Theory and design of optimum FIR compaction filters

Energy compaction filters have attracted considerable attention due in part, to the fact that they are the building blocks of optimal orthonormal (paraunitary) filter banks. In this paper we introduce some new design techniques for optimum M-channel FIR compaction filters for a given input power spectrum. Some properties of the optimum FIR compaction filters and the corresponding maximum compaction gains are also derived. For the design part, a modification of the well-known linear programming technique is considered. We also consider multistage (WIR) designs of compaction filters. A new, efficient design method called the window method is then introduced. The method generates M-channel FIR compaction filters for any given power spectrum. Although it is suboptimal, no optimization tools of any kind are involved and the algorithm terminates in a finite number of elementary steps. As the filter order increases, the window method produces compaction gains that are very close to the optimal ones. We give a necessary condition for a compaction filter to be optimum and provide some bounds on the maximum compaction gains. Finally we propose an analyical method for the two-channel case which finds the optimum FIR compaction filters for a class of random processes. Mm m DTIC QUALITY IFSPICTED 2 $ Work supported in parts by Office of Naval Research grant N00014-93-1-0231, Tektronix, Inc., and Rockwell Intl. DISTRIBUTION STATEMENT A Approved for public release; Distribution Unlimited /. INTRODUCTION Energy compaction filters have attracted a great deal of attention due in part, to the fact that they are the building blocks of optimal orthonormal (paraunitary) filter banks [1, 2, 3, 4]. This connection is made for the case where the filters are allowed to be ideal. More recently a number of authors have considered the FIR energy compaction problem for the two-channel case [1, 5, 6, 7, 8, 9, 10] and for the M-channel case [11]. The FIR compaction filters have applications in many areas including data compression, signal analysis, signal modeling, and data transmission [1, 3, 12]. An M-channel FIR compaction filter can be considered as one of M filters of a maximally decimated M-channel FIR orthonormal filter bank. Hence, one can view the problem as compaction of most of the signal energy into one channel of an orthonormal filter bank. In this paper we consider some new design techniques for optimum FIR compaction filters. We give analytical solutions in the two-channel case for a class of random processes. Some properties of optimum FIR compaction filters and corresponding gains are also considered. Detailed outline is provided in Sec. 1.4. 1.1. Notations and Terminology 1. Bold faced upper and lower case letters represent matrices and vectors respectively. 2. X(z) and X(e?) stand for z-transform, and Fourier transform, respectively of a sequence x(n). The notation X(z) denotes the ^-transform of x'(-n) where * stands for complex conjugation. If x(n) is real, then X(z) = X{z~). Notice that X(z) = X*(l/z*), and the Fourier transform of x*(-n) is X'{e>). 3. The symbols I M and t M denote M-fold decimation and expansion as defined in [13]. The notation X(Z)\IM denotes the z-transform of the decimated sequence x(Mn). 4. Nyquist(M) property. A sequence x(n) is said to be Nyquist(M) if x{Mn) = 6(n) or equivalent^ X(Z)\IM = 1This can be rewritten in the form [13]: M-l £ X(zW) = M (!) *=o where W = e~^l. In an M-channel orthonormal filter bank {Hk(z)}, each |ff*(e^)| 2 is Nyquist(M). So the integer M is often referred to as the number of channels. 5. The notation xL(n) stands for a periodic sequence with periodicity L. If there is a reference to a finite sequence x(n) as well, then it is to be understood that xL(n) is the periodical expansion of x(n) with period L, i.e., xL(n) = ES-oo ( + ^ The Fourier series coefficients of xL(n) is denoted by XL(k). 6. For L a multiple of M, a periodic sequence xL(n) is said to be Nyquist(M) if CO xL(Mn) = SK(n)= £ S(n + Ki) (2) where K = L/M. The equivalent form of this property in terms of the Fourier series coefficients XL{k) is M-l Y4XL{k + iK)=M, k = 0,...,K-l (3) t=0 (see Lemma 2 in Sec. III). 7. Positive definite sequences. Let a sequence {x(n),n = 0,.... AT} be given and let P be the Hermitian Toeplitz matrix whose first row is [x(fi) x{l) ... x(N)], The sequence {x(n),n = 0,...,N} is called positive (negative) definite or semidefmite if P is positive (negative) definite or semidefmite respectively. Let [a(0) a(l) ... a(N)] denote the corresponding eigenvector. Then the filter A[z) = £n=0 a{n)z~ n will be called a maximal eigenfilter of P. If we consider the minimum eigenvalue instead, we will call the corresponding filter a minimal eigenfilter of P. 1.2. The FIR energy compaction problem A filter H(z) of order TV will be called a valid compaction filter for the pair (M, A) if the product G(z) = H(z)H(z) is Nyquist(M). We will refer to G{z) as the product filter corresponding to H{z). Conversely, G(z) is the product filter of a valid compaction filter for the pair (M,N) if it is of symmetric order N, that is G(z) = Yln=-N 9()~ and ifc satisfies the following conditions: g(Mn) = 6(n) and G(e) > 0. (4) Now consider Fig. 1 where H(z) is an FIR filter of order N applied to an input x(n) which is a zero-mean WSS random process with the power spectral density S„(e*). The output of the filter is decimated by M to produce y{n). The optimum FIR energy compaction problem is to find a valid compaction filter H(z) for the pair (M, N) such that the variance o\ of y(n) is maximized. Since decimation of a WSS process does not alter its variance, we have