CHAPTER 5 MINIMAX MEAN SQUARE ESTIMATION

Transcription

1 110 CHAPTER 5 MINIMAX MEAN SQUARE ESTIMATION 5.1 INTRODUCTION The problem of parameter estmaton n lnear model s pervasve n sgnal processng and communcaton applcatons. It s often common to restrct attenton to lnear estmators, hch smplfes the mplementaton as ell as the mathematcal dervatons. The smplest desgn scenaro s hen the second order statstcs of the parameters to be estmated are knon and t s desrable to mnmze the Mean Squared Error (MSE). The optmal lnear estmator for ths problem s straghtforard to derve and s the ell-knon Mnmax Mean Square Estmator (MMSE). Hoever, f the estmated parameters are determnstc, then the MSE depends explctly on the unknon parameter, and therefore cannot be optmzed drectly. In an estmaton context, the objectve typcally s to mnmze the sze of the estmaton error, rather than that of the data error. In fact, n many practcal scenaros the least square estmator s knon to result n a large MSE. To overcome ths lmtaton of the least square estmator, t s desrable to develop estmators that optmze a crteron based on the estmaton error rather than the data error.

2 LITRATURE REVIEW The estmated parameters are determnstc, MSE depends explctly on the unknon parameters, and therefore cannot be optmzed drectly by Eldar et al (004). The problem of estmatng n the presence of model uncertantes, a random vector that s observed through a lnear transformaton and corrupted by addtve nose. The assumpton are made that both the covarance and the transformaton are not completely specfed and the lnear estmator that mnmzes the MSE across all possble covarance matrces and transformatons n the regon of uncertanty dscussed by Eldar and Ner merhav (004). An alternatve method as developed (Eldar 005) to mnmze the orst case regret,.e., the orst case dfference beteen the MSE and the best possble MSE achevable th a lnear estmator. Kassam and Lm (1977) dscussed robust estmators of ths form extensvely n the context of ener estmaton, here the sgnal to be estmated s an nfnte length statonary process th uncertan second order statstcs. The problem of estmatng an unknon parameter vector n a lnear model that may be subject to uncertantes, here the vector s knon to satsfy a eghted norm constrant s knon to be robust Mnmax MSE estmator. The Mnmax MSE estmator can sgnfcantly ncrease the performance over the conventonal least squares estmator examned by Eldar et al (005). Zvka Ben-Ham and Eldar (005).dealt th any bounded parameter set, a lnear Mnmax estmator acheves loer MSE than the LS estmator, over the entre parameter set. The performance of estmators evaluated for estmatng a determnstc parameter vector th the MSE as

3 11 the performance measure. A frameork for examnng the MSE of dfferent approaches based on the concepts of admssble and domnatng estmators (Eldar 005). For an arbtrary choce of eghtng matrx, the Mnmax MSE estmator can be formulated as a soluton to a SDP. Zvka Ben-Ham and Eldar (005) concentrated on the lnear regresson problem s used for estmatng an unknon, determnstc parameter vector based on measurements corrupted by colored Gaussan nose. The estmators based on the blnd Mnmax approach, a technque hereby a parameter set s estmated from measurements and then used to construct a Mnmax estmator. Tradtonal Cramer-Rao type bounds provde benchmarks on the varance of any estmator of a determnstc parameter vector. A drect measure of the estmaton error that takes both the varance and the bas as presented (Eldar 005). Another nterestng problem (Eldar 006) of estmatng an unknon determnstc parameter vector n a lnear model th a random model matrx as dscussed. The Mnmax MSE estmator can mprove the performance over both a Bayesan approach and a least squares method n hch the norm of the parameter vector s also bounded. The problem of estmatng a vector n the lnear model here block crculant matrx s assumed to have a eghted norm bound s subjected to nose. For an arbtrary choce of eghtng, the Mnmax MSE estmator can be formulated as a soluton to a SDP by Amr Beck et al (007). A nonlnear Mnmax estmator has been developed for the classcal lnear regresson model assumng that the true parameter vector les n an ntersecton of ellpsods that mnmzes the orst case estmaton error over the gven parameter set by Eldar et al (008).

4 113 To mprove the total MSE of the Least Squares estmator n the lnear regresson model over bounded norm sgnals, under all eghted MSE measures and suggest a Mnmax estmator that mnmzes the orst case MSE (Eldar 008). A popular approach s to bound the MSE achevable thn the class of unbased estmators. MSE bounds that are loer than the unbased Cramer-Rao bound for all values of the unknon parameters ( Eldar 008). Ths ork appears to have been the startng pont for the study of alternatves to LS estmators. Among the more promnent alternatves are the rdge estmators dscussed by Hoerl and Kennard (1970) also knon as Tkhonov regularzaton. Another alternatve estmator s sad to be shrunken estmator also dealt by Mayer and Wllke (1973). Estmaton of a multvarate normal mean vector under sum of squared error loss and a Mnmax admssble estmator hch s generalzed as Bayes th respect to a pror dstrbuton. These generalzed Bayes Mnmax estmators move on to the James-Sten estmator (Yuzo Maruyama 004). The necessary condtons for an estmator to domnate the James-Sten estmator hch s admssble and t s the generalzed Bayes estmator relatve to the fundamental harmonc functon n three and hgher dmensons dscussed by Yuzo Maruyama and Straderman (005). A lnear htenng transformaton that mnmzes the MSE beteen the orgnal and htened data hch results n a hte output that s as close as possble to the nput. The optmal htenng transformaton s developed both fnte length data vectors and nfnte length sgnals (Eldar 003). In ths chapter, the performance of lnear, nonlnear and adaptve estmators n estmatng the speech sgnal through Quantum parameters have been studed. The objectve s to estmate the speech sgnal by a set of lnear, nonlnear and adaptve estmators have been proposed to be effcent

5 114 n performance, and compare t th Mnmax Estmator. The Mnmax Mean Square Error (MMSE) estmator s desgned to mnmze the orst case MSE. Hoever, n an estmaton context, the objectve typcally s to mnmze the sze of the estmaton error, rather than that of the data error. The estmaton of speech sgnal subject to nose by MMSE has been proposed. The comparatve analyss of the performance of MMSE estmator th that of lnear estmators such as Least Square (LS), Weghted Least Square (WLS) has been presented. The MMSE acheved very lo MSE compared to other lnear estmators for SNR ranges from -10 db to 60 db and partcularly 0 db to 10 db. The comparatve analyss of the performance of MMSE estmator th that of nonlnear estmators such as James sten, Shrunken, MAP estmators has been dscussed. The MMSE acheved very lo MSE compared to other nonlnear estmators for SNR ranges -10 db to 60 db and partcularly 0 db to 10 db. The comparatve analyss of the performance of MMSE estmator th that of adaptve estmators such as normalzed Least Mean Square (n-lms), Recursve Least Square (RLS) estmators are also presented. The MMSE acheved very lo MSE compared to other adaptve estmators for SNR ranges -10 db to 60 db and partcularly 0 db to 10 db. 5.3 LINEAR ESTIMATORS Least Square Estmator Least square estmaton, also knon as ordnary least square analyss s a method for lnear regresson that determnes the values of unknon quanttes n a statstcal model by mnmzng the sum of the

6 115 resduals (dfference beteen the predcted and observed values) squared. The least square approach to regresson analyss has been descrbed n the Least Mean Squares (LMS) method. It occurs hen the number of measured data s 1 and the gradent descent method s used to mnmze the squared resdual. LMS s knon to mnmze the expectaton of the squared resdual, th the smallest number of operatons per teraton. Hoever, t requres a large number of teratons to converge. Furthermore, many other types of optmzaton problems can be expressed n a least square form ether by mnmzng energy or by maxmzng entropy Weghted Least Square Estmator Weghted least square s a method of regresson smlar to least square n that t uses the same mnmzaton of the sum of the resduals n S 1 y f x. (5.1) Hoever, nstead of eghtng all ponts equally, they are eghted such that ponts th a greater eght contrbute more to the ft n 1 y f x. S (5.) Often, s the nverse of the varance, gvng ponts th a loer varance a greater statstcal eght: 1/. (5.3) In a lnear regresson context, X T X. f (5.4) denoted as y 1/ y (5.5) 1/, X X, then mnmzng the eghted least squares

7 116 n S 1 T y x, s the same as mnmzng the ordnary least squares (5.6) n S 1 T y x, (5.7) Let consder the class of estmaton problems represented by the lnear model y Hx here x s a determnstc vector of unknon parameters, H s a knon vector th covarance C. n m matrx and s a zero mean random chosen such that The least square (LS) estmate of x hch s denoted xˆ LS s yˆ Hxˆ LS HGy as close as possble to y n a (eghted) LS sense so that mnmzes the total squared error n the observatons. Thus, the LS estmate IS ˆ Gy s chosen to mnmze the total squared error x LS y HGy W y HGy (5.8) here W s an arbtrary postve defnte eghtng matrx and choosew C 1, then the LS estmate s gven by, (5.9) 1 H H H y x IS Snce the MSE depends explctly on the unknon parameters x and t cannot choose an estmate to drectly mnmze the MSE. A common approach s to restrct the estmator to be lnear and unbased. Then to seek the estmator of ths form that mnmzes the varance or the MSE, hch leads to the LS estmator. In further development, the estmator s not constraned to be unbased. In ths approach, choosng the estmator s motvated by the observaton and the data vector y s not very senstve to changes n x. so that a large error n estmatng x may translate nto a small

8 117 error n estmatng the data vector y, n hch case, xˆ LS may result n a poor estmate of x. To desgn a lnear estmator xˆ of x for the observatons y, so that xˆ Gy for some m n matrx G. A popular desgn strategy s to choose G to mnmze the MSE beteen the estmator xˆ and x, hch s gven by xˆ x Tr( GC G ) x ( I GH ) ( I GH x E ) (5.10) the MSE of xˆ depends on the unknon vector and the unknon covarance C. To elmnate the dependence of the desgn crteron on x, an alternatve approach s to mnmze the data error yˆ y ( yˆ y) ( yˆ y) here yˆ Hxˆ. Clearly ths crteron no longer depends on, and s straghtforard to optmze. If the covarance of the nose C s knon and the estmator performance has been mproved by mnmzng a eghted data 1 error ( yˆ y) C ( yˆ y), resultng n the eghted LS (WLS) estmator. xˆ WLS ( H C H ) H C y (5.11) 5.4 NON LINEAR ESTIMATORS James Sten Estmator The James Sten estmator s a nonlnear estmator hch can be shon to domnate the ordnary least square technques. Suppose s an unknon parameter vector of length m and y be an observatons of the parameter vector, such that y ~ N (, I) (5.1)

9 118 ^ To obtan an estmate of y) of, based on the observatons of y. The set of parameters s measured and the measurements are corrupted by ndependent Gaussan nose. Snce the nose has zero mean, t s very reasonable to use the measurements themselves as an estmate of the parameters equal to the least squares estmator. Ths aspect demonstrates n terms of MSE. The James-Sten estmator s gven by M js 1 Y. Y (5.13) ^ ( Three or more unrelated parameters are measured, ther total MSE can be reduced by usng a combned estmator such as the James- Sten estmator. When each parameter s estmated separately, the least square (LS) estmator s admssble. James and Sten demonstrated that the estmator presented above can stll be used hen the varance σ s unknon, by replacng t th the standard estmator of the varance 1 y y (5.14) n Ths domnant result stll holds under the same condton namely m > Maxmum A Posteror (MAP) Estmator In statstcs, the method of Maxmum A Posteror (MAP) estmaton can be used to obtan a pont estmate of an unobserved quantty based on the bass of emprcal data. Let us assume that estmate of an unobserved populaton parameter θ, on the bass of observatons x. Let f be the samplng dstrbuton of x, so that f (x θ) s the probablty of x hen the underlyng populaton parameter s θ. Then the functon ll be f x (5.15)

10 119 and s knon as the lkelhood functon and the estmate s arg ML x m a x f x (5.16) as the maxmum lkelhood estmate of θ. Assume that a pror dstrbuton g over θ exsts. Ths allos θ as a random varable as n Bayesan statstcs. Then the posteror dstrbuton of θ s as follos g f x (5.17) ' ' f x ' g d here g s densty functon of θ, Θ s the doman of g. Ths s a straghtforard applcaton of Bayes theorem. The method of MAP estmaton then estmates θ as the mode of the posteror dstrbuton of ths random varable: MAP x g f x arg m a x arg m a x f x g (5.18) x ' ' f ' g d The denomnator of the posteror dstrbuton does not depend on θ. The MAP estmate of θ concdes th the ML estmate hen the pror g s unform. Whle MAP estmaton shares the use of a pror dstrbuton th Bayesan statstcs, t s not generally seen as a Bayesan method. Ths s because MAP estmates are pont estmates, hereas Bayesan methods are characterzed by the use of dstrbuton to summarze data and dra nferences. Bayesan methods tend to report the posteror mean or medan together th posteror ntervals rather than the posteror mode. Ths s especally so hen the posteror dstrbuton does not have a smple analytc form. In ths case, the posteror dstrbuton can be smulated usng Markov

11 10 chan Monte Carlo technques, hle optmzaton to fnd ts mode(s) may be dffcult or mpossble Shrunken Estmator The matrx of the MSE of a shrunken s not bgger than matrx MSE of the LS estmator and satsfed the follong matrx nequalty XX ' ( ) ( C' v C) ( C' v C) (5.19) Ths requres that β be chosen such that 1 X '( C' v C) X 1 1 (5.0) 1 X '( C' C) X 1 v Notce that for β=1, the shrunken estmator concdes th the LS estmator and for 1 X '( C' v C) X 1 (5.1) 1 X '( C' C) X 1 v gves a bas estmate th the same MSE of LS. 5.5 ADAPTIVE FILTERS An adaptve flter s a flter hch self-adjusts ts transfer functon accordng to an optmzng algorthm. Most adaptve flters are dgtal flters that perform dgtal sgnal processng and adapt ther performance based on the nput sgnal. For some applcatons, adaptve coeffcents are requred snce some parameters of the desred processng operaton (for nstance, the propertes of some nose sgnal) are not knon n advance. In these stuatons feedback uses to refne the values of the flter coeffcents.

12 11 Generally, the adaptng process nvolves the use of a cost functon, hch s a crteron for optmum performance of the flter (for example, mnmzng the nose component of the nput), to feed an algorthm, hch determnes ho to modfy the flter coeffcents to mnmze the cost on the next teraton. The block dagram, as a foundaton for partcular adaptve flter realsatons, such as Least Mean Squares (LMS) and Recursve Least Squares (RLS) shon n Fgure.. The dea behnd the block dagram s that a varable flter extracts an estmate of the desred sgnal.the nput sgnal s the sum of a desred sgnal d(n) and nterferng nose v(n) here x(n) s gven by x(n) = d(n) + v(n) (5.) The varable flter has a Fnte Impulse Response (FIR) structure. For such structures the mpulse response s equal to the flter coeffcents. The coeffcents for a flter of order p are defned as W 0, 1,... p n n n n T (5.3) The error sgnal or cost functon s the dfference beteen the desred sgnal and estmated sgnal e n d n d n (5.4) The varable flter estmates the desred sgnal by convolvng the nput sgnal th the mpulse response. In vector notaton ths s expressed as d n here T n W X n (5.5)

13 1, 1,... T (5.6) x n x n x n x n p s an nput sgnal vector. Moreover, the varable flter updates the flter coeffcents at every tme nstant W n 1 Wn Wn (5.7) here Wn s a correcton factor for the flter coeffcents. The adaptve algorthm generates ths correcton factor based on the nput and error sgnals. LMS and RLS defne to dfferent coeffcent update algorthms Least Mean Square (LMS) Flter The LMS algorthm s used n adaptve flters to fnd the flter coeffcents that relate to produce the least mean squares of the error sgnal (dfference beteen the desred and the actual sgnal). In a stochastc gradent descent method, the flter s only adapted based on the error at the current tme. The man draback of the pure LMS algorthm s that t s senstve to the scalng of ts nput x(n). Ths makes t very hard to choose a learnng rate μ that guarantees stablty of the algorthm. The normalsed Least Mean Squares flter (n-lms) s a varant of the LMS algorthm that solves ths problem by normalsng th the poer of the nput Recursve Least Square (RLS) Flter RLS algorthm s used n adaptve flters to fnd the flter coeffcents that relate to produce the recursvely least squares of the error sgnal (dfference beteen the desred and the actual sgnal). Consder the lnear tme seres model

14 13 1 y n x n e n (5.8) here e(n) N(0,1) s a hte nose and estmate the parameter va least squares at each tme N refer to the ne least squares estmate byˆ. As tme evolves, avod completely redong the least squares algorthm to fnd the ne estmate for ˆ N 1, n terms ofˆ N, and then update usng varous technques.the beneft of usng the RLS algorthm s that there s no need to nvert extremely large matrces, thereby savng computng poer. Another advantage s gettng some ntuton behnd such results as the Kalman flter. N 5.6 MINIMAX MSE ESTIMATION To develop an estmator that mnmzes an objectve drectly related to the MSE, t s suggested to seek the lnear estmator that mnmzes the orst-case MSE over all possble values of x, assumng that C s knon. The Mnmax estmator for the case n hch x Tx U, here T s an arbtrary postve defnte eghtng matrx. The arbtrary postve defnte eghtng matrx T I the estmator reduces to, xˆ MX ( C U ) U 0 ( H C 1 H ) 1 H C 1 y (5.9) 1 1 here Tr{ H C H s the varance of the WLS estmator. The 0 ) comparson shos that the Mnmax estmator s shrnkage of the WLS estmator. The advantage of the Mnmax estmator x ˆ ( C ) s establshed and proven that ts MSE s smaller than that of xˆ WLS for all x U. MX Snce xˆ MX ( C ) depends explctly on C, t cannot be mplemented f C s not completely specfed. Instead, the estmator that

15 14 mnmzes the orst case MSE over all possble choces of x and C that are consstent th pror nformaton on these unknons. To reflect the uncertanty of the true covarance matrx C, an uncertanty regon ths resembles the band model dely used n the contnuous tme case. Specfcally, snce Q that jontly dagonalzes C s postve defnte, there exsts an nvertble matrx C and HH so that HH ~ ~ Q Q (5.30a) C ~ Q Q (5.30b) Here ~ s an n m matrx defned by ~ 0 (5.31) here dag,,... ) th 0, 1 m, and ( 1 m dagonal matrx defned by ~ 0 0 ~ s an n n (5.3) here dag,,... ) th 0, 1 m, s an arbtrary dagonal ( 1 m matrx of sze ( n m) ( n m). In gven uncertanty model the dagonalzng matrx Q s knon. The uncertanty n C s due entrely to the dagonal matrx ~ hch s constraned to an arbtrary convex set U, such that 0. Wth HH follos that H has decomposton, H ~ V Q (5.33) for some m m untary matrxv. Let denote the unknon determnstc parameter vector n the model y Hx, here H s a knon n m matrx and s a zero-mean random vector th covarance C. Let Q be an nvertble matrx that

16 15 jontly dagonalzes C and HH, so that H V for some untary ~ V matrxv. Then the soluton to problem for any convex set U such that 0, s xˆ MX ( C ) U U m 1 V 1 Z Q 1 y (5.34) If C s any postve defnte covarance matrx then the estmator x ˆ ( C ) can be expressed as a shrnkage of the WLS estmator xˆ MX ( C MX ) ( H C H ) H C y (5.35) th dag,,... ) and s an arbtrary dagonal matrx. The ( 1 m shrnkage factor represented as U (5.36) m U 1 In partcular, choosng ^ C here C 0 0 Q ^ Q C (5.37) here xˆ MX ( C ) s the Mnmax MSE estmator for fxed C. Next, for any C here s any postve defnte covarance matrx of the form ~ ~ { Q Q U} (5.38) C m 1 Tr{ HC 1 H ) 1 (5.39) hch s the varance of the WLS estmator th eghtc. 1

17 16 matrx To estmate the unknon flter, frst estmate the covarance C usng N=50 of the nose W and then fnd a matrx Q that jontly dagonalzes HH and the estmate of C. The dagonal elements of HH and the estmate of C represented by and respectvely. To estmate the vector from the observatons, dfferent estmators such as LS estmator, WLS estmator, Mnmax MSE estmator, James-sten estmator, MAP estmator are consdered. To mplement Mnmax MSE estmator, the norm of x s assumed to be knon, so thatu x. 5.7 RESULTS AND DISCUSSION The analog nput s obtaned through the channel 1 at a sample rate of 8000 and duraton of 1.5 seconds and number of samples obtaned from the speech sgnal s about as shon n Fgure.5. The sgnal s obtaned as a column vector. Ths column vector s converted nto a square matrx. No Hlbert transform s performed on ths matrx so that the numercal values of the sgnal can be obtaned. FFT s performed on the sgnal so that the spectral values of the sgnal can be obtaned. As the concept of QSP s to be satsfed, no the spectral matrx s beng converted nto orthogonal matrx usng Gram- Schmdt orthogonalzaton procedure. In the orthogonal matrx, hte nose th zero mean and unt standard devaton s added to the sgnal. The Fgure 5.1 shos the plot of the nput speech sgnal. Then comes the nose corrupted sgnal, Fgure 5. shos the nput sgnal corrupted th nose. Input s a contnuous speech sgnal gven through mcrophone. After gettng the orgnal speech sgnal and the nose corrupted

18 17 speech sgnal, the orgnal sgnal has been estmated usng the estmaton technques. Fgure 5.1 Input speech sgnal Fgure 5. Nose corrupted sgnal Fgure 5.3 and Fgure 5.4 shos the result of lnear estmaton of the speech sgnal hch s compared th the result of the Mnmax MSE estmator for -10 db to 60 db and partcularly 0 db to 10 db. Fgure 5.5 and Fgure 5.6 shos the result of nonlnear estmaton of the speech sgnal hch s compared th the result of the Mnmax MSE estmator for -10 db to 60 db and 0 db to 10 db. Fgure 5.7 and Fgure 5.8 shos the result of adaptve estmaton of the speech sgnal hch s compared th the result of the Mnmax MSE estmator for -10 db to 60 db and partcularly 0 db to 10 db. The comparatve ork for the performance of dfferent estmaton schemes n estmatng the orgnal speech sgnal from the nose corrupted sgnal has been presented. Here the Mean Square Error that result at the end of the estmaton process as crtera to analyze the performance of the estmators. Mean square error s calculated from the dfference beteen the true value and the estmated value of each estmator. The performance of the Mnmax estmator th that of the other estmators has been

19 18 compared that are proposed to be effcent n ther performance th ther on constrants. Here consder three dfferent estmators they are lnear, nonlnear and the adaptve estmators. Fgure 5.3 Lnear Estmaton and MMSE (-10 db to 60 db) Fgure 5.4 Lnear Estmaton and MMSE (0 db to 10 db) Table 5.1 MSE of Lnear Estmaton and MMSE SNR(dB) LS WLS MMSE E E E E E E E E E-05 Table 5. MSE of Lnear Estmaton and MMSE SNR(dB) LS WLS MMSE E E E E E E E E E E-05

20 19 Fgure 5.5 Non Lnear Estmaton and MMSE (-10 db to 60 db) Fgure 5.6 Non lnear Estmaton and MMSE (0 db to 10 db) Table 5.3 MSE of Nonlnear Estmaton and MMSE SNR(dB) MAP MMSE JS SH E E E E E+0 6.4E E E E E E E Table 5.4 MSE of Nonlnear Estmaton and MMSE SNR(dB) MAP MMSE JS SH E E E E E E E E E E The MSE averaged over 400 nose realzatons as a functon of the SNR defned by 10log x,usng each of the methods above (for

21 130 the LS, WLS, and MMSE. so the 400 realzatons of the nose are relevant only for the Mnmax MSE estmators). Fgure 5.7 Adaptve Estmaton and MMSE (-10 db to 60 db) Fgure 5.8 Adaptve Estmaton and MMSE (0 db to 10 db) Table 5.5 MSE of Adaptve Estmaton and MMSE SNR(dB) MMSE RLS n-lms E E E E Table 5.6 MSE of Adaptve Estmaton and MMSE SNR(dB) n-lms MMSE RLS E E E E

22 131 From the Fgure 5.3 and Fgure 5.4 the estmate s approxmately the MSE value of these estmators. Fnally, the comparson analyss done usng MSE value, the MMSE estmator s best than the other to lnear estmators. The Fgure 5.3 and Fgure 5.4 shos the plot of Mean Square Error obtaned for estmaton of speech sgnal through Least Square (LS), Weghted Least Square (WLS) and the Mnmax Mean Square Estmator for -10 db to 60 db and 0 db to 10 db. From the Fgure 5.3 and Fgure 5.4 the nference s that Mnmax estmator s more effcent than the other to estmators. In Fgure 5.5 and Fgure 5.6, plot the MSE averaged over 400 nose realzatons as a functon of the SNR defned by 10log x, usng each of the methods above (James-Sten, Shrunken, MAP and MMSE. so the 400 realzatons of the nose are relevant only for the Mnmax MSE estmators). From the Fgure 5.5 and Fgure 5.6 estmate s approxmately the MSE value of these estmators s gven by MMSE estmator. Fnally, the comparson analyss done usng these MSE values, the MMSE estmator s best than other non lnear estmators. The MSE averaged over 400 nose realzatons as a functon of the SNR defned by10log x,usng each of the methods above (for the RLS, n-lms and MMSE. so the 400 realzatons of the nose are relevant only for the Mnmax MSE estmators). From the Fgure 5.7 and Fgure 5.8, the estmate s approxmately the MSE value of these estmators. Fnally the comparson analyss done usng MSE value, the MMSE estmator s best than the other to adaptve estmators. The Fgure 5.7 and Fgure 5.8 shos the plot of Mean Square Error obtaned for

23 13 estmaton of speech sgnal through normalzed Least Mean Square (n- LMS), Recursve Least Square (RLS) adaptve flters and the Mnmax Mean Square Estmator for -10 db to 60 db and 0 db to 10 db. From the Fgure 5.7 and Fgure 5.8 the nference s that Mnmax estmator s more effcent than the other adaptve estmators. 5.8 CONCLUSION The comparatve ork can be performed th dfferent estmaton schemes n estmatng the orgnal speech sgnal from the nose corrupted sgnal. The Mean Square Error s calculated from the dfference beteen the true value and the estmated value of each estmator. The comparson of the performance of the Mnmax estmator th that of the other lnear estmators such as Least square (LS), Weghted Least Square (WLS) estmators that are proposed to be effcent n ther performance th ther on constrants. The comparatve analyss of the performance of MMSE estmator th that of nonlnear estmators such as James sten, Shrunken, MAP estmators has been dscussed and MMSE estmator acheved loer MSE than other nonlnear estmators for all SNR values. The comparatve analyss also extend to performance of MMSE estmator th that of adaptve estmators such as normalzed Least Mean Square (n- LMS), Recursve Least Square (RLS) estmators has been dscussed. The MMSE estmator acheved loer MSE than other adaptve estmators for all SNR values.