0 where w denotes coefficient vector at bloc iteration he step-size is bonded in (0) for convergence [7] he segments the filter into L sb-filters each with an order N and is implemented in freqency domain with FF size N Withot loss of generality we assme that M = NL he ses a positive integer α to control the overlap between the sccessive inpt blocs Conseqently it pdates the coefficients every R = N α samples In the th iteration define reference inpt vector x and desired response vector d respectively as x [ ( ) ( 1) ( 1) ] = R R+ R+ N (5) d [ ( ) ( 1) ( 1) ] = dr dr+ dr+ N (6) Freqency-domain inpt vector for l th sb-filter denoted as X l = 1 L is compted as l l FF l α ( l 1) α X = x x (7) he corresponding freqency-domain coefficient vector H l is defined accordingly as Hl = FF hl 0 (8) th where h l is the l sb-filter s time-domain coefficient vector Filter otpt vector ˆd is calclated as dˆ = second part of L FF 1 Hl X l (9) l = 1 where denotes element-wise mltiplication In practice any weighted overlap and add reconstrction algorithm cold be sed to constrct the final R -sample otpt at each bloc iteration Freqency-domain error vector E is obtained as follows e = d d ˆ (10) = FF E 0 e (11) Most NPFLMS algorithms employ althogh might be slightly different freqency bin power normalization We present a sb-filter based normalization scheme as follows he freqency power of the l th sbfilter at th iteration is calclated as Zl =β Zl 1 + (1 β) Xl X l (1) where X l denotes the complex conjgate of X l and β is a forgetting factor he coefficient vector H l is pdated as H l + 1 = Hl + Φ l M (13) where is a fixed common n-normalized step-size parameter of the filter In (13) Φ l the new information for pdating is obtained as Φl = FF φl 0 (14) where φ l = first part of ( l ) 1 FF E X Z l (15) where denotes element-wise division III SEP-SIZE OUNDS ANALYSIS We present a range of step-size that spports a stable filter in this section A recent paper claimed that for sfficiently large bloc size N NPFLMS algorithm oght to perform similarly regardless of whether the inpt process is white or correlated [10] hat is as a conseqence of the freqency-bin normalization process the eigenvale spread of the atocorrelation matrix wold approach nity However [10] did not derive a range of the fixed common n-normalized step-size it simply referred to the bonds presented in [1] In this paper we derive a bond for the fixed common step-size parameter Note the Parseval s theorem relates power compted in time domain and in freqency domain Now assming n ( ) is a white process with zero mean and variance σ the expected vale of averaged freqency bin power wold eqal Nσ Using this Nσ for each freqency bin power associated with (13) and noting that (13) is a freqency-bin normalized version of (4) we have the following eqality for bonding step-size parameter = (16) Mσ M( Nσ ) ransforming the bond on to yields 0< < N (17)

3 WCE 007 Jly London UK he bond in (17) is good for spporting a stable algorithm for all types of inpt processes provided the FF size is large enogh to de-correlate the transformed inpt samples Contrary to a recent paper that explicitly states the limitation of freqency domain filters owing to the very restrictive step-size bonds [5] or reslts give a very good range of vales that spport a stable In the following we compare or step-size bond with other researchers wor Since the strctres of the NPFLMS algorithms presented in the literatre are somewhat different we rewrite the associated coefficient pdating eqations in the form of (13) for comparison he pdating eqation in [1] is rewritten as H 1 L + 1 = H + M 4 Φ (18) where H is the M 1coefficient vector and 1 is the fixed common n-normalized step-size [1] derived the range of convergence as 0 < < 4 (1 + L) (19) 1 his is eqivalent to confine as 0 < < L (1 + L) (0) Obviosly this bond is too restrictive compared to or derivation in (17) Sommen [6] had the eqivalent pdating eqation H l + 1 = H l + l M Φ (1) and gave a range of convergence 0 < < 1 () ransform this bond to or version will reslt in 0 < < 1 (3) Regarding the convergence properties it is well nown that the NLMS and the NLMS algorithms exhibit similar convergence performance if = N when NLMS adjsts the coefficients once per N samples of data [7] Follow the wor in [7] and note that the modifies the weights every R samples we can show that NLMS and wold perform similarly if = R = N α (4) for small vale of and N is large enogh Even thogh the was first presented over a decade ago to or best nowledge this is the first one that derives relations of step-size parameters for NLMS and to have similar convergence properties IV SIMULAION RESULS In this section we present the reslts of several experiments that validate or analysis on step-size bonds as well as verify the convergence analysis he adaptive filter was sed to identify a 51-tap acostic echo system h opt measred in a small office he acostic echo system was ept nchanged for the first 4 seconds Dring the next 4 seconds the system trned to be time-varying he evoltion of coefficients is described by hopt ( n) = hopt + g ( n) (5) where g ( n) is a Gassian random vector with mean 0 and variance 01 he system was switched bac to be time-invariant with coefficients h opt for the final 36 seconds Several types of inpt signals (white Gassian processes moving average (MA) processes and atoregressive (AR) processes) were sed For all experiments the power of the acostic echo system was set to be nit (dring the first 4 seconds and the last 36 seconds) and the additive white Gassian noise has variance 001 he reported mean sqared error (MSE) crves are reslts of ensemble averages over 0 independently rns followed by 64-sample time averages A Example 1: Step-size bonds We have performed extensive simlations to validate or stable step-size bonds in (17) We observed satisfactory reslts with = for all experiments with parameters L = 4 N = 18 and α = 1 We also performed with L eqals and 64 respectively We observed stable step-size bonds getting closer to N where N is the bloc size associated with that particlar L he extreme case wold be L = 51 N = 1 and stable step-size seem will approach to herefore we conclde that Molines [1] and Sommen [6] gave a too restrictive bond he MSE crves associated with AR processes inpt signal for L = 3 α = L = 64 α = are depicted in Fig 1 and Fig respectively Fig 3 shows the MSE crves of white inpt signal for L = 3 α = We never had any good reslt with step-size larger than N Example : Convergence properties Extensive experiments were condcted to spport the seflness of or convergence analysis in (4) De to the limitation of space we only provide some MSE plots of the setp: L = 4 N = 18 NLMS with = 04 with α = 1 4 and the corresponding fixed common step-size parameter = α he MSE crves associated with AR MA and white Gassian inpts are depicted in Figs 4 5 and 6 respectively We observed the MSE crves of algorithms are fairly close in all experiments he reslts verified or performance analysis presented in the previos section he NLMS performed comparably to the for AR and white Gassian inpts However the NLMS converged qite slowly for MA inpts It shold be noted that the NLMS did a better job in tracing the changing of the acostic echo system his might de to its sample-processing characteristic Experimental reslts of NLMS with = 08 and filters with = α are shown in Figres 7 8 and 9 for AR MA and white Gassian inpts respectively he filters perform similarly bt the MSE crves are not that close as for the case = α his is becase the associated is not small enogh

4 WCE 007 Jly London UK V CONCLUSIONS In this paper we derived the bonds of the fixed common step-size parameter for the generalized mltidelay adaptive filter Contrary to most wor crrently available in the literatre or reslts gave a very good range of vales that spport a stable Extensive simlation reslts were provided to validate the analysis We also derived relations of step-size parameters for NLMS and to have similar convergence properties he performance analysis was verified by extensive simlations REFERENCES [1] E Molines O Ait Amrane and Y Grenier he generalized mltidelay adaptive filter: strctre and convergence analysis IEEE ransactions on Signal Processing Vol43 No1 pp 14-8 Jan 1995 [] E Ferrara Fast implementations of LMS adaptive filters IEEE ransactions on Acostics Speech and Signal Processing Vol 8 pp Ag 1980 [3] J J Shyn Freqency-domain and mltirate adaptive filtering IEEE Signal Processing Magazine pp Jan 199 [4] J S Soo and K K Pang Mltidelay bloc freqency domain adaptive filter IEEE ransactions on Acostics Speech and Signal Processing Vol 38 No pp Feb 1990 [5] Y endel D rshtein O Shalvi and E Weinstein Delayless freqency domain acostic echo cancellation IEEE ransactions on Speech and Adio Processing Vol 9 No 5 pp Jly 001 [6] R M M Derx G P M Egelmeers and P C W Sommen New constraining method for partitioned bloc freqency-domain adaptive filters IEEE ransactions on Signal Processing Vol 50 No 9 pp Sep 00 [7] G A Clar S A Mitra and S R Parer loc implementation of adaptive digital filters IEEE ransactions on Acostics Speech and Signal Processing Vol ASSP-9 No 3 pp Jne 1981 [8] J C Lee and C K Un Performance analysis of freqency domain bloc LMS adaptive digital filters IEEE ransactions on Circits and Systems Vol 36 No pp Feb 1989 [9] K S Chan and Farhang-orojeny Analysis of the partitioned freqency-domain bloc LMS (PFLMS) algorithm IEEE ransactions on Signal Processing Vol 49 No 9 pp Sep 001 [10] Farhang-orojeny and K S Chan Analysis of the freqency-domain bloc LMS algorithm IEEE ransactions on Signal Processing Vol 48 No 8 pp Ag 000 Figre 1 MSE crves (Example 1) of algorithms with step-size = and AR inpt signal ( L = 3 α = ) Figre MSE crves (Example 1) of algorithms with step-size = and 16 8 AR inpt signal ( L = 64 α = ) Figre 3 MSE crves (Example 1) of algorithms with step-size = and White Gassian inpt signal ( L = 3 α = )

5 WCE 007 Jly London UK Figre 4 MSE crves of algorithms (with step-size = ) and NLMS (with = 04 ) (AR inpt L = 4 Figre 7 MSE crves of algorithms (with step-size = ) and NLMS (with = 08 ) (AR inpt L = 4 Figre 5 MSE crves of algorithms (with step-size = ) and NLMS (with = 04 ) (MA inpt L = 4 Figre 8 MSE crves of algorithms (with step-size = ) and NLMS (with = 08 ) (MA inpt L = 4 N = 18 ) Figre 6 MSE crves of algorithms (with step-size = ) and NLMS (with = 04 ) (White Gassian inpt L = 4 Figre 9 MSE crves of algorithms (with step-size = ) and NLMS (with = 08 ) (White Gassian inpt L = 4