An Efficient RLS Algorithm For Output-Error Adaptive IIR Filtering And Its Application To Acoustic Echo Cancellation

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1 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) A Efficiet RLS Algoithm Fo Output-Eo Adaptive IIR Filteig Ad Its Applicatio o Acoustic Echo Cacellatio Jafa Ramadha ohammed 1 ad Guam Sigh 2 1 Uivesity Of osul, Electoics Egieeig College, Commuicatio Eg. Dept., osul, IRAQ. {obile No , E-ail: jafaam@yahoo.com} 2 Pujab Egieeig College, Electical Eg. Dept., Chadigah 1612, INDIA. {obile No , E-ail: guamsighpec@yahoo.com} Abstact- I this lette, a efficiet ecusive least squaes (RLS) algoithm usig ifiite impulse espose (IIR) filte fo acoustic echo cacellatio () is poposed. he RLS adaptive filte is atually exteded fom the fiite impulse espose (FIR) stuctue to the IIR stuctue. Oe of the mai advatages of a IIR RLS filte is that a log-delay echo ca be sythesized by a elatively small umbe of filte coefficiets leadig to lesse computatioal complexity. I additio it is moe suitable fo modelig physical systems, due to its pole-zeo stuctue, ove thei FIR coutepats. o ivestigate the tackig pefomace ad the stability of the poposed IIR RLS filte i a pactical implemetatio, eal data fo fa ed speech sigal ad its echo wee used i ca eviomet. he poposed algoithm is show to have global covegece. o demostate the effectiveess of the poposed IIR RLS filte, it is compaed to the usual FIR RLS filte. he good pefomace of the poposed IIR RLS algoithm fo have bee veified via compute simulatios. I. INRODUCION I mode hads-fee commuicatio systems such as hadsfee mobile phoes, audio ad video cofeece systems, it is ecessay to pefom a acoustic echo cacellatio of the fa-ed speake sigal. A umbe of efficiet algoithms has bee poposed fo that pupose [1, 2]. I a acoustic echo cacellatio algoithm, a model of the oom impulse espose is idetified. Sice the coditios i the oom may vay cotiuously, the model eeds to be updated cotiuously. his is doe by meas of adaptive filteig algoithms. he oom acoustics ca be modeled by a FIR filte o IIR filte. I pactice, the legth of the oom acoustics ad by cosequece also the filte legth of the adaptive cacelle is typically 5-2 filte taps. Log filtes imply a lage computatioal bude ad slow covegece ate[3]. hese poblems have motivated a ew appoach: adaptive filteig i subbads with the double pupose of educig the computatioal complexity ad of impovig the covegece speed of the algoithm. O the othe had, subbad pocessig itoduces tasmissio delays caused by the filtes i the filte bak ad sigal degadatios due to aliasig effects [4]. I this lette, a IIR RLS algoithm based o the output-eo fomulatio fo acoustic echo cacellatio is ivestigated. Oe of the mai advatages of a IIR RLS filte is that a log delay echo ca be sythesized by a elatively small umbe of filte coefficiets leadig to lesse computatioal complexity. I additio it is moe suitable fo modelig physical systems due to its pole-zeo stuctue. Ufotuately, these good chaacteistics come alog with some possible dawbacks iheet to IIR adaptive filtes such as algoithm istability ad local miimum solutios. Cosequetly, seveal algoithms fo adaptive IIR filteig have bee poposed to ovecome these poblems [2, 5]. Futhemoe, the study of algoithms othe tha stochastic gadiet-based algoithms has edeed it possible to guaatee that the statioay poits of adaptive IIR algoithms ae close to the global miimum of the least-squaes output eo pefomace suface [6]. I additio, this lette study the tackig pefomace of the poposed IIR RLS algoithm fo time-vayig system. Simulatio esults show that the poposed algoithm poduces esults that ae sigificatly favoable tha usual FIR RLS algoithm fo. oeove the poposed algoithm has good ability to tack the time-vayig ukow system ad emai stable. his lette is ogaized as follows. Sectio II povides the theoetical fomulatio of the poposed algoithm. I Sectio III, simulatio esults ad aalysis will be peseted to illustate the impoved pefomace. I Sectio IV, bief coclusios will be daw. II. DESIGNING RLS ALGORIH USING IIR FILER he block diagam i Fig. 1 depicts the poposed filte achitectue. he RLS algoithm based o output eo appoach was used to deive the feed fowad ad feedback coefficiets which ae used to idetify the time-vayig ukow system. A simila techique has bee adopted by Yeay ad Giswold [7] to desig adaptive IIR filte that employs a sigle iput seso ad ove samplig fo speech ehacemet /7/$ IEEE 139

2 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) Fa Ed Speech Usig IIR RLS Filte Nea Ed Room u ( a i b i ime Vayig impulse espose h ( k, e ( - v ˆ ( ) Sythesis Echo + x ( Fig. 1 Block Diagam Of Poposed IIR RLS Filte Fo. Echo Fom Fa-Ed v( ( Noise With efeece to Fig. 1, the least squae algoithm is desiged to miimize the sum of squaed eos (assumig hee fo simplicity o ea ed speech) as defied by 2 ε ( = e ( (1) whee e( = x( vˆ( = x( hˆ (2) ad h ˆ = [ a ], a1,... an, b1, b2,..., b (3) is the coefficiet vecto, z ( i ) = [ u(, u( 1),..., u( N), vˆ( 1), vˆ( 2),...ˆ( v )] (4) is the sigal vecto cotaiig the elemets of fa ed speech vecto u( ) ad past samples of sythesis echo sigal v ˆ(. he vecto has N elemets, idexed fom to z ( ( N + )). he vecto ĥ cotais a, a, an which ae kow as the feed fowad coefficiets, ad b, b, , b which ae kow as the feed back coefficiets. his vecto has N elemets, ad these elemets ae idexed as h ˆ( j ), whee j =,1,..., N +. o miimize the sum of the squaed eos, the patial deivative of ε ( is evaluated as: ε ( 2 ei e i ( ) = ( ) = 2 e( (5) =2 e(, whee j=,1..., N+ o miimize this eo fuctio, the patial deivative is set to zeo ε ( = 2 e( =, j =,1,..., N + (6) By combiig (2) ad (6), it follows x( N+ l= N+ l= hl zi l ˆ( ) ( ) = l) l) = x( j=,1,..., N+, j=,1,... N + Recogizig oute poducts, yields ( ˆ h= xz (. heefoe, the optimum coefficiets ae ˆ h= 1 ( ) (, (9) xz whee ( = z (, ad ( = x( (1) xz Rathe tha solvig equatio (9) by computig the ivese of (, the ivese will be ecusively computed by makig use of the matix ivesio lemma. he weight vecto ĥ will become a fuctio of discete time, ad will assume the otatio h ˆ ( ). he followig ecusive equatio is the fist step towads detemiig a ecusive fomula that will allow the weight vecto h ˆ ( ) to be updated at each. ( = z ( = z ( + z ( (11) = + z ( Similaly, ( ) = ( 1) (12) xz xz he ivese of ( ca be ecusively computed usig the matix ivesio lemma [7, 8] ( A + BCD) = A A B( C + DA B) DA (13) of allowig A=, B =, C = 1, ad D = z (, equatio (11) takes the fom (7) (8) 14

3 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) z ( ( =. (14) z ( 1 ( At ay istat of time, ) may also be detemied by ( = (15) z ( o ecusively update h ˆ ( ), the set of equatios (9), (12), (13) may be used. Hece = ( [ xz ] (16) = ( xz ( heefoe it follows that z ( = z ( (17) xz ( A updatig equatio fo IIR filte coefficiets may be obtai as = ) z ( (18) z ( ˆ( h) ( By substitutig 1 ( fom equatio (15) i the thid tem o the ight had side of the equatio (18), the followig expessio may be obtaied = ) z ( (19) ( ) ( ) z z ( ˆ( h) z ( By ecogizig the backeted tem i the above equatio as a time vayig gai tem that modulates how much the eo iflueces the magitude of the update at each iteatio, the followig update equatio is ealized: ˆ h ( = 1) + G( x( z ( ˆ( h 1), (2) [ ] whee G( =. (21) z ( Sice the IIR RLS filte is used fo o statioay sigals, the eo tem e( that iflueces the weights may be modified so that oly elatively ecet values of e( will be sigificat. heefoe, (1) is modified to eflect this chage [7], as is typically doe fo a FIR RLS filte [8] i 2 ε ( = λ e (, (22) this cosequetly iflueces the time vayig filte gai defied by (21), which becomes G( =. (23) λ+ z ( III. SIULAION RESULS o ivestigate the pefomace of the poposed IIR RLS filte coside the followig two cases: Case 1: he tasfe fuctio of the ukow liea system to be idetified by poposed algoithm is give by [9] (.4z.5z +.7z H z) = (24) z +.8z.88z +.64z hus N=5, =5, ad = N + + 1= 9. he system is L excited by the coloed iput sigal y (t), which is geeated by passig white oise though the secod ode IIR filte. he filte coefficiets of secod ode IIR filte ae {-1.126,.64}. he system output is coupted by additive white Gaussia oise (t) with zeo mea ad uit vaiace, the (t) ad y(t) ae mutually idepedet. I the simulatio, the fogettig facto is chose to be.98. I ode to examie the tackig behavio of the poposed algoithm i a ostatioay eviomet, suppose that the secod pole ad fist zeo of the tasfe fuctio give by (24) ae suddely chaged fom value (.8) to (.4) ad fom () to (1.) espectively, which ae idicates the effect of timevayig ukow system. he evolutio cuves of the estimated paametes to idetify the ew tasfe fuctio of the timevayig system ae plotted i Fig. 2. It ca be see that the poposed algoithm has good ability of tackig i ostatioay eviomet. oeove, the stability of the IIR RLS algoithm has bee tested by checkig the pole-zeo locatios at each iteatio of the coefficiet vecto update equatio (2). Fig. 3 (a) shows the pole-zeo locatios of the tasfe fuctio give by (24), while Fig. 3 (b) shows the pole-zeo locatios of the ew tasfe fuctio fo time-vayig ukow system. Case 2: o veify these esults i a physical eviomet, the fa ed speech sigal ad its eal echo i ca eviomet wee used [1] at a sample ate 8kHz with 16 bit esolutio. he fa ed speech sigal ad its eal echo (assumig o ea ed speech) wee fed ito poposed IIR RLS filte ad usual FIR RLS filte. Fo the give eal test sigals, the pefomace of the FIR RLS filte is show i Fig. 4. he mai disadvatage of this algoithm is high computatioal complexity as give i able 1, which is o the ode of L ultiplicatios pe samplig iteval 2 (UL s pe ), wheel is legth of the RLS filte. he pefomace of the desiged IIR RLS filte, accodig to the algoithms developed i Sectio II, is show i Fig. 5. o compae the effectiveess of the poposed IIR RLS ad FIR RLS filtes, the mea squae eo (SE) of both filtes fo diffeet filte legth ae depicted i Figs. 6, 7 ad 8. Note that the SE of poposed IIR RLS filte (see Fig. 8) with filte legth equal to 141

4 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) 115 taps (1 feedfowad taps ad 15 feedback taps) is appoximately equal to the SE of the usual FIR RLS filte (see Fig. 7) with filte legth equal to 256 taps. I this case the computatioal complexity of the poposed IIR RLS filte is lesse whe compaed to usual FIR RLS filte as give i able 1. I aothe case, if both adaptive filtes have same filte legth (see Figs. 6 ad 8) ad cosequetly same computatioal complexity. it is obseved that the poposed IIR RLS filte is supeio fo echo suppessio. his is also evidet fom able 1. System 1 FIR RLS ( L = 115) 2 FIR RLS ( L = 256) 3 FIR RLS ( L = 512) 4 IIRRLS (N =1, =15) Echo Suppessio [db] Computatioal Complexity UL s pe able 1: Echo Suppessio ad Complexities Of Diffeet Algoithms. able 2 illustates the effect of the umbe of feedback coefficiets o the echo suppessio pefomace fo the poposed IIR RLS algoithm. It ca be see, the addig feedback to the IIR RLS filte will allow a impovemet i echo suppessio while keepig the umbe of oveall filte coefficiets costat. Poposed IIR RLS Filte Filte Legth Echo Suppessio [db] N =114 taps, =1 taps N =11 taps, =5 taps -2.7 N =15 taps, =1 taps N =1 taps, =15 taps N =58 taps, =57 taps able 2: Echo Suppessio fo Diffeet Numbe of Feedback Coefficiets. IV. CONCLUSIONS I this lette, a adaptive IIR RLS filte based o output eo appoach fo has bee ivestigated. he RLS adaptatio algoithm was used to deive the feed fowad ad feedback coefficiets o a sample by sample basis. Fom simulatio esults, it seems that the poposed IIR RLS algoithm has good tackig pefomace fo time-vayig ukow system ad emai stable at each iteatio of the coefficiet update equatio. Fo eal data test sigals i ca eviomet, it is obseved that the IIR models of acoustic echo paths, is outpefom ove thei FIR coutepats. Futhemoe, umeical compaisos show that the poposed IIR RLS equies fewe umbe of filte coefficiets ad by cosequece lesse computatioal complexity to obtai the specified echo suppessio pefomace. I additio the othe mai advatage of the poposed algoithm is that it pocesses data o sample by sample basis, which leds itself to a moe efficiet eal time implemetatio. REFERENCES [1] S. Beiig, et al., Acoustic Echo Cotol: A Applicatio of Vey-High-Ode Adaptive Filtes, IEEE Sigal Pocessig agazie, July [2] Segio L. Netto, Paulo S. Diiz ad Paajotis A., Adaptive IIR Filteig Algoithms Fo System Idetificatio: A Geeal Famewok, IEEE as. O Educatio, Vol. 38, No. 1, Feb [3] S.. Kuo, B.H. Lee ad W. ia, Real-ime Digital Sigal Pocessig, Joh Wily&Sos Ltd, 26. [4] S. Weiss, A. Stege, R. W. Stewat, ad R. Rabestei, Study-State Pefomace Limitatios Of Subbad Adaptive Filtes, IEEE as. O Sigal Pocessig, Vol. 49, No. 9, Septembe 21. [5] Jiog Zhou ad Gag Li, A New Stuctue Fo Adaptive IIR Filtes With Guaateed Stability, IEEE Iteatioal Cofeece o Commuicatios, Cicuits ad Systems, Vol. 2, 27-3 ay 25. [6] P. A. Regalia ad. boup, Udemodeled Adaptive Filteig: A A Pioi eo Boud Fo he Steiglits- cbide ethod, IEEE as. Cicuts Syst. II, Vol. 43, PP , Feb [7]. B. Yeay ad N. C. Giswold, Adaptive IIR Filte Desig Fo Sigle Seso Applicatios, IEEE as o Istumetatio ad easuemet, Vol. 51, No. 2, Apil 22. [8] S. Hayki ad. Kailath, Adaptive Filte heoy, Fouth Editio, Petice-Hall, Peaso Educatio, Ic, 22. [9] Da-Zheg feg ad W. Xig, Fast RLS ype Algoithm Fo Ubiased EquatioEo Adaptive IIR Filteig Based O Appoximate Ivese-Powe Iteatio, IEEE as. O Sigal Pocessig, Vol. 53, No. 11, Nov. 25. [1] Gillia. Davis, Noise Reductio I Speech Applicatios, CRC Pess LLC,

5 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) (a) Fig. 2 Evolutio Cuves Of he Estimated Paametes o Idetify he Liea ime-vayig System, (a) Poles. (b) Zeos. (b) X Pole O Zeo Fig. 3 Pole-Zeo Plot, (a) Fo Liea ime-ivaiat System. (b) Fo Liea ime-vaiat System. 143

6 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) Amplitude Amplitude Echo Suppessio [db] ( a ) x 1 4 ( b ) x 1 4 ( c ) Sample Numbe x 1 4 Fig. 4 FIR RLS Filte, (a) Echo Fom fa-ed Speech. (b) Residual Echo. (c) Echo Suppessio. ( L = 256 taps, fogettig facto λ =. 98 ) Fig. 5 Acoustic Echo Cacellatio By Usig Poposed IIR RLS Filte With Compae o Real Fa-Ed Echo I Ca Eviomet. 144

7 Poceedigs of the 27 IEEE Symposium o Computatioal Itelligece i Image ad Sigal Pocessig (CIISP 27) -1-2 SE Of FIR RLS -3 ea Squae Eo [db] Sample Numbe Fig. 6 ea Squae Eo (SE) Of FIR RLS Algoithm, Filte Legth L =115 taps. x SE Of FIR RLS -3 ea Squae Eo [db] Sample Numbe x 1 4 Fig. 7 ea Squae Eo (SE) Of FIR RLS Algoithm, Filte Legth L =256 taps. Fig. 8 ea Squae Eo (SE) Of Poposed IIR RLS Algoithm, (Feed Fowad Coefficiets N =1, Feed Back Coefficiets =15). 145