AN IMPROVED VARIABLE STEP-SIZE AFFINE PROJECTION SIGN ALGORITHM FOR ECHO CANCELLATION * Jianming Liu and Steven L Grant 1

Transcription

1 AN IMPROVED VARIABLE SEP-SIZE AFFINE PROJECION SIGN ALGORIHM FOR ECHO CANCELLAION * Jiamig Liu ad Steve L Grat Departmet of Eectrica ad Computer Egieerig, Missouri Uiversity of Sciece ad echoogy, Roa, Missouri ABSRAC his paper proposes a improved variabe step-size (VSS) agorithm for the recety itroduced affie projectio sig agorithm (APSA) based o the recovery of the ear-ed siga eergy i the error siga. Simuatio resuts demostrate that, compared to the previous VSS for APSA, the proposed approach provides both more robustess to impuse iterferece ad better trackig abiity of echo path chage. * Idex erms Variabe step-size, affie projectio sig agorithm, acoustic echo caceatio. INRODUCION he probem of acoustic echo caceatio (AEC) is usuay approached by modeig the echo path impuse with a adaptive fiter ad subtractig the estimated echo from the microphoe output siga []. However, due to the ear-ed siga ad echo path chage, it is very importat that the adaptive fiter agorithm shoud be robust to impusive iterferece ad sesitive to echo path chage. Recety, for acoustic echo caceatio, the famiy of sig agorithms has become popuar i the iterature because they are robust to impusive iterferece []-[3]. It is we kow that the performace of most adaptive fiters is govered by the choice of the step-size parameter [4]-[6]. A few variabe step-size agorithms for sig agorithms have bee reported i [7]-[]. hese variabe step-size agorithms, however, are ot robust to impusive iterferece or sesitive to echo path chage. he variabe step-size agorithm i [] depeds o a priori iformatio of the iterferece. Meawhie, VSS i [7] ad the deta sequece i [8]-[9] coud ot react to the chage of echo path. I this paper, we propose a variabe step-size agorithm the affie projectio sig agorithm. Meawhie, the proposed VSS APSA agorithm is robust to impusive iterferece ad tracks echo path chages quicky. * his work is performed uder the Wikes Missouri Edowmet. Formery Steve L Gay. his paper is orgaized as foows. Sectio reviews the recety proposed VSS for APSA, ad i Sectio 3 we preset the proposed VSS-APSA. he simuatio resuts ad compariso to the previous agorithms are preseted i Sectio 4. Fiay cocusios are draw i Sectio 5. he far-ed siga impuse respose. REVIEW OF VSS APSA x is fitered through the room h to get the echo siga y. y( ) x( )* h( ) x h () where x [ ] x x x L, h [ ] h h h L, ad L is the egth of echo path. his echo siga is added v (icudig both speech ad to the ear-ed siga back-groud oise, etc.) to get the microphoe siga d, x * h d v xh v ( ). Groupig the P most recet iput vectors gives the iput siga matrix: P X [ x x x ] () x together We defie the a priori ad a posteriori error vectors as ˆ e d X h, ad (3) d X h ˆ. (4) his error, e is used to adapt the AEC fiter h APSA agorithm updates the fiter coefficiets as foows: h Xsg e sge X X sg e hˆ ˆ. (5). he

2 i which is the variabe step-size. I [7], a variabe step-size was proposed for APSA as foows: mi e,. sg e X X sg e It shoud be oted that, i the variabe step-size (6) uses the absoute vaue of error siga which icudes the oise. However, this wi cause arger step-size ad higher misaigmet at steady-state. Meawhie, (6) is very simiar to the deta sequece i [8]-[9], which has the decreasig property. Athough the agorithm becomes more robust agaist perturbatios, it aso oses its trackig capacity. Ufortuatey, this is ot practica whe echo path chages are possibe ad additioa ad hoc cotro has to be icuded, which icreases the compexity of cotro ogic. Meawhie, i [], aother variabe step-size was proposed for ormaized sig agorithm as foows E e E v x x Athough (7) was ot proposed for affie projectio sig agorithm, we wi show that this coud be exteded to affie projectio sig agorithm i Sectio 3. However, the mai disadvatage is that it suffers from requirig the estimatio of E v, sice the ear-ed siga is buried by residua error siga. herefore, this agorithm depeds o the priori iformatio about the ear-ed siga. Compared to previous VSS i [7]-[], our proposed agorithm overcomes the above probems through the estimatio of the ear-ed siga eergy ad provides both robustess to impusive iterferece ad trackig abiity of echo path chage. 3. PROPOSED VSS-APSA We wi derive our proposed variabe step-size agorithm as foows. Simiar to [6], we rewrite the update of APSA i (5) as the foowig form: h X sg e sg hˆ ˆ where e X X e (6) (7) (8). sg diag,,, P (9) is a P P diagoa matrix. It is obvious that the orma. APSA is obtaied whe P Substitutig (8) ito (3) ad (4), we have e sg sg X X e e X X e sg () Accordig to [5] ad [6], acoustic echo caceatio ca be viewed as the recovery of the usefu siga (i.e., the ear-ed siga) from the error siga of adaptive fiter. herefore, a more reasoabe target shoud be v,where v v v P v,,, the ear-ed siga vector. I order to simpify the aaysis, we use the foowig diagoa assumptio: diag{,,, x P x P }. X X x x x x ad defie the symbo as sg e X X sge. herefore, we have e where the variabes x x e eemets of the vectors sg. ad e ad () deote the (+) th e,,,, P. We ca geeraize the variabe step-size i [] to APSA based o the foowig coditio. E E. () he variabe step-size is the see to be x E e E E x (3) However, as metioed before, it is difficut to estimate E i rea time, ad i order to overcome this difficuty, we propose to use the foowig criterio as i [5] ad [6]. E E (4) Squarig () ad takig expectatios resut i E x x E e x x E e E v. (5)

3 i which we deote x x x for brevity. Sovig this quadratic equatio we get: E e x x E x x (6) E e E x x x x E e E v. where we have kept the smaer soutio as the step-size to esure stabiity. he eergy of the ear-ed siga E i (6) is sti ot directy accessibe, therefore simiar to the ear-ed siga eergy estimator i [], we propose to estimate E e E v for APSA as foows. r xe, rx sg e, E e E (7) E x x i which rx sg e, E x sg (e ) (8) rxe, E x e. (9) herefore, the proposed variabe step-size for APSA is: E e x x E x x () E xe, xsg e, x x r r E e x x E x x Meawhie, for practica impemetatio, we compute the expectatios i the foowig recursive maer. x e, ˆ E x e, e x x x x e ˆ, ˆ x, E x x x, x x ˆ, E x x ˆ x, rˆ x, x x ˆ, rˆxe x e xe,,, () () (3) (4) rˆ ˆ r x e xsg e, xsg e, sg( ). (5) where is a smoothig factor. Meawhie, i order to further smooth the proposed variabe step-size ad avoid cacuatig the square root of egative umber, we propose to use the foowig smoothed step-size i practice: ˆ x e, ˆ x, ˆ, ˆ ˆ (6) x e rxe, rx sg e, max, ˆ ˆ ˆ x, x, x, 4. SIMULAION RESULS We do computer simuatios i the sceario of acoustic echo caceatio. We use a radom echo path with egth, L=8, ad the adaptive fiter is with the same egth. he projectio order for the affie projectio sig agorithm is P=5. he coored iput sigas are geerated by fiterig white Gaussia oise (WGN) through a first order system with a poe at.8. Idepedet white Gaussia oise is added to the system backgroud with a siga-to-oise ratio, SNR = 3 db. he impusive oise is geerated as a Beroui-Gaussia (BG) distributio with siga-toiterferece ratio (SIR). he Beroui-Gaussia distributio was geerated as a product of a Beroui z k w k k, process ad a Gaussia process, i.e., where k was WGN with zero mea ad variace ad, wk was a Beroui process with the probabiity mass fuctio give as Pw Pr Pw Pr for w for w, ad. he average power of the BG process was P ad i our simuatio, the probabiity for r Beroui process is. [3]. he covergece state of adaptive fiter is evauated with the ormaized misaigmet which is defied as og ( h hˆ h ) For the VSS i [], cosiderig we are usig coored iput sigas, the far-ed ad echo siga are aways there, thus we coud ot estimate the priori iformatio E v of ear-ed siga durig the siece period of echo siga. I our simuatio, we estimate E v directy from the aready kow backgroud WGN ad BG sigas which are added i the microphoe

4 siga. However, sice this ear-ed siga is ot avaiabe i practice, we oy take this as the theoreticay optima VSS we coud obtai. Meawhie, we aso compare proposed VSS with two fixed step-size APSAs (. ad.), oe fixed step-size APA (.) ad Shi s VSS APSA i [7]. At first, whe there is o impusive iterferece but oy backgroud oise with SNR =3 db, we compare the performace of above agorithms as i Fig. (a). he parameters of VSS APSAs are chose to aow a the VSS APSAs have both simiar covergece rate ad steady-state misaigmet (about db) as APA with fixed step-size.. he APSA with fixed step-size. is chose because it has the simiar covergece rate with VSS APSAs, ad APSA with fixed step-size. wi have about db steady-state misaigmet too. We further demostrate the variatio of step-size i Fig. (b) at the same time. We coud observe that the VSS APSAs provide a good trade-off betwee the covergece rate ad steady-state misaigmet compared with the two fixed step-size APSAs. Secody, whe there are both strog BG impusive iterferece with SIR = db ad backgroud oise with SNR = 3 db, we compare the ormaized misaigmet ad VSS i Fig.. We coud ceary observe the advatage of APSAs over APA sice APA wi diverge due to the strog impuse oise. At the same time, Shi s VSS i [7] wi have a higher steady-state misaigmet sice it ivoves the oise i the step-size at steady-state. However, the proposed VSS APSA has a ower misaigmet at steady-state due to the recovery of the impusive iterferece ad amost approaches the theoreticay optima performace of Shao s agorithm i []. Fiay, we wi compare the trackig abiity of echo path chage for the above differet agorithms as i Fig. 3. We simuate the echo path chage at sampe by switchig to aother radom echo path, ad resuts demostrate that the proposed VSS APSA provides a good approximatio of theoretica performace, which meas good trade-off betwee fast trackig abiity ad ower steady-state misaigmet. o sum up, our improved variabe step-size for APSA coud amost approach the theoreticay optima performace of Shao s VSS agorithm ad outperforms both the Shi s VSS ad fixed step-size APSAs i terms of the covergece rate ad steady-state misaigmet. 5. CONCLUSION We have proposed a improved variabe step-size for the recety proposed affie projectio sig agorithm based o the ear-ed siga eergy recovery from the error siga. Simuatio resuts demostrate that our proposed agorithm is robust to impusive iterferece ad provides better tradeoff betwee ower steady-state misaigmet ad fast trackig abiity for echo path chage compared to the previous oes. 6. REFERENCES [] Jacob Beesty, omas Gäser, Deis R. Morga, M. Moha Sodhi, ad Steve L. Gay. Advaces i etwork ad acoustic echo caceatio. Spriger,. []. Shao, Y. R. Zheg, ad J. Beesty, A affie projectio sig agorithm robust agaist impusive iterfereces, IEEE Siga Proces.s Lett., vo 7, o. 4, pp , Apr.. [3] Z, Yag, Y. R. Zheg, ad S. L. Grat, Proportioate Affie Projectio Sig Agorithms for Network Echo Caceatio, IEEE ras. Audio, Speech, Lag. Process., vo.9, o. 8, pp.73-84, Nov.. [4] H.-C. Shi, A. H. Sayed, ad W.-J. Sog, Variabe step-size NLMS ad affie projectio agorithms, IEEE Siga Proces.s Lett., vo, o., pp. 3-35, Feb. 4. [5] Jacob Beesty, Hera Rey, Leoardo Rey Vega, ad Sara resses, A oparametric vss ms agorithm, IEEE Siga Processig Letters, vo. 3, pp. 5884, October 6. [6] C. Paeoogu, J. Beesty, ad S. Ciochia, A variabe stepsize affie projectio agorithm desiged for acoustic echo caceatio, IEEE ras. Audio, Speech, Lag. Process., vo.6, o. 8, pp , Nov. 8. [7] Shi, J., Yoo, J., ad Park, P.: Variabe step-size affie projectio sig agorithm, Eect. Lett.,, 48, (9), pp [8] L. R. Vega, H. Rey, J. Beesty, A ew robust variabe stepsize ms agorithm, IEEE ras. Audio, Speech, Lag. Process., vo.9, o. 8, pp.73-84, Nov.. [9] L. R. Vega, H. Rey, J. Beesty, A robust variabe step-size affie projectio agorithm, Siga Process.,, 9, (9), pp []. Shao, Y. R. Zheg, ad J. Beesty, A variabe step-size ormaized sig agorithm for acoustic echo caceatio, IEEE Iteratioa Coferece o Acoustics, Speech, ad Siga Processig,. pp ,. [] Mohammad Asif Iqba ad Steve L. Grat, Nove variabe step size ms agorithms for echo caceatio, IEEE Iteratioa Coferece o Acoustics, Speech, ad Siga Processig, 8. pp. 4-44, 8.

5 Normaized Misaigmet Variabe Step-size Normaized Misaigmet Variabe Step-size Normaized Misaigmet Variabe Step-size - APA with =.5 APSA with =. APSA with =. Shi VSS heoretica VSS APSA with =. APSA with =. Shi VSS heoretica VSS Iteratios Fig. (a) Compariso of ormaized misaigmet with SNR=3dB..5.5 Iteratios Fig. (b) Compariso of variabe step-size with SNR=3dB. 5 APA with =.5 APSA with =. APSA with =. Shi VSS heoretica VSS APSA with =. APSA with =. Shi VSS heoretica VSS Iteratios Fig. (a) Compariso of ormaized misaigmet with impusive iterferece SIR=dB ad SNR=3dB..5.5 Iteratios Fig. (b) Compariso of variabe step-size with impusive iterferece SIR=dB ad SNR=3dB. 5 APA with =.5 APSA with =. APSA with =. Shi VSS heoretica VSS APSA with =. APSA with =. Shi VSS heoretica VSS Iteratios Fig.3 (a) Compariso of ormaized misaigmet with echo path chage at, SIR=dB ad SNR=3dB..5.5 Iteratios Fig.3 (b) Compariso of variabe step-size with echo path chage at, SIR=dB ad SNR=3dB.