DESIGN AND ANALYSIS OF SPEECH PROCESSING USING KALMAN FILTERING

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1 5 JATIT & LLS All righs reserved wwwjaiorg DESIGN AND ANALYSIS OF SPEECH PROCESSING USING KALMAN FILTERING VINEELA MURIKIPUDI, KPHANI SRINIVAS DSRAMKIRAN, PROFHABIBULLA KHAN, GMRUDULA, KSUDHAKAR BABU, TRAGHAVENDRA VISHNU Deparmen of ECE, K L Universiy, Gunur DT, AP, India Head of he deparmen, Depof ECE, K L Universiy, Gunur DT, AP, India vineelamuriipudi@gmailcom ABSTRACT Speech processing is used widely in every day s applicaions ha mos people ae for graned, such as newor wire lines, cellular elephony, elephony sysem and elephone answering machines Due o is populariy and increasing of demand, engineers are rying various approaches of improving he process One of he mehods for improving he process is Kalman filering Kalman filering now become a popular filering echnique for esimaing and resolving redundan errors conaining in speech The objecive of his paper is o generae a reconsruced oupu speech signal from he inpu signal involving he applicaion of a Kalman filer esimaion echnique In his paper, Kalman filer is used o esimae he parameers of he auoregressive (AR) process and represened in he saespace domain Keywords: Kalman Filer, Noise Reducion, Speech Processing INTRODUCTION Speech is a form of communicaion in every day life I is essenial o now how we produce and perceive i and how speech echnology may assis us in communicaion Therefore in his projec, we will be looing more ino speech processing wih he aid of an ineresing echnology nown as he Kalman Filer [] One of he common adapive filering echniques ha are applied o speech is he Wiener filer This filer is capable of esimaing errors however a only very slow compuaions On he oher hand, he Kalman filer suppresses his disadvanage As widely nown o he world, Kalman filering echniques are used on GPS (Global Posiioning Sysem) and INS (Inerial Navigaion Sysem) Noneheless, hey are no widely used for speech signal coding applicaions The reason why Kalman filer is so popular in he field of radar racing and navigaing sysem is ha i is an opimal esimaor, which provides very accurae esimaion of he posiion of eiher airborne objecs or shipping vessels Due o is accurae esimaion characerisic, engineers are picuring he Kalman filer as a design ool for speech, whereby i can esimae and resolve errors ha are conained in speech afer passing hrough a disored channel Due o his moivaing fac, here are many ways a Kalman filer can be uned o sui engineering applicaions such as newor elephony and even saellie phone conferencing Knowing he fac ha preserving informaion, which is conained in speech, is of exreme imporance, he availabiliy of signal filers such as he Kalman filer is of grea imporance [35] KALMAN FILTER The Kalman Filer is an esimaor for wha is called he linear quadraic problem, which focuses on esimaing he insananeous sae of a linear dynamic sysem perurbed by whie noise Saisically, his esimaor is opimal wih respec o any quadraic funcion of esimaion errors I is a Recursive Daa Processing Algorihm The bloc diagram of ypical Kalman filer applicaion is shown in Figure [67] In pracice, his Kalman Filer is one of he greaer discoveries in he hisory of saisical esimaion heory and possibly he greaes discovery in he wenieh cenury I has enabled manind o do many hings ha could no have been done wihou i, and i has become as indispensable as silicon in he maeup of many elecronic sysems

2 5 JATIT & LLS All righs reserved wwwjaiorg In a more dynamic approach, conrolling of complex dynamic sysems such as coninuous manufacuring processes, aircraf, ships are he mos immediae applicaions of Kalman filer In order o conrol a dynamic sysem, one needs o now wha i is doing firs For hese applicaions, i is no always possible or desirable o measure every variable ha you wan o conrol, and he Kalman filer provides a means for inferring he missing informaion from indirec (and noisy) measuremens Some amazing hings ha he Kalman filer can do is predicing he liely fuure courses of dynamic sysems ha people are no liely o conrol, such as he flow of rivers during flood, he rajecories of celesial bodies or he prices of raded commodiies [8] Conrols Sysem error sources Sysem Measuri ng Devices Measuring error sources opimal esimaion of sysem sae Figure Bloc Diagram of ypical Kalman Filer applicaion Process of esimaion The process commences wih he addresses of a general problem of rying o esimae he sae o f a discreeime conrolled process ha is governed by a linear sochasic difference equaion: X = Ax +Bu +W () Wih a measuremen m z R ha is Kalma n filer Z = Hx +V () The random variables W and V represen he process and measuremen noise (respecively)we assume ha hey are independen of each oher, whie, and wih normal probabiliy disribuions P(W)~N(O,Q) (3) P(V)~N(O,R) (4) Ideally, he process noise covariance Q and measuremen noise covariance R marices are assumed o be consan, however in pracice, hey migh change wih each ime sep or measuremen In he absence of eiher a driving funcion or process noise, he n n Marix A in he n l difference equaion, relaes he sae a he previous ime sep K o he sae a he curren sep In pracice, change wih each ime sep, however here i is assumed consan The marix B relaes he I u R o he sae X H opional conrol inpu which relaes he sae o he measuremen Z In pracice H migh change wih each ime sep or measuremen; however we assume i is consan Compuaional origins of he filer n Le say we define xˆ R (noe: super minus )o be our priori sae esimae a sep,given nowledge of he process prior o sep and xˆ R n (noe: wih ou he super minus ) o be our poseriori sae esimae a sep,given measuremen z The priori and poseriori esimae errors can be defined as: [3] e = x xˆ (5) e = x xˆ The priori esimae error covariance is hen P = E[e e T ] (7) And he poserior esimae error covariance is (6) P =E[e e T ] (8) Afer deriving he equaions for he Kalman filer, he goal is o find an equaion ha compues a poseriori sae esimae xˆ, as a linear combinaion of a priori esimae xˆ and a weighed difference beween an acual measuremen Z and a measuremen predicion Hˆ x as shown below in e q (9)Some jusificaion for eq (9) is given in he secion The probabilisic Origins of he Filer found below

3 5 JATIT & LLS All righs reserved wwwjaiorg x = xˆ + K(Z Hx ˆ ) ˆ (9) The gain, K or oherwise nown as he blending facor, minimizes he poseriori error covariance in equaion 8 and is a n m marix in eq 9 This minimizaion can be accomplished by firs subsiuing equaion 9 ino he above definiion for e, afer which subsiuing e ino eq, performing he indicaed expecaions, aing he derivaive of he race of he resul wih respec o K, seing ha resul equal o zero, and hen solving for K One form of he resuling K ha minimizes eq (8) is given by eq () as follow: T K = P H ( HP H + R) K T PH = T HP H + R T By looing a eq (), i can be seen ha as he measuremen error covariance R approaches zero, () he gain K weighs he residual more heavily Specifically, LimR K = H () On he oher hand, as he priori esimae error covariance P approaches zero, he gain K will weigh he residual less heavily Specifically, Lim K = P () Anoher way of describing he weighing by K is ha as he measuremen error covariance R approaches zero, he acual measuremen Z will be depended on more and more, whereas he Hˆ x is depended on less prediced measuremen and less On he oher hand, as he priori esimae error covariancep approaches zero, he acual measuremen Z is depended on less and less, and he prediced measuremen more and more [46] Hˆ x is depended on 3 Probabilisic Origins of he Filer This secion is a shor secion describing he jusificaion as menioned in he previous secion for eq (9) This jusificaion is rooed in he probabiliy of a priori esimae Xˆ condiioned on all prior Z measuremens (Bayes' rule) For now i is suffice o poin ou ha he Kalman filer mainains he firs wo momens of he sae disribuion, [ X] = X E ˆ (3) [ T X Xˆ )( X Xˆ ] = P E ) ( (4) The poseriori sae esimae of (9) reflecs he mean (he firs momen) of he sae disribuion i is normally disribued if he condiions of () and (3) are me The poseriori esimae error covariance of (8) reflecs he variance of he sae disribuion (he second noncenral momen)in oher words, ˆ ˆ T p( X Z) NE ( [ X], E[( X X)( X X) ]) = N( Xˆ, P ) (5) 3 IMPLEMENTATION OF KALMAN FILTER TO SPEECH From a saisical poin of view, many signals such as speech exhibi large amouns of correlaion From he perspecive of coding or filering, his correlaion can be pu o good use The all pole, or auoregressive (AR), signal model is ofen used for speech The AR signal model is inroduced as: [7 9] y = N W i aiz i= (6) Equaion (3) can also be wrien in his form as shown below: y = ay + ay + + anyn where, + w (7) = Number of ieraions;

4 5 JATIT & LLS All righs reserved wwwjaiorg y = curren inpu speech signal sample; y N = (N) h sample of speech signal; an = N h Kalman filer coefficien and w = exciaion sequence (whie noise) In order o apply Kalman filering o he speech expression shown above, i mus be expressed in sae space form as: H = X H + w z (8) y = g H (9) Where X= H = a a a N an y y y yn+ W = w g = [ ] () X is he sysem marix, H consiss of he series of speech samples; w is he exciaion vecor and g, he oupu vecor The reason of (N+) h ieraion is due o he sae vecor, H, consiss of N samples, from he h ieraion bac o he( N+) h ieraion The above formulaions are suiable for he Kalman filer As menioned in he previous chaper, he Kalman filer funcions in a looping mehod Referring o as a guide in implemening Kalman filer o speech, I denoe he following seps wihin he loop of he filer Define marix H T as he row vecor [4] H T = [y y yn ] () Z = H T X + w () where X will always be updaed according o he number of ieraions, Noe ha when he =, he marix H is unable o be deermined However, when he ime z is deeced, he value in marix H is nown The above purpose is hus sufficien enough for defining he Kalman filer, which consiss of: X = [ I KH T ] X + K z (3) Where I= K = PH[ H T PH + R ] (4) Where K is he Kalman gain marix, P is he a priori error covariance marix, R is measuremen noise covariance, and P = P PH [H T PH + R] H T P + Q (5) Where P is he a poseriori error covariance marix; And Q= Thereafer he reconsruced speech signal, Y 3

5 5 JATIT & LLS All righs reserved wwwjaiorg afer Kalman filering will be formed in a manner similar o eq (7) Y = ay + ay + + anyn + w (6) Since he value of y is he inpu a he beginning of he process, here will be no problem forming H T In ha case a quesion rises, Y formed by he parameers w and{a i } N i are deermined from applicaion of he Kalman filer o he inpu speech signal Y Tha is in order o consruc Y,we will need marix X ha conains he Kalman coefficiens and he whie noise, w which boh are obained from he esimaion of he inpu signal This informaion is enough o deermine HH Where Y Y HH = Y 3 Y N+ 3 Cross Correlaion Cross Correlaion is acually a mehod for measuring he similariy of wo waveforms based upon he amoun of common componens conained wihin he wo waveforms The purpose of his echnique is o compare he differences in he ime frame of wo signals Before we move on o he resuls of cross correlaion beween he inpu speech signals and he oupu speech signals, le us briefly ae a loo a cross correlaion Cross correlaion of f () and f () is defined as: C = f ( f ( d ) ) ( ) f d f d ( ) (8) The magniude of he inegral in he numeraor of (4) is an indicaion of he similariy of hose wo signals If, f ) f ( ) d = ( (9) I means ha he wo signals will have no similariy over he ime inerval (, ) In general, he cross correlaion funcion will be modified accordingly o C(τ)= f ( ) f( τ) d= (3) Wheher Then, C will be a measuremen of he similariy of wo paricular signals over an enire inerval (, ), whereby τ is he ime shif parameer This inegral deermines f is shifed in ime relaive o f 4 RESULTS The following resuls are obained by seing 5 Kalman coefficiens of a 5h order Kalman filer 8,, 6, o7 and 4 Differen coefficiens a differen ieraions are shown in Table ( 8 ) y + ( ) y + ( 6 ) y 3+ ( 7 ) y 4+ ( 4) y w y + = 5 Table Differen Coefficiens a Differen Ieraions Coefficien/ s nd 3 rd 4 h 5 h Ieraions Se Se Se Se Se Speech Samples of s8od Figure Inpu Signal of s8od 4

6 5 JATIT & LLS All righs reserved wwwjaiorg Figure 3 Oupu Signal of s8od Figure 6 Oupu Signal of s68od Figure 4 Cross Correlaion of s8od 4 Speech Samples of s68od Figure 7 Cross Correlaion of s68od 43 Speech Samples of s8od Figure 5 Inpu Signal of s68od Figure 8 Inpu Signal of s8od 5

7 5 JATIT & LLS All righs reserved wwwjaiorg As a consequence, parameer Q has o be uned in order o mee he objecive However, parameer R is of superfluous o be uned Moreover, a es for cross correlaion had also been conduced during his paper for measuring he similariy of he inpu and oupu speech signals This es is of necessiy for he reason ha differen signals are bound o be similar bu no idenical By and large, his paper has been quie successful in erms of achieving he objecives Consequenly, percepion on signal processing and Kalman filer had also been reasured hroughou he process 7 REFERENCES Figure 9 Oupu Signal of s8od [] R G Brown and P Y C Hwang, Inroducion o Random Signals and Applied Kalman Filering nd Ediion, John Wiley & Sons, Inc, 99 [] A Gelb, Applied Opimal Esimaion, MIT Press, Cambridge, MA, 974 [3] M S Grewal and AP Andrews, Kalman Filering Theory and Pracice, Upper Saddle River, NJ USA, Prenice Hall, 993 [4] Jacobs, O L R Inroducion o Conrol Theory nd Ediion, Oxford Universiy Press, 993 Figure Cross Correlaion of s8od 5 CONCLUSIONS In his paper, an implemenaion of employing Kalman filering o speech processing had been developed As has been previously menioned, he purpose of his approach is o reconsruc an oupu speech signal by maing use of he accurae esimaing abiliy of he Kalman filer Furhermore, he resuls have also shown ha Kalman filer could be uned o provide opimal performance Wih he inroducion of uning parameers Q & R, oupu speech signals can be obained similar o he inpu speech signals Addiional esing on differen orders of he Kalman filer when applied o speech had also been conduced [5] Julier, Simon and Jeffrey Uhlman "A General Mehod of Approximaing Nonlinear Transformaions of Probabiliy Disribuions," Roboics Research Group, Deparmen of Engineering Science, Universiy of Oxford, Nov 995 [6] S J Julier, J K Uhlmann, and H F Durran Whye, "A New Approach for Filering Nonlinear Sysems", Proceedings of he 995 American Conrol Conference, Seale, Washingon, pp 683 [7] Kalman, R E 96 "A New Approach o Linear Filering and Predicion Problems," Transacion of he ASMEJournal of Basic Engineering, pp 3545, March 96 [8] Lewis, Richard, Opimal Esimaion wih an Inroducion o Sochasic Conrol Theory, John Wiley & Sons, Inc, 986 [9] Maybec, Peer S 979 Sochasic Models, Esimaion, and Conrol, Volume, Academic 6

8 5 JATIT & LLS All righs reserved wwwjaiorg Press, Inc [] Sorenson, H W 97 "LeasSquares esimaion: from Gauss o Kalman,"IEEE Specrum, vol 7, pp 6368, July 97 [] A Gelb, J F Kasper, Jr, R A Nash, Jr, C F Price, and A A Suherlandd, Jr,Applied Opimal Esimaion, MIT Press, Cambridge, MA 974 [] MS Grewal and APAndrews, Kalman Filering Theory& Pracice Using MATLAB nd ediion,john Wiley&Sons, Canada,, pp [3] MS Grewal and AP Andrews, Kalman Filering Theory and Pracice Using MATLAB nd ediion, John Wiley & Sons, Canada,, pp 67 [4] RG Brown and PYC Hwang, Inroducion o Random Signals & Applied Kalman Filering wih MATLAB EXERCISES & Soluions 3 rd ediion, John Wiley & Sons, Canada, 997, pp 45 [5] MS Grewal and AP Andrews, Kalman Filering Theory and Pracice UsingMATLAB nd eiion, John Wiley & Sons, Canada,, pp 57 [7] AN Ince, Digial Speech Coding, Kluwer Academic Publishers, Massachuses, USA, 99 [8] AN Ince, Overview of Voice Communicaions and Speech Processing,Digial Speech Coding, Kluwer Academic Publishers, Massachuses, USA,99, pp 4 [9] C Wheddon & R Linggard, Low Bi Rae Speech, Speech and Language Processing, Chapman and Hall, 99, pp 333 [] RV Cox, Curren Mehods of Speech Coding, Inernaional Journal of High Speed Elecronics and Sysems, Vol 8, No, World Scienific Publishing Company, 997, pp3 68 [] S Crisafulli, JD Mills, and RR Bimead, Kalman Filering Techniques in Speech Coding In Proc IEEE Inernaional Conference on Acousics, Speech, and Signal Processing, San Francisco, March 99 [] BDO Anderson and JBMoore, The Kalman Filer, Opimal Filering, Prenice Hall Inc, Englewood Cliffs, NJ, 979, pp 55 [3] C R Wains, Pracical Kalman Filering in Signal Coding, New Techniques in Signal Coding, ANU, Dec 994 [6] R P Ramachandran and R Mammone, Modern Mehods of Speech Processing, Kluwer Academic Publishers, Massachuses, USA, 994 [4] S Saio and K Naaa, Digiizaion, Fundamenal of Speech Processing, Academic Press, Japan, 985, pp 7