An Improved Proportionate Normalized Least Mean Square Algorithm with Orthogonal Correction Factors for Echo Cancellation

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1 202 Iteratioal Coferece o Electroics Egieerig ad Iformatics (ICEEI 202) IPCSI vol. 49 (202) (202) IACSI Press, Sigapore DOI: /IPCSI.202.V49.33 A Improved Proportioate Normalized Least Mea Square Algorithm with Orthogoal Correctio Factors for Echo Cacellatio JiHye Seo ad PooGyeo Par,2 WCU (Divisio of I Covergece Egieerig, POSECH) 2 Departmet of Electrical Egieerig, POSECH Abstract. Recetly, the proportioate ormalized least mea square (LMS) algorithm was developed i the cotext of etwor echo cacellatio to get fast covergece rate for the sparse impulse respose. Ufortuately, the LMS coverges much slower tha the ormalized least mea square (NLMS) algorithm whe the impulse respose becomes dispersive. hus, the improved LMS (ILMS) algorithm was developed which is less sesitive to the sparseess of the echo path ad outperforms both the NLMS ad LMS. I this paper, we propose a improved proportioate NLMS algorithm with orthogoal correctio factors. he proposed algorithm exteds the cocept of the ILMS algorithm to NLMS with orthogoal correctio factors (NLMS-OCF) algorithm ad it realizes the geeralized ILMS algorithm. Experimetal results show that the proposed algorithm performs better tha the NLMS-OCF ad proportioate NLMS- OCF algorithm whatever the ature of the impulse respose is. Keywords: Adaptive filter, improved proportioate, NLMS-OCF, echo cacellatio.. Itroductio I the etwor echo cacellatio scheme, a adaptive filter plays a importat role o idetifyig the echo path, ad may adaptive filterig algorithms are developed to improve the performace of the filter []. Recetly, proportioate ormalized least mea square (LMS) algorithm has bee proposed to achieve fast covergece rate whe the impulse respose is sparse [2]. he idea of LMS is to update each coefficiet of the filter by adjustig the step sizes i proportio to the estimated filter coefficiets. From this cocept, the geeralizatio of LMS algorithm is realized by applyig the structure of LMS algorithm to NLMS- OCF [3] which is proposed to solve the problem of slow covergece rate of the NLMS, ad it is referred as proportioate NLMS-OCF (LMS-OCF) algorithm [4]. he LMS ad LMS-OCF algorithms are desiged to perform well i the sparse impulse respose. However, whe the impulse respose is dispersive, they coverge slower tha the NLMS ad NLMS-OCF, respectively. It meas that the rule of adjustig the step sizes of proportioate-type algorithms is far from optimal. o overcome this problem, a improved LMS (ILMS) algorithm is suggested [5]. It presets more optimal way of determiig the step sizes ad shows better performace tha the NLMS ad LMS regardless of the ature of the impulse respose. I this paper, the improved proportioate NLMS-OCF algorithm (ILMS-OCF) is proposed. he proposed algorithm adopts the idea of the ILMS to the NLMS-OCF algorithm, so the geeralizatio of ILMS algorithm ca be established. he method of developig ILMS-OCF is similar to that of the LMS-OCF, so the LMS-OCF is itroduced i sectio 2 to describe the structure of NLMS-OCF algorithm i the case of the proportioate idea is applied. Experimetal results show that the proposed algorithm achieves fast covergece rate tha other existig algorithms whatever the ature of the impulse respose is. Correspodig author. el.: , Fax: address: bmclubhs@postech.ac.r 80

2 2. Proportioate Normalized Least Mea Square Algorithm with Orthogoal Correctio Factors I this sectio, the proportioate NLMS-OCF (LMS-OCF) algorithm used for echo cacellatio is briefly explaied. Fig. shows structure of echo caceller, ad the followig otatios are used. x : Far-ed sigal (iput sigal) y : Echo ad bacgroud oise (desired sigal) x [, x,, x ] : Excitatio vector (iput vector) x L h [ h0, h,, h L ] : rue echo path (impulse respose) h [ ˆ h0,, hˆ,,, ˆ hl, ] : Estimated echo path (estimated weight vector) where L is the legth of the echo path, ad is the time idex. Fig. : Structure of echo caceller. he echo caceller filters x by a echo-path estimate h ˆ, to obtai a echo estimate y ˆ. If h ˆ estimates h well, the yˆ ca cacel the echo portio of y so that the retur sigal (or error sigal) e is about equal to the ear-ed sigal v. Proportioate-type algorithms such as LMS ad LMS-OCF have bee proposed [2],[4]. hese algorithms are desiged to apply a adaptive idividual step size to each coefficiets of the filter. he step sizes are calculated from the last estimate of the filter coefficiets i such a way that a larger coefficiet receives a larger icremet, thus icreasig the covergece rate of that coefficiet. It meas that active coefficiets are adjusted faster tha o-active coefficiets, so the algorithms show fast covergece rate for sparse impulse resposes (i.e., resposes for which oly a small percetage of coefficiets is sigificat). Most impulse resposes i the telephoe etwor have this characteristic, so it is very useful i echo cacellatio scheme. he weight update equatio of the LMS-OCF is as follows: M h h x x x () 0 M where x (, 2,, M ) are the compoets of xd which is called orthogoal correctio factors (OCFs) because they are chose to be orthogoal to each other, D is the delay betwee iput vectors used for updates. he correspodig step ( 0,,, M) is calculated accordig to G e for 0, xg x G e for,2,, M, x G x 0 otherwise (2) where (0 2) is the adaptatio step, is a regularizatio parameter, ad D D e y x h (3) e y x h (4) h h x x x (5) 0 G diag{ g,, g }. (6) 0, L, 8

3 he compoets of a diagoal matrixg adjust the step sizes of the idividual taps of the filter ad they are determied as follows: ˆ l, max{ max[, h0,,, hˆ L, ], h ˆ, l, } (7) l, gl, L, 0 l L, (8) i0 i, where parameter ad are positive umbers with typical values 0.0 ad 5 / L. he parameter prevets hˆl, from stallig whe it is much smaller tha the largest coefficiet ad regularizes the updatig whe all coefficiets are zero at iitializatio. 3. Improved Proportioate Normalized Least Mea Square Algorithm with Orthogoal Correctio Factors I this sectio, a ILMS-OCF algorithm is proposed. For dispersive impulse respose, the covergece rate of the LMS-OCF algorithm is slower tha that of the NLMS-OCF algorithm. hus, equatio (8) has to be modified to overcome this problem. Recetly, a ILMS algorithm has bee proposed which uses modified rule of adjustig the step sizes of the idividual taps of the filter. his algorithm has similar weight update equatio as the LMS algorithm, but adopts the modified rule of adjustig the step sizes. Based o this cocept, the objective of the ILMS-OCF algorithm is applyig the idea that was firstly itroduced i the ILMS algorithm to NLMS- OCF algorithm. he weight update equatio of the ILMS-OCF is give by M h h x x x (9) 0 M where x (, 2,, M ) are the OCFs. he correspodig step ( 0,,, M) is calculated accordig to G e for 0, xg x I G e for,2,, M, (0) x G x I 0 otherwise where (0 2) is the adaptatio step, I is a regularizatio parameter of the ILMS-OCF, ad the other parameters used i equatio (0) is accordig to equatio (3)-(6). Whe the umber of OCFs, M, is set to 0, equatio (9) is idetical to the weight update equatio of the ILMS, so the ILMS-OCF algorithm ca be said to be a geeralized ILMS algorithm. he compoets of a diagoal matrix G are determied as follows [5]: hˆ l, g l, ( ), 0 l L, () 2L 2 h where is a small positive umber, ad L h ˆ hl,. (2) l0 he parameter is a positive umber with rage betwee - ad. For, it ca be easily oticed that the ILMS-OCF algorithm ad NLMS-OCF algorithm are idetical. For goes close to, the ILMS- OCF algorithm behaves lie the LMS-OCF algorithm. he good choices for are 0 or -0.5 i practice. With proper choice o parameter, ILMS-OCF algorithm always behaves better tha NLMS-OCF ad LMS-OCF, whatever the impulse respose is. 82

4 4. Experimetal Results I this sectio, we compare the NLMS-OCF, LMS-OCF, ILMS, ad proposed ILMS-OCF algorithm i the cotext of a etwor echo caceller. he echo path h ad its estimate h are assumed to have the same legth L =024. he iput sigal x is either a white Gaussia radom sequece through the system G( z) 0. 9z or a speech sigal that is sampled at 8Hz, ad the sigal-to-oise ratio (SNR) is set to 30dB. he parameter values for each algorithm are set to 0.0, 5 / L (LMS-OCF), 0.5, 0.00 (ILMS, ILMS-OCF), ad the regularizatio parameter of each algorithm is set to 2 N x, N / L, ad I N /2L where N is a regularizatio parameter used for the NLMS-OCF ad I is also used for both ILMS ad ILMS-OCF. he mea square deviatio (MSD) is used as the performace idicator. Figure 2 shows the MSD learig curves of the ILMS ad ILMS-OCF algorithm for dispersive impulse respose. We choose for two algorithms, M 0,, 2,3 ad D 20 for ILMS-OCF. he value of D is set to clearly show the improvemet of the covergece rate. he iput sigal is a white Gaussia through chael G( z ). We ca otice that covergece rate of the ILMS-OCF algorithm becomes faster as the umber of OCFs ad the delay icrease, ad the ILMS-OCF with M 0 is idetical to the ILMS. hus, the ILMS-OCF algorithm geeralizes the ILMS, ad it ca cotrol the covergece rate by adjustig the umber of OCFs or delay. Fig. 3: MSD learig curves of the ILMS ad ILMS-OCF algorithm (dispersive impulse respose, iput sequece: white oise through a system G( z )) Figures 3-4 show the MSD learig curves of the NLMS-OCF, LMS-OCF, ad ILMS-OCF for sparse impulse respose ad dispersive impulse respose, respectively. We choose, M 3, ad D for all algorithms. he iput sigal is a white Gaussia through chael G( z) i these cases. We ca see that the covergece rate of ILMS-OCF is similar to that of the LMS-OCF, ad faster tha that of the covetioal NLMS-OCF for sparse impulse respose. For dispersive impulse respose, the covergece rate of the ILMS-OCF is the fastest amog three algorithms, where the LMS-OCF coverges slowly tha the NLMS-OCF. he ILMS-OCF algorithm achieves fast covergece rate whatever the impulse respose is. Fig. 3: MSD learig curves of the NLMS-OCF, LMS-OCF, ad ILMS-OCF algorithm (sparse impulse respose, iput sequece: white oise through a system G( z )) 83

5 Fig. 4: MSD learig curves of the NLMS-OCF, LMS-OCF, ad ILMS-OCF algorithm (dispersive impulse respose, iput sequece: white oise through a system G( z )) Figure 5 shows the MSD learig curves of NLMS-OCF, LMS-OCF, ad ILMS-OCF for dispersive impulse respose whe the iput sigal is a speech. We choose, M 2, 3 ad D 24 for all algorithms. he covergece rate of the ILMS-OCF is better tha that of the LMS-OCF ad the NLMS-OCF. 5. Coclusios Fig. 5: MSD learig curves of the NLMS-OCF, LMS-OCF, ad ILMS-OCF algorithm (dispersive impulse respose, speech iput sigal) I this paper, we proposed ILMS-OCF algorithm which adopted the idea of the ILMS to the NLMS-OCF. he experimetal results showed that the ILMS-OCF algorithm was established as a geeralized ILMS algorithm ad it outperformed the covetioal NLMS-OCF algorithm ad LMS- OCF algorithm regardless of the ature of the impulse respose. 6. Acowledgemets his research was supported by the MKE (he Miistry of Kowledge Ecoomy), Korea, uder the IRC (Iformatio echology Research Ceter) support program (NIPA-202-H ) supervised by the NIPA (Natioal I Idustry Promotio Agecy) 7. Refereces [] S. Hayi, Adaptive Filter heory, 4 th ed. Upper Sadler River. NJ:Pretice-Hall, [2] D. L. Duttweiler, Proportioate Normalized Least-Mea-Squares Adaptatio i Echo Cacelers, IEEE rasactios o speech ad audio processig, vol.8, o.5, pp , September [3] S. G. Saara ad A. A. Beex, Normalized LMS Algorithm with Orthogoal Correctio Factors, Proc. 3 st Asilomar Coferece o sigals, systems, ad Computers, pp , November 997. [4] M. Borhai ad V. Sedghi, Proportioate NLMS with Orthogoal Correctio Factors for Stereophoic Acoustic Echo Cacellatio, 3 rd Iteratioal IEEE-NEWCAS coferece, pp , Jue [5] J. Beesty ad S. L. Gay, A improved LMS algorithm, i Proc. IEEE ICASSP,