A FAMILY OF VARIABLE STEP-SIZE NLMS ALGORITHMS FOR ECHO CANCELLATION

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1 A FAMILY OF VARIABLE STEP-SIZE NLMS ALGORITHMS FOR ECHO CANCELLATION CONSTANTIN PALEOLOGU 1, JACOB BENESTY, SILVIU CIOCHINĂ 1 Ky words: Adapti filtrs, Echo cancllation, Normalizd last-man-squar (NLMS), Variabl stp-siz NLMS (VSS-NLMS). Th normalizd last-man-squar (NLMS) algorithm is on of th most common choics for cho cancllation. Nrthlss, an NLMS algorithm has to compromis btwn sral prformancs critria (.g., conrgnc rat rsus misadjustmnt, tracking capabilitis rsus robustnss). Thus, a ariabl stp-siz NLMS (VSS-NLMS) algorithm rprsnts a mor rliabl solution. Rcntly, sral VSS-NLMS algorithms that tak into account th xistnc of th nar-nd signal (in trms of powr stimat) ha bn proposd with th objcti of rcoring th nar-nd signal from th rror signal. Sinc this is th basic goal in cho cancllation, this class of VSS-NLMS algorithms can b ry suitabl for such an application. Th main issu rmains th stimation of th nar-nd signal powr, in trms of accuracy and othr practical aspcts (.g., aailabl paramtrs, computational complxity). This papr analyzs diffrnt solutions for this problm, making a first unifid approach or th prformancs of this family of VSS-NLMS algorithms. 1. INTRODUCTION In th contxt of cho cancllation, an adapti filtr gnrats at its output a rplica of th cho, which is furthr subtractd from th microphon signal [1]. In othr words, it has to modl an unknown systm, i.., th cho path, lik in a systm idntification problm []. En if this formulation is straightforward, th spcific faturs of cho cancllation (.g., high lngth and tim-ariant cho paths, th prsnc of th nar-nd signal) challng any adapti algorithm. Diffrnt typs of algorithms wr inold in this contxt [1, ], but th work-hors rmains th normalizd last-man-squar (NLMS) algorithm. Th classical NLMS algorithm uss a constant stp-siz to control its prformanc. A compromis should b mad whn choosing this paramtr; a larg alu is propr 1 Faculty of Elctronics, Tlcommunications and Information Thory / Tlcommunications Dpartmnt, Unirsity Polithnica of Bucharst, 1-3 Iuliu Maniu, ZIP 6171, Bucharst, Romania, {pal, siliu}@comm.pub.ro. Unirsité du Qubc, INRS-EMT, Montral, QC HA 1K6, Canada, bnsty@mt.inrs.ca. R. Roum. Sci. Tchn. Élctrotchn. t Énrg.,, 1, p. 49 8, Bucarst, 1

2 Constantin Palologu, Jacob Bnsty, Siliu Ciochină for conrgnc rat and tracking, whil a small alu lads to low misadjustmnt and good robustnss faturs. Sinc in cho cancllation thr is a nd for all ths prformanc critria, th stp-siz should b controlld. Hnc, a ariabl stp-siz NLMS (VSS-NLMS) algorithm rprsnts a mor propr choic. Diffrnt typs of VSS-NLMS algorithms wr dlopd (.g., s [3] and th rfrncs thrin). In gnral, most of thm rquir th tuning of som paramtrs which ar difficult to control in practic. Rcntly, a non-paramtric VSS-NLMS () algorithm was proposd in [3]. It was dlopd in a systm idntification contxt, aiming to rcor th systm nois (i.., th nois that corrupts th output of th unknown systm) in th rror of th adapti filtr. In th contxt of cho cancllation, this systm nois is th nar-nd signal. If only th background nois is considrd at th nar-nd, its powr stimat can b obtaind during silncs. This was th framwork of th rsults rportd in [3], showing that th algorithm is ry simpl and asy to control in practic. Nrthlss, th main challng is whn th nar-nd signal contains not only th background nois but also th nar-nd spch (i.., doubl-talk scnario). Inspird by th original ida of th algorithm, sral approachs ha focusd on finding othr practical mthods for th stimation of th nar-nd signal powr [4, ]. Th objcti of this papr is to mak a first unifid approach or this family of VSS-NLMS algorithms, in th contxt of cho cancllation. Thir capabilitis and prformancs ar analyzd in trms of th classical critria (.g., conrgnc rat, tracking, robustnss), but also from othr practical points of iw (.g., aailabl paramtrs, computational complxity). In th following, all signals ar ral-alud and th tim indx is dnotd by n.. VSS-NLMS ALGORITHMS FOR ECHO CANCELLATION In th framwork of cho cancllation, th far-nd signal x(n) gos throw th cho path h, proiding th cho signal y(n). This signal is addd with th nar-nd signal (n) [which can contain both th background nois, w(n), and th nar-nd spch, u(n)], rsulting th dsird signal d(n). Th adapti filtr, dfind by th ctor ĥ(n), aims to produc at its output an stimat of th cho, ŷ(n), whil th rror signal (n) should contain an stimat of th nar-nd signal. Th gnral updat quation of th VSS-NLMS algorithms is ( ) = hˆ ( n 1) µ x hˆ n + x, T (1) x whr µ(n) is th ariabl stp-siz paramtr (a positi scalar btwn and 1) and x(n) = [x(n) x(n 1) x(n L + 1)] T is th input signal ctor; th lngth of

3 3 A family of ariabl stp-siz NLMS algorithms 1 th adapti filtr is L, whil suprscript T dnots transposition. It should b notd that th classical NLMS algorithm uss a fixd stp-siz, i.., µ(n) = = constant, for any tim indx n. Th a-priori rror signal is dfind as ( ) = ( ) ˆ ( ) = ˆT ( ) ( 1) x( ) n d n y n d n h n n. () Using th adapti filtr cofficints at tim n, th a-postriori rror signal is d( n ˆT ) ε = h x. (3) Introducing (1) in (3), and taking () into account, it rsults ( n ) = [ µ ] ε 1. (4) A straightforward mthod for dtrmining th stp-siz paramtr is to cancl th a-postriori rror, i.., ε(n) =. Imposing this condition in (4) [assuming that (n) ], w obtain µ(n) = 1. This is th cas of th classical NLMS algorithm; apparntly, this could b th optimum alu of th stp-siz paramtr. But this approach is tru only in th absnc of th nar-nd signal. Lt us rwrit (3) as T ˆ ε = x h h +. () If (n) = thn th condition ε(n) = maks sns, bcaus it lads to h = = ĥ(n) (if w assum that th rror and far-nd signals ar indpndnt). Othrwis, th annihilation of th a-postriori rror from () implis that x T (n)[h ĥ(n)] = (n), which will bias th adapti filtr stimat. Sinc th xistnc of th nar-nd signal can not b omittd in th contxt of cho cancllation, a mor rasonabl condition to impos is ε(n) = (n). Consquntly, using (4), th basic rlation bcoms [ 1 µ ( n )] =. (6) At a first glanc, th thortical solution is µ(n) = 1 (n)/(n), but this rsult is not practical from many points of iw (.g., (n) is unaailabl, instantanous alus ar usd, µ could bcom ngati). Rlatd to this framwork, an algorithm, calld st mmbrship NLMS (SM- NLMS) was proposd in [6]. Its stp-siz is aluatd as µ SM 1 η ( ) =, if > η n, (7), othrwis with η bing a bound on th signal (n). En if it is basd on (6), this algorithm achi only a slight prformanc impromnt as compard to th NLMS algorithm, sinc thr is no araging on (n).

4 Constantin Palologu, Jacob Bnsty, Siliu Ciochină 4 Th algorithm proposd in [3] uss a mor rliabl approach, imposing th condition E{ε (n)} = E{ (n)}, whr E{ } dnots mathmatical xpctation. Squaring and taking th xpctation in both sids of (6), th solution of th rsultd quadratic quation is { } E µ =1. (8) E { } It should b notd that th scond solution of th quation lads to a alu btwn 1 and, whil th stp-siz from (8) is btwn and 1. En if both solutions ar stabl, th on from (8) producs lss stady-stat man squar rror with th sam conrgnc spd. From practical considrations, (8) has to b rwrittn as µ = In a gnral mannr, th paramtr 1. (9) σ dnots th powr stimat of th signal α(n). It can b aluatd in a rcursi mannr as ˆ α = λ ( n 1) + ( 1 λ) α α α, (1) whr λ is a wighting factor chosn as λ = 1 1/(KL), with K > 1. Rgarding (9), it is asy to stimat th powr of th rror signal (n), sinc it is aailabl. On th othr hand, th stimation of th nar-nd signal powr is mor problmatic. In [3], only th background nois is considrd at th nar-nd, i.., (n) = = w(n). Its powr can b stimatd during silncs and can b assumd constant; thus, σ ˆ ( ) = σ ˆ. Consquntly, th stp-siz of th algorithm is n w µ NPVSS w 1, if > w n, (11), othrwis ( ) = ζ + whr th small positi constant ζ prnts diision by zro. It can b noticd that th SM-NLMS algorithm is obtaind whn choosing λ = and η = w. Th rsults rportd in [3] (in th contxt of acoustic cho cancllation) showd that th algorithm outprforms th classical NLMS and SM-NLMS algorithms. Only a singl-talk scnario was considrd in [3]. Nrthlss, two main situations can bias this approach. First, th background nois w(n) can b tim-ariant; scond, th nar-nd spch u(n) can b prsnt (i.., doubl-talk cas).

5 A family of ariabl stp-siz NLMS algorithms 3 A ry simpl way to roughly stimat th nar-nd signal powr is to us th rror signal (n), but with a largr alu of th wighting factor, i.., = γ ( n 1) + ( 1 γ) σ ˆ, (1) with γ > λ. Lt us call this algorithm as th simpl VSS-NLMS () algorithm. Sinc ˆσ can aris around ˆσ, its stp-siz is computd using th absolut alus, i.., µ SVSS = 1. (13) ζ + Th alu of γ influncs th orall bhaiour of th algorithm. Whn γ is clos to λ, th stp-siz of th algorithm is small, which affcts th conrgnc rat and tracking capabilitis, but offrs a good robustnss against nar-nd signal ariations. Whn γ is larg as compard to λ, th conrgnc rat is improd, but th robustnss issus ar dgradd. A mor laboratd mthod was rcntly proposd in [4]. It was dmonstratd that th powr stimat of th nar-nd signal can b aluatd as ( ) T ˆ x n 1 σ ˆ = rˆ ˆ x r x, (14) σ whr th powr stimats of (n) and x(n) ar computd basd on (1), whil th last trm is stimatd in a similar mannr, i.., x = λrˆ ( n 1) + ( 1 λ) x rˆ. (1) x Using (14), th NEW- algorithm proposd in [4] computs its stp-siz as µ NEW NPVSS 1 = ζ + 1 if ξ < ς, (16) othrwis whr ξ(n) is th conrgnc statistic and ς is a small positi quantity. Th conrgnc statistic is aluatd as rˆ rˆ d ξ =, (17) d whr th powr of d(n) is stimatd basd on (1), and th corrlation trm is computd as d

6 4 Constantin Palologu, Jacob Bnsty, Siliu Ciochină 6 = λrˆ ( n 1) + ( 1 λ) d rˆ. (18) d d It can b noticd that th stp-siz of th NEW- algorithm is computd using signals that ar aailabl, xcpt th constant ς. This paramtr is rlatd to th conrgnc stat. In th initial phas of th algorithm, or whn thr is an cho path chang, th condition ξ(n) < ς is not fulfilld; consquntly, according to (16), th stp-siz is qual to 1, forcing a fast adaptation. Othrwis, in th stady-stat of th algorithm or whn th nar-nd signal fluctuats (.g., doubl-talk cas), th condition ξ(n) < ς is fulfilld and th algorithm uss th first lin of (16). Th main problm is how to choos this conrgnc thrshold, in trms of th alu of ς. Th xprimntal rsults will show that this paramtr is important for th orall prformanc of th algorithm. Both prious algorithms rquir th tuning of som paramtrs that influnc thir orall prformancs. In practic, it is not always asy to control such paramtrs. A mor practical solution was rcntly proposd in []. Th dsird signal of th adapti filtr is xprssd as d(n) = y(n) + (n). Sinc th cho signal and th nar-nd signal can b considrd uncorrlatd, th prious rlation can b rwrittn in trms of powr stimats as = + σ σ ˆ. (19) d y ˆ us Assuming that th adapti filtr has conrgd to a crtain dgr, w can y σ σ. () yˆ Consquntly, σ σ ˆ, (1) d ˆ yˆ whr th paramtrs from th right hand sid of (1) ar aluatd as in (1). Th stp-siz of th practical VSS-NLMS () algorithm is aluatd as yˆ ˆ d µ PVSS = 1. () ζ + σ Th absolut alus from () prnt any minor diations (du to th us of powr stimats), which can mak th stp-siz ngati or complx. It is a nonparamtric algorithm, sinc all th paramtrs from () ar aailabl. Also, good robustnss against nar-nd signal ariations is xpctd. Th main drawback is rlatd to (). This assumption will b biasd in th initial conrgnc phas or

7 7 A family of ariabl stp-siz NLMS algorithms whn thr is a chang of th cho path. Concrning th first problm, w can us a rgular NLMS algorithm in th first stps (.g., in th first L itrations). Also, th ida of th algorithm can b xtndd to th affin projction algorithm (APA), improing th conrgnc rat and tracking capabilitis. Tabl 1 Computational complxitis of th diffrnt ariabl-stp siz algorithms Algorithms Additions Multiplications Diisions Squar-roots NEW- L + 8 3L Th computational complxitis rquird by th ariabl-stp sizs of th prsntd algorithms ar shown in Tabl 1. In th cas of th algorithm, th computational amount rlatd to th stimation of th background nois powr (during silncs) is not includd, so w cannot conclud that it is th chapst algorithm. Fairly, th algorithm is th last complx on. Also, th algorithm has a modrat complxity. Th most xpnsi on is th NEW- algorithm, sinc its computational complxity dpnds on th filtr s lngth [du to (14) and (1)]; it is known that in th contxt of cho cancllation, this alu can b ry larg (hundrds in th cas of ntwork cho cancllrs or n thousands in th cas of acoustic cho cancllrs). 3. SIMULATION RESULTS Th simulations wr prformd in th contxt of acoustic cho cancllation. Th lngth of th adapti filtr is L = 1. Th acoustic impuls rspons (i.., th cho path) was truncatd to th sam lngth. Th sampling rat is 8 khz. Th input signal, x(n), is a spch squnc. An indpndnt whit Gaussian nois w(n) is addd to th cho signal y(n), with db signal-to-nois ratio (SNR) for most of th xprimnts. Th masur of prformanc is th normalizd misalignmnt, dfind as log 1 ( h ĥ(n) / h ), whr dnots th l norm. Sral paramtrs ar st for all th prsntd VSS-NLMS algorithms, as follows. Th rgularization factor [i.., a positi constant that nds to b addd to th dnominator from (1), in ordr to aoid diisions by small numbrs] is δ = σ x, whr σ x is th powr of th input signal. Th wighting factor from (1) is computd using K = 6 [3]. Th small positi constant from th dnominator of th stp-sizs is ζ = 1 8. First, w aluat th influnc of th paramtrs γ and ς or th prformancs of th and NEW- algorithms, rspctily. Th simulations wr prformd in a singl-talk scnario (Fig. 1). In

8 6 Constantin Palologu, Jacob Bnsty, Siliu Ciochină 8 ordr to strss th algorithms, an abrupt chang of th cho path was introducd at tim 1, by shifting th acoustic impuls rspons to th right by 1 sampls; also, th SNR dcrass from db to db (i.., background nois incrass) at tim. (a) (b) 1 with γ = 1-1/(3KL) with γ = 1-1/(KL) with γ = 1-1/(1KL) 1 NEW- with ς =.1 NEW- with ς =.1 NEW- with ς = Fig. 1 Singl talk cas, cho path changs at tim 1, SNR dcrass from db to db at tim : a) misalignmnt of th algorithm for diffrnt alus of γ; b) misalignmnt of th NEW- algorithm for diffrnt alu of ς. (a) NEW- (b) NEW Fig. a) Misalignmnt of th, [with γ = 1 1/(3KL)], NEW-NPVSS- NLMS (with ς =.1), and algorithms; singl talk-cas, SNR = db; b) cho path changs at tim 1 (othr conditions as in (a)). 1 (a) NEW (b) NEW Fig. 3 Misalignmnt of th VSS-NLMS algorithms. SNR dcrass btwn tim 1 and : a) from db to 1 db; b) from db to 1 db (othr conditions as in Fig. a).

9 9 A family of ariabl stp-siz NLMS algorithms 7 1 (a) NEW- 1 (b) NEW Fig. 4 Misalignmnt of th VSS-NLMS algorithms during doubl-talk (without DTD). Nar-nd spch appars btwn tim 1 and 16; a) FNR = db; b) FNR = 3 db (othr conditions as in Fig. a). Th bhaiour of th algorithm is prsntd in Fig. 1a. At a first glanc, a larg alu of γ rprsnts a mor propr choic. But in this situation th stimation of th nar-nd signal powr from (1) will also ha a larg latncy, which is not suitabl for th cas whn thr ar nar-nd signal ariations. Consquntly, a compromis should b mad whn choosing th alu of γ. In th cas of th NEW- algorithm (Fig. 1b), a ry small alu of ς will mak th first lin of (16) futil, sinc ξ(n) > ς; thus, th algorithm acts lik a rgular NLMS with th stp-siz qual to 1. Th prformanc is improd for a largr alu of ς. Nrthlss, it should b takn into account that a ry larg alu of ς could affct th tracking capabilitis of th algorithm (and th initial conrgnc), bcaus th condition ξ(n) > ς could b inhibitd. Nxt, th paramtrs of th NEW- and th algorithms ar st in ordr to obtain similar initial conrgnc rats. Th original algorithm (assuming that th powr of th background nois is known) and th algorithm ar also includd for comparisons. Thir prformancs ar analyzd in a singl talk-scnario (Fig. a). In trms of final misalignmnt, th algorithm is clos to th algorithm; th NEW- algorithm slightly outprforms th SVSS- NLMS algorithm. In Fig. b, th abrupt chang of th cho path is introducd at tim 1. As xpctd, th algorithm has a slowr raction as compard to th othr algorithms, sinc th assumption (1) is strongly biasd. A ariation of th background nois is considrd in Fig. 3. Th SNR dcrass from db to 1 db (Fig. 3a) and to 1 db (Fig. 3b), btwn tim 1 and. En if th NEW- algorithm is robust against th first ariation, it fails in th last situation. For such a low SNR, a highr alu of ς is rquird, which could dgrad th conrgnc rat and th tracking capabilitis of th algorithm. For improing th robustnss of th algorithm, th

10 8 Constantin Palologu, Jacob Bnsty, Siliu Ciochină 1 alu of γ should b incrasd, but with th sam unwantd ffcts. Dspit th alu of SNR, th algorithm is ry robust in all th situations. Finally, th doubl-talk cas is considrd in Fig. 4, without using any doubl-talk dtctor (DTD). Diffrnt alus for th far-nd to nar-nd spch ratio (FNR) wr usd, i.., FNR = db (Fig. 4a) and FNR = 3 db (Fig. 4b). Dspit th small ariation of th FNR, it can b noticd that th NEW-NPVSS- NLMS algorithm is snsiti to this modification. In trms of robustnss, th algorithm outprforms th othr VSS-NLMS algorithms. 4. CONCLUSIONS A family of VSS-NLMS algorithms was analyzd in th contxt of cho cancllation. Th basic ida of ths algorithms is to rcor th nar-nd signal in th rror of th adapti filtr. For this purpos, thr is a nd for th stimation of th nar-nd signal powr, which is not always asy in practic. Each of th prsntd algorithms uss a diffrnt mthod for this problm. Among ths, th algorithm is th most non-paramtric on, sinc it dos not rquir th tuning of any additional paramtrs. Th simulation rsults indicat a robust bhaiour of this algorithm. Thus, it is a good candidat for ral-world applications. ACKNOWLEDGEMENT This work was supportd by th UEFISCSU Romania undr Grants PN-II- Idi no. 6/1.1.7 and no. 331/ Rcid on 8 Nombr 8 REFERENCES 1. J. Bnsty and Y. Huang (Eds.), Adapti Signal Procssing Applications to Ral-World Problms, Brlin, Grmany, Springr-Vrlag, 3.. S. Haykin, Adapti Filtr Thory, Fourth d., Uppr Saddl Rir, NJ, Prntic-Hall,. 3. J. Bnsty, H. Ry, L. Ry Vga, S. Trssns, A nonparamtric VSS NLMS algorithm, IEEE Signal Procssing Lttrs, 13, 1, pp , M.A. Iqbal, S.L. Grant, Nol ariabl stp siz NLMS algorithm for cho cancllation, IEEE ICASSP 8, Las Vgas, U.S.A., pp C. Palologu, S. Ciochină, J. Bnsty, Doubl-talk robust VSS-NLMS algorithm for undrmodling acoustic cho cancllation, IEEE ICASSP 8, Las Vgas, U.S.A., pp S. Gollamudi, S. Nagaraj, S. Kappor, Y.-F. Huang, St-mmbrship filtring and a st-mmbrship normalizd LMS algorithm with an adapti stp-siz, IEEE Signal Procssing Lttrs,,, pp , 1998.