Fast Recursive Least Squares Algorithm for Acoustic Echo Cancellation Application

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1 SEI 7 4 h Inernaional Conferene: Siene of Eleroni, ehnologie of Informaion and eleommuniaion Marh 5-9, 7 UNISIA Fa Reurive Lea Square Algorihm for Aoui Eho Canellaion Appliaion Farid Ykhlef, A. Gueoum and D. Berkani LASI laboraory, Deparmen of Eleroni, Fauly of Engineering Siene, Univeriy of Blida, Algeria. f_ykhlef@yahoo.fr Abra: Adapive filering i ued in a wide range of appliaion inluding eho anellaion, noie anellaion and equalizaion. In hee appliaion, he environmen in whih he adapive filer operae i ofen non-aionary. For aifaory performane under non-aionary ondiion, an adapive filering i required o follow he aiial variaion of he environmen. raking analyi provide inigh ino he abiliy of an adapive filering algorihm o rak he hange in urrounding environmen. he raking behavior of an algorihm i quie differen from i onvergene behavior. While onvergene i a ranien phenomenon, raking i a eady-ae phenomenon. Over he la deade a la of equivalen algorihm uh a he Normalized Lea Mean Square algorihm () and he Fa Reurive Lea Square algorihm () ha been developed o aelerae he onvergene peed. In aoui eho anellaion onex, we propoe in hi paper o ue numerially able Fa Reurive Lea Square algorihm o improve he qualiy and he inelligibiliy of he enhaned peeh. Key word:,, abiliy, aoui eho anellaion, peeh ignal. INRODUCION In hand-free eleommuniaion [HÄN 9] uh a full-duplex eleonferening he aoui ehoe of far-end peeh need o be removed. he mehod for doing hi i alled Aoui Eho Canellaion and i uually performed uing ime domain Finie Impule Repone filer o model he aoui eho pah and o predi he ehoe. Ideally, he ehoe an hen be removed by removing he predied ehoe from he mirophone ignal. Sine he algorihm for aoui eho anellaion hould be feaible o implemen on a andard Digial Signal Proeor, i i herefore very imporan ha he requiremen of he filer eimaion algorihm in erm of orage and ompuaional omplexiy hould no be oo large. Sine he eho pah are ime-varying i i alo very imporan ha he algorihm rapidly adap o hange in hee. A va number of mehod have been eed for he aoui eho anellaion problem [AHG ] of whih probably he mo ommonly ued mehod, due o i low ompuaional omplexiy, i he Normalized Lea Mean Square () algorihm. I adap, however, very lowly and wha would be deired i an algorihm ha adap faer uh a he Reurive Lea Square (RLS) algorihm. he Reurive Lea Square algorihm i, however, oo ompuaionally omplex and require oo muh memory o be implemened in a real-ime appliaion on a andard Digial Signal Proeor. Inead algorihm uh a he Fa Reurive Lea Square () algorihm have been propoed. he Fa Reurive Lea Square algorihm uffer, however, from problem wih numerial inabiliy and i ha been hown [SLO 9] ha all known algorihm are unable when uing a forgeing faor le han one. here exi an exenive lieraure addreing he abiliy oluion of he Fa Reurive Lea Square algorihm under a aionary environmen. We inrodue in Seion he numerially able verion of algorihm propoed in [YKH 6]. In hi paper, we will propoe o ue he numerially able algorihm for aoui eho anellaion appliaion. he re of he paper i organized a follow. Seion onain he definiion of he aoui eho anellaion by adapive algorihm follow-up of a preenaion of he algorihm numerially able in eion. We preen alo in hi eion a abiliy ondiion of hi numerially able algorihm. We preen in Seion 3 he aoui eho anellaion baed on algorihm. Simulaion reul are illuraed in eion 4. Alo, we will ompare in hi eion he algorihm wih algorihm. Finally, Seion 5 onlude our paper.

2 SEI7. Aoui eho anellaion by adapive algorihm An eho aneller [MOR 98] remove he undeired eho ignal ha feed bak from he loudpeaker o he mirophone in full-duplex handfree eleommuniaion. he anellaion i done by modeling he eho pah impule repone wih an adapive finie impule repone (FIR) filer and ubraing he eho from he mirophone oupu ignal. Aoui eho anellaion deal wih adapive yem idenifiaion wih known inpu-oupu. A bai hema of idenifiaion by an adapive algorihm i hown in figure. he far-end peeh ignal x goe hrough he eho pah repreened by a filer H ( and add o he mirophone ignal y ogeher wih he near-end alker v and noie n : y = H ( X v n () he oeffiien of he adapive filer W uing adapive algorihm are adjued aording o he following equaion: ε = y W X () W = W ε G (3) where denoe marix or veor ranpoed, N i he lengh of he filer, i he diree ime index, X N, denoe a veor whih ummarize he pa of he ignal x over lengh of N poin, y i he eho ignal, ε N, i he a priori error oupu and G N, denoe he adapaion gain veor. x. Loudpeaker Eho pah ( H ( ) Filering Adapive algorihm n ŷ v y Mirophone Figure. Aoui eho anellaion by an Adapive Algorihm. _. ε. Fa Reurive Lea Square Algorihm For he algorihm, he adapaion gain veor G i equal o he quaniy N, N,, where i he likelihood variable and N, i he normalized Kalman gain veor, alulaed independenly of he filering par, by a Fa Reurive Lea Square () algorihm uing linear forward/bakward prediion analyi on he inpu ignal x [BEN 89]. Here i he fir diffiuly for algorihm when applied o he idenifiaion of (Moving Average) MA model: he adapaion gain alulaed on he inpu ignal x i more eniive o hi non-aionary ignal han o he yem o be idenified [BEN 89]. In hi ae, he veor W N, i alulaed by minimizing he rierion of lea quare aording o: where λ faor. i= i ( y W X ) N,i J = λ (4) i ( < λ ) i an exponenial forgeing A wa menioned before, everal numerial oluion of abilizaion, wih aionary inpu, are propoed in lieraure [SLO 9], [BEN 88], [BEN 89] and [ARE 4]. On he oher hand, he algorihm i nooriouly unable, bu i i poible o mainain abiliy by uing a few equaion. Here, we ue he numerially able verion of algorihm propoed in [YKH 6]. he general priniple i o modify he numerial proprieie of he algorihm wihou modifying he heoreial behavior of he algorihm. he numerially able of hi verion i lied on able. he onan E (rily poiive) i he only one whih mu be orrely eleed. During he fir ieraion, he value aken by he inernal variable of he algorihm are loely linked o he hoie of he onan E. In praie, i i neeary o enure he aring of he algorihm. We an hooe for example he onan E whih verifie he following inequaliy [CIO 84]: where N E σ x (5) σ x i he energy of he ignal o enure he numerial abiliy of hi verion in he aionary ae, i i neeary o hooe a forgeing faor λ [YKH 6]: λ > (6) pn where he parameer p i a real number higher han o enure he numerial abiliy of he algorihm. he ale parameer ( µ ) onrol he x

3 SEI7 propagaion of he numerial error in he algorihm [YKH 6]. Aording o he imulaion made in [YKH 6], he be reul are given for µ =.75. able Numerially Sable Algorihm. - Prediion par: e = x A X α = λ α e N, e, = λα A = A e N, r = x N B X f N N, r = λβ N, r N = λ α f, f ξ N, = r [( µ ) r µ r = r ξ λ α, = α =,, r, A N, f N, B = N N,, B B r N, = β N, = λ β (r N, ) - Filering par: ε = y W X W W ε = 3. Aoui eho anellaion uing numerially able algorihm In onferening yem, uh a eleonferening and dekop onferening, aoui eho aneler (AEC) are needed o redue he eho ha reul from he aoui oupling beween he loudpeaker and he mirophone. he AEC idenifie he eho pah and imulaneouly redue he eho by mean of adapive filering. For he real-world inpu ignal uh a peeh, he numerially able algorihm an ill exhibi unable behavior due o he non aionary haraer of he peeh ignal. In aoui yem, he eho pah i uually long and he number of ap in he adapive filer i r ] herefore large. Furhermore, he raining ignal i ofen narrow-banded like peeh ignal. he inroduion of he fa reurive lea quare algorihm, whih an provide a fa onvergene rae wih a ompuaional o of omplexiy. he inabiliy problem of hi algorihm, however, grealy impair i praial appliaion. In aoui eho aneller appliaion, a periodi re-iniializaion proe for overoming he inabiliy problem wa propoed [BEN 89] [SLO 9]. he main of hi ehnique oni o re-iniialize he algorihm (re-iniialize of he prediion par forward/bakward) on he parameer W already N, idenified. he proedure of re-iniializaion i a follow: = B = = ; W W ; A N = N α N, = E λ ; β = E ; =. where: E = Ep( ) i he onan E, i mu be quie eleed o enure he rearing of he algorihm. hu, we eimae permanenly energy Ep () of he inpu ignal: Ep () = wep( ) (7) wih onan w <. x 4. Simulaion reul Fir, we deribe he aoui of he loudpeakero-mirophone ignal pah where he peakerphone i loaed. We an ue a meaured impule repone in a room of lengh equal o ap. he repone i illuraed by he figure Room Impule Repone Sample Figure. Room Impule Repone In he following, he inpu ignal are generaed uing peeh equene ampled a 8 khz. he far-end peeh ignal x goe hrough he eho pah repreened by he room impule repone and add o he mirophone ignal y ogeher wih he near-end alker v and noie n a Signal-o-Noie Raio equal o 4 db are repreened in he figure 3.

4 SEI7 Near-End alker Mirophone Signal Frequeny(Hz) Frequeny(Hz) Figure 3. ime evoluion. For a omparaive udy, we will inrodue he algorihm defined by he equaion () and (3), where G N, i: G δ = X (8) X X N, he ep-ize parameer (< δ <) onrol he abiliy and onvergene properie of algorihm. Figure 4 and figure 5 how he ime evoluion and he perogram of he reul afer proeing by he wo algorihm for N=, λ =.9998 (: abiliy ondiion (6) verified) and δ = () Figure 5. Sperogram of he oupu of Aoui Eho Caneller. However, i appear learly ha he ue of algorihm an aelerae he onvergene of he impule repone oward i limi. In order o ae he robune of he udied algorihm o noie power, a performane analyi wa alo provided. he figure 6 preen he reidual ehoe of he wo algorihm for variou SNR. he reul of hee algorihm are uperimpoed o verify advanage of he algorihm. 5. Conluion hi paper illurae he appliaion of adapive filer o aoui eho anellaion. Aoui eho anellaion i imporan for audio eleonferening when imulaneou ommuniaion (or full-duplex ranmiion) of peeh i neeary. In aoui eho anellaion, a meaured mirophone ignal onain wo ignal: he near-end peeh ignal and he far-end ehoed peeh ignal. In hi ae, he goal i o remove he far-end ehoed peeh ignal from he mirophone ignal o ha only he near-end peeh ignal i ranmied. A numerially able verion of algorihm wa ued for aoui eho anellaion appliaion. A omparaive udy beween and algorihm ha been inrodued o verify he high qualiy aoui eho anellaion, fa onvergene and good raking apabiliie of algorihm. Figure 4. ime evoluion of he oupu of Aoui Eho Caneller.

5 SEI Reidual eho ignal for inpu SNR = 6dB Reidual eho ignal for inpu SNR = 5dB Reidual eho ignal for inpu SNR = 4dB Reidual eho ignal for inpu SNR = 35dB Reidual eho ignal for inpu SNR = 3dB Reidual eho ignal for inpu SNR = 5dB REFERENCES [ARE 4] M. Arezki, A. Benallal, F. Ykhlef, A. Gueoum, and D. Berkani, New mehod of omparion NS- and Algorihm for he aoui eho anellaion, Proeeding of he 4h inernaional ympoium on CSNDSP 4, Univeriy of Newale, U.K, pp , July 4. [AHG ] P. Ahgren, E. G. Laron, Eho Canellaion Uing he Conjugae Gradien Algorihm, publihed a he Reglermöe, May 9-3. [BEN 88] A. Benallal, A. Gilloire, A New Mehod o Sabilize Fa RLS Algorihm Baed on a Fir-Order Model of he Propagaion of Numerial Error, 988 Inl. Conf. on Aoui, Speeh, and Signal Pro., New York, N.Y., April 988. [BEN 89] A. Benallal, Sudy of he ranvere Fa Reurive Lea Square algorihm and appliaion o he idenifiaion of aoui impule repone, Phd hei, univeriy of Renne I, Frane, January 989. [MOR 98] R. Morgan, J. Beney, M. Sondhi, On he evaluaion of eimaed impule repone, IEEE Signal proeing leer, vol-5, N 7, pp , July 998. [CIO 84] J. M. Cioffi,. Kailah, Fa Reurive Lea Square ranveral Filer for Adapive Proeing, IEEE ran. Aou., Speeh, Signal Proe., Ap-34, pp , April 984. [HÄN 9] E. Hänler, he hand-free problem, An annoaed bibliography. Signal Proeing, 3(7):59-7, 99. [SLO 9] D.. M. Slok,. Kailah, Numerially Sable ranveral Filer for Reurive Lea Square Adapive Filering, IEEE ran. on Signal Proeing, Vol. 39, No., pp. 9-4, January 99. [YKH 6] F. Ykhlef, M. Arezki, A. Benallal, A. Gueoum, and D. Berkani, Sabiliy Analyi of Fa Reurive Lea Square Algorihm: Appliaion o Adapive Filering, Proeeding of he 5h inernaional ympoium on Communiaion Syem, Nework and Digial Signal Proeing (CSNDSP 6), Para Univ. Conferene Cenre, Para, Greee, pp , July 6. Reidual eho ignal for inpu SNR = db Reidual eho ignal for inpu SNR = 5dB Figure 6. Reidual eho ignal.