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Thus, the coefficient solutions are related not only to the system under identification but also to the FB which underlies the proposed STR. For specific values of its parameters the new STR degenerates into the GSD-STR, the transform domain LMS (TD-LMS), 0; the transversal full-band LMS (FB-LMS) FIR filter. Introduction: The generalised subband decomposition (GSD)-based adaptive FIR STRucture (GSD-STR) has been successfully applied in a number of applications, including system identification, adaptive line enhancement, and adaptive channel equalisation [I]. The GSD- STR is based upon the GSD model [2] K-1 ."=O zyxwvutsrqponmlkjihgfedcbaZY S(Z) = I,(z)G,(#) = z;T$G,(#) = G;(Z)T,I~Z, (1) for the FIR system S(z) of length P under modelling, denoted in the time domain as s[n], 0 5 n <e or s In (l), (.)T denotes the vector-matrix transpose, ZKA [l z-' . . . z-(~-')]: TK an arbitrav non-singular K x K matrix, f,(z) an interpolator filter with coeffi- cients given by the rth column of TKI and Cdz) the vector of the K generalised polyphase components (GPCs) of S(z) given by GAz)=T,SK(z), or explicitly by [Go(z) . . . GK-,(z)lT= TK [So(z) . . . S~-l(z)]? with S,.(z) the polyphase components (PCs) s,[n] A-s[Kn + r], 0 p r<K, of s[n] in the Z domain. The coefficients s,,,, g,, of S,(z), G,(z), 0 5 r< K, are related as [so . . . [gr,clKx LP/KJ TK[sr,clKx[P/Kj (2) where [g,,] is a K x LP/K] matrix with the element g,,, in its rth row and cth column and accordingly for [ s ~ , ~ ] . The GSD-STR [l] results from the rightmost part of (1) and is given by the lower section of the diagram of Fig. 1. For the identification of S(z), adaptive weights are used in lieu of g,,, 0 5 r < K, 0 5 c< lP/K]. The transform domain LMS (TD-LMS) [3] and the transversal full-band LMS (FB-LMS) FIR filter [4] result as special cases of the GSD-STR [I]. Specifically, for K=P the GSD-STR yields the TD-LMS algorithm. It then employs one adaptive coefficient per band applies the transformation T;'' to the tap-input-vector zyxwvutsrqpo xp[n] 2 [x[n] . . x[n - P + I]' and identifies s through its transform T,s. Moving to the other extreme when K= 1, the GSD-STR degenerates into a transversal FB-LMS FIR filter of length J! ELECTRONICS LE7TERS 24th July 2003 Vol. 39 T.1' Fig. 1 Proposed STR (upper section) and GSD-STR (lower section) In this Letter, an extension of the GSD-STR is presented and a new STR is designed based upon the DFT algorithm [5]. The proposed STR identifies S(z) through the DFTs, which form the coefficient solutions, that correspond to the subband sequences resulting from the decom- position of the system s[n] or its blocks using the analysis section of an FIR perfect reconstruction (PR) filter bank (FB). It is noted that this decomposition is hypothetical, since the FB is only used to define a sparse transformation matrix QK, which when incorporated in the GSD- STR yields the extension to the new STR. If no decomposition is performed OK = IK and hence the proposed STR degenerates into the GSD-STR [ 11 and therefore is a further generalisation of the TD-LMS [3] and the transversal FB-LMS FIR filter [4]. Proposed structure: The proposed STR is based upon the DFT algorithm [5]. For this algorithm to be applicable here, the choice T,= [(e-"2"'"'c)r,,] is made, namely TK is the DFT matrix. Accord- ing then to (2), for modelling S(z) the coefficient matrix [g,,,] is obtained by DFT transforming the columns of [s?,~], each column being a block of K consecutive samples of s[n]. The DFT algorithm [5] has as follows: Step 1: Consider a signal of length K multiple of M, denoted in vector form xK. Analyse its samples x[n], 0 p n < K, with an M-band cosine modulated (CM) PR FB {Hi(z), Fi(z), 0 p i<M} [6] of filters of length L, to obtain the subband signals x,[n] A {hi[n] *x[n]}JM, Osn, OsiiM, oflength L(K+L-I)/M]=K/M+L(L-l)/MJ. Step 2: Form the vectors 1,, 0 5 i<M, of length KIM as .fi[n] 4 E; ' !; ' xi[n + lK/h.il, 0 5 n < K/M, where nb = L(K/M+L(L - l)/MJ) x Step 3: Take the KIM-point X,LDFT{1,}, Osi<M, also denoted as Step 4: Form the K-point X,~DFT{XK} as X[k] = c%il C[k]F,[k] x X, M/K] = 1 + [[(L - I)/MJM/K]. Xj[k], 0 5 k< K/M. 0 5 k< K, where (k)K,Mg k modulo(K/M), C[k] , and 5 [k] = $(z) for z = e'(Za/Qk. ei(2a/K)(L- I)k Defining F,c[k] C[k]F,[k], XK can be expressed as 4F:d "' 4F:M-I) ] [ 7 ] xK=[ ; (3 1 [gr,cl = QK ; j (4) [ Xb-I.0 ". XM-l,LP,K,-, KXLP,K, 4G-1,d ... ~(F;-I,M-I) XM-I K ~ I where within the K x K matrix in (3) denoted in the sequel as RK, d(F:j, 0 p ij<M, is a diagonal matrix with diagonal the ith block of F,c[k] of length KIM. If (3) is used to yield the K-point LP/KJ DFTs of (2), the [g,,] appearing therein can be expressed as XO,O ' ' ' XO, LP/KJ -1 and compactly as [gr,c] = C&[X,,], 0 5 r < K, 0 5 r' < M, 0 5 c < LP/K]. The M vectors Xr,,c of length KIM that form the cth column of [X,,], correspond to the vectors appearing at the right-hand side of (3). No. 75 1157