AnAdaptive Algorithm forFastIdentification ofFIR Systems
Da‐Zheng Feng · 2006
Inthis paper, wedevelop afast recursive algorithmKalmangainvector, butsuchapproaches havethemore withaviewtofinding thetotal least squares (TLS) solution for complex structure withincreased computational complexity. adaptive FIRfiltering withinput andoutput noises. We Interestingly, thenumerically stable fast transversal filter introduce anovel butapproximate inverse poweriteration in algorithms areestablished in(10), together withthefast combination withGalerkin methodsothattheTLSsolution canbeupdated adaptively ata lowercomputational cost. We g . W further reduce thecomputational complexity ofthedevelopediversion lemma, a fast RTLSalgorithm isestablished in algorithm bymakingefficient computation ofthefastgain (11). LikeDavila's RTLSalgorithms, thealgorithms in(6) vector. We thenmakea careful investigation intoglobaland (11)findtheTLS solution through searching the convergence ofthedeveloped algorithm. Simulation results are eigenvector associated withthesmallest eigenvalue ofthe provided that clearly illustrate appealing performances ofthe augmented datavector andbyusinggradient-descent developed algorithm. method, sotheycandecrease thecomputational complexity. Inthis paper, we study theproblem ofadaptive FIR I. INTRODUCTION filtering intheTLSframework. Wedevelop afast recursive algorithm byapproximating thewell-known inverse power DuetoitSattract1ve features, therecursive least squaresiteration along theinput data vector. Weexploit theregular (RLS)algorithms (1)areoneofthemostpowerfulformoftheTLSsolution andcompute thegainvector by techniques that implement adaptive parameter estimation. virtue oftheFGV soastoenhance thecomputational However, whenthere exists white Gaussian noise inboth efficiency ofthedeveloped algorithm. Inaddition toits thesystem input andthesystem output, theRLSalgorithms significantly reduced computational complexity, the usually fails toyield anunbiased estimate ofthesystem developed algorithm isalsoexpected topossess thehighly parameters, thereby causing theperformance ofadaptive numerical stability. We illustrate performance ofthe filtering tobesignificantly degenerated. Instead, thetotaldeveloped algorithm via computer simulations. least squares (TLS)methods (2)maybetheefficient technique usedtoachieve theunbiased estimate ofthe II.SIGNAL MODELANDINVERSE POWERITERATION system parameters whenboththesysteminput andthe LettheFIRvector ofanunknownsystem bedescribed output arecontaminated bynoise. Several recursive TLS(RTLS) algorithms havebeen byh=(h0,,h,, h__ ,TAssumethat bothinput andoutput established andarenormally ofcomputational complexity ofthesystem arecorrupted bywhite Gaussian noise. Then O(M)periteration. Suchalgorithms, including theinverse-theexpectation output isgiven byd(t) =XT(t)h +n(t), powermethod(3), theconjugate-gradient method(4), and wherethenoise-free inputvectoris denotedby theleast squares-like method(5), alsorequire O(M) x(t) =(x(t), x(t -1), .., x(t -M +1))T andtheoutput noise multiplication operations pereigenvector update. Foron- line solving TLSproblems inadaptive FIRfiltering, afast i o RTLSalgorithm isproposed in(6)based ongradient searchinput vector oftheFIRfilter isgiven byi(t) =x(t) +n, (t), fortheRayleigh quotient along Kalmangain vector (7), and itscomputational complexity isO(M)periteration. Note wheren(t) =(n(t), n(t-1),., n(t-M +1))andtheinput thattheRTLSalgorithms in(6)aredependent onfast noisen(t)iswhiteGaussian noisewithvariance a. computation ofKalmangain vector. Itisshownin(8)that __V 1-- ~~~~~~Moreover, theaugmented datavectoriSdefined as computation ofKalmangain vector might bepotentially Moevr h umne aavco sdfnda unstable. Someefficient solution approaches (9)are -(t) =LT(t), d())T e( l)XHereassume thatx(t),(t), developed forovercoming thepotential instability of ann(tarmualyidpdetoechte.