Statistical Approach for Noise Removal in Speech Signals Using LMS, NLMS, Block LMS and RLS Adaptive filters

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1 Statistical Approach for Noise Removal i Speech Sigals Usig LMS, NLMS, Block LMS ad RLS Adaptive filters D. Hari Hara Satosh, Member, IACSI, Vusl Sravya Pedyala, V. N. Lakshma Kumar, ad N. Shamukh Rao Abstract I this paper the applicatio of differet adaptive filters i removig the oise preset i the speech sigals is preseted. o aalyze the performace of differet adaptive filter family members, the parameters like covergece, output PSNR ad CPU cosumptio time are cosidered. Results sho that NLMS filter shos the better performace i CPU time cosumptio ad output PSNR. Block LMS has the highest Covergece factor amog all the members of the adaptive filter family. Idex erms Adaptive filters, LMS, RLS, NLMS, NLMS I. INRODUCION Adaptive oise cacellatio is beig used as a promiet solutio i a ide rage of fields. From the stadpoit of performace, it is idely ko [1] that the Recursive Least- Squares (RLS) algorithm offers fast covergece ad good error performace i the presece of both hite ad coloured oise. his robustess makes this algorithm highly useful for adaptive oise cacelatio. Ufortuately eve as the computatioal poer of devices today icreases, it remais largely difficult to utilize the RLS algorithm for real-time sigal processig. All the members of the Adaptive filter family are see i terms of output PSNR, covergece ad CPU time cosumptio. Sectio II gives the brief itroductio of LMS algorithm, Sectio III gives the overvie of Block LMS algorithm, Sectio IV gives the RLS algorithm ad Sectio V gives the results ad discussios. II. LMS ALGORIHM Least mea squares (LMS) algorithms[] are a class of adaptive filters used to mimic a desired filter by fidig the filter coefficiets that relate to producig the least mea squares of the error sigal (differece betee the desired ad the actual sigal. he Least Mea Square (LMS) algorithm is a adaptive algorithm, hich uses a gradiet-based method of steepest decet. LMS algorithm uses the estimates of the gradiet vector from the available data. LMS icorporates a iterative procedure that makes successive correctios to the eight vector i the directio of the egative of the gradiet vector hich evetually leads to the miimum mea square error he LMS algorithm is a liear adaptive filterig algorithm, hich i geeral cosists of to basic processes: 1) A filterig process, hich ivolves (a) Computig the output of a liear filter i respose to a iput sigal (b) Geeratig a estimatio error by comparig this output ith a desired respose ) A adaptive process, hich ivolves the automatic adjustmet of the parameters of the filter i accordace ith the estimatio error LMS algorithm is importat because of its simplicity ad ease of computatio ad because it does ot require off-lie gradiet estimatio or repetitios of data. A. LMS Algorithm Formulatio: N 1 y( ( x( 1) (1) i0 e( d( y( () We assume that the sigals ivolved are real valued he LMS algorithm chages (adapts) the filter tap eights so that e( is miimized i the mea square sese. he the process x( ad d( are joitly statioary, this algorithm coverges to a set of tap-eights hich o average are equal to the Wieer-Hopf solutio he covetioal LMS algorithm is a stochastic implemetatio of the steepest descet algorithm. E[ e ( ] by its istataeous coarse estimate e ( Substitutig e ( for ζ i the steepest descet recursio, e obtai W( 1) W( e ( here (3) W [ ( (... N ( ] ( [ he i th elemet of the gradiet vector e ( is he e i e e( e( e( x( 1) ( i e( x( ] (4) Mauscript received April 10, 01; revised May 30, 01. he authors are ith departmet of ECE. ( 8at08h-ece005@yahoo.com). Fially e obtai W( 1) W( e( x( (5) 351

2 he idea behid RLS filters is to miimize a cost fuctio C by appropriately selectig the filter coefficiets updatig the filter as e data arrives. he error sigal e( ad desired sigal d( are defied i the diagram : Fig. 1. Block diagram of LMS filterig scheme III. BLOCK LMS ALGORIHM he block LMS (BLMS) algorithm orks o the basis of the folloig strategy. he filter tap-eights are updated oce after the collectio of every block of data samples. he gradiet vectors, - e(x(, used to update the filter tap-eights are calculated durig the curret block. Fig.. Block diagram of Block LMS Algorithm Usig k to deote the block idex, the BLMS recursio is obtaied as [3][4] here is the block legth ad is the step - size parameter. Also, for e( kl i) d( kli) y( kli) i 0,1,,... l IV. RLS ALGORIHM he Recursive least squares (RLS) adaptive filter [5], [6] is a algorithm hich recursively fids the filter coefficiets that miimize a eighted liear least squares cost fuctio relatig to the iput sigals. his i cotrast to other algorithms such as the least mea squares that aim to reduce the mea square error. I the derivatio of the RLS, the iput sigals are cosidered determiistic, hile for the LMS ad similar algorithm they are cosidered stochastic. Compared to most of its competitors, the RLS exhibits extremely fast covergece he RLS algorithm exhibits the folloig properties: Rate of covergece that is typically a order of m magitude faster tha the LMS algorithm. Rate of covergece that is ivariat to the Eige value spread of the correlatio matrix of the iput vector. RLS Algorithm Formulatio: Fig. 3. Block diagram of RLS filterig scheme he error implicitly depeds o the filter coefficiets through the estimate e( d( d( (6) he eighted least squares error fuctio C the cost fuctio e desire to miimize beig a fuctio of e( is therefore also depedet o the filter coefficiets[7],[8]: 1 ( ( ) i0 C e (7) his form ca be expressed i terms of matrices as R ( r ( (8) x dx here R x ( is the eighted sample correlatio matrix for x(, ad r dx ( is the equivalet estimate for the cross-correlatio betee d( ad x(. Based o this expressio e fid the coefficiets hich miimize the cost fuctio as [9] R 1 ( r ( (9) We have x dx P( R x 1 ( 1 1 p( 1) g( x ( p( 1) (10) here g ( is gai vector With the recursive defiitio of P( the desired form follos e derive g( p( x( p( r ( dx g( [ d( x ( 1] ( g( )...(11) 1 1 Where ( d( x ( is a priori error. 1 Compare this ith the a posterior error; the error calculated after the filter is updated e ( d( x ( hus e have correctio factor as 35

3 1 g( ( (1) RLS. I coclusio the ormalized LMS adaptive filter shos the highest performace as per the CPU time cosumptio ad output PSNR. Ad the Block LMS shos the highest performace for covergece. All these results are see i MALAB[10]. Fig. 4. Performace of differet Adaptive Filters i deoisig of Speech Sigals Fig. 6. Performace of differet Adaptive Filters i Covergece time. Fig. 5. Performace of differet Adaptive Filters i CPU time cosumptio for differet orders V. RESULS AND CONCLUSIONS o see the performace of the adaptive filters the parameters covergece, output PSNR ad the CPU time are cosidered. he performace of differet adaptive filters amely LMS, BLMS, DLMS ad RLS filters are sho for speech sigals. he speech sigals ith iput PSNR ragig from -0dB to 5dB as iput to differet Adaptvie filters ad the output PSNR are dra i the figure 4. he ormalized adaptive filter has the best performace amog all the filters. Figure 5 sho that Normalized Least Mea Square adaptive filter has the better performace. As per the Covergece parameter is cocered the Block LMS took feer umber of iteratios to reach miimum error. his effect has bee sho i Figure 6. Figure 7, 8 ad 9 are the origial oisy ad deoised of oe of the speech sigal hich has 117,000 coefficiets. o give more clarity i the deoisig procedure the part of speech sigal from 75,00 to 76,000 is take. hese figures are sho i figure 10, 11 ad 1. Although i theory RLS suppose to have highest covergece factor, but from the results it is sho that RLS is ot a stable filter. Whereas the Block LMS has better performace tha eve Fig. 7. Origial Speech Sigal. Fig. 8. Noisy Speech Sigal. 353

4 Fig. 9. Deoised Speech Sigal. Fig. 1. Deoised Speech Sigal ith speech coefficiets from 75,00 to 76,000. Fig. 10. Origial Speech Sigal ith speech coefficiets from 75,00 to 76,000. REFERENCES [1] S. Hayki, Adaptive Filter heory, 4th ed. Pearso Educatio, 00. [] P. S. R. Diiz, Adaptive Filterig: Algorithms ad Practical Implemetatio. Kluer Academic Publishers, [3] J. Cioffi ad. Kailath, Fast Recursive-Least-Squares, rasversal Filters for Adaptive Filterig, IEEE ras. Acoust., Speech, Sigal Processig, vol. 3, pp , [4] D.. M. Slock ad. Kailath, Numerically Stable Fast rasversal Filters for Recursive Least Squares Adaptive Filterig, IEEE ras. O Sigal Processig, vol. 39, pp , Ja [5] Adaptive filterig algorithms ad practical implemetatio by Poulo.S.R. diiz, third Editio, spriger 008. [6] Least mea squares adaptive filters by simo hayki ad Berard idro, joh iley ad sos publishers, 009. [7] Fudemetals of adaptive filterig by Ali H Sayed, iley iter sciece 008. [8] Advaces i etork ad acoustive oise cacelatio by J.Beesty, D. R. Morga, M. Sodhi ad S. L Gay, spriger publicatios, 001. [9] Priciples of adative filtrs ad self learig systems by A Zacich, spriger publishers. [10] Adaptive Filter Algorithm Research ad Matlab Realizatio, W Lubi, Z Jigchu, H XIONG - Moder Electroics echique, 007. Mr. D. Hari Hara Satosh obtaied his B. ech. ad M. ech degrees from JN Uiversity, Hyderabad i the year 005 ad 010. He is presetly orkig as Assistat Professor i MVGR College of Egieerig, Viziaagaram, AP. He has publicatios i both Iteratioal ad Natioal Jourals ad preseted 16 papers at various Iteratioal ad Natioal Cofereces. His areas of iterests are Image Processig, Speech processig. Fig. 11. Noisy Speech Sigal ith speech coefficiets from 75,00 to 76,000. Ms. P.V.U.S.L.Sravya completed her B. ech. from MVGR College of Egg., JN Uiversity, Kakiada. She is presetly doig MS i USA. She has preseted may papers at various studet symposiums o latest techologies. Her areas of iterests are Image Processig, Speech processig. 354

5 Mr. N. Shamukha Rao obtaied his B. ech. ad M. ech degrees from JN Uiversity, Hyderabad ad AU College of Egierig, Visakhapatam i the years 005 ad 010. He is presetly orkig as Assistat Professor i MVGR College of Egieerig, Viziaagaram, AP. He preseted 16 papers at oe Iteratioal. His areas of iterests are Image Processig, Speech processig. Mr. V. N. Lakshmaa Kumar obtaied his B. ech. ad M. ech degrees from JN Uiversity, Hyderabad i the year 005 ad 010. He is presetly orkig as Assistat Professor i MVGR College of Egieerig, Viziaagaram, AP. He has publicatios i both Iteratioal ad Natioal Jourals ad preseted 8 papers at various Iteratioal ad Natioal Cofereces. His areas of iterests are Image Processig, Speech processig. 355