A state space approach to the design of globally optimal FIR energy compaction filters

We introduce a new approach for the least squared optimization of a weighte d FIR filter of arbitrary order N under the constraint that its magnitude s quared response be Nyquist(M). Although the new formulation is general enou gh to cover a wide variety of applications, the focus of the paper is on op timal energy compaction filters. The optimization of such filters has recei ved considerable attention in the past due to the fact that they are the ma in building blocks in the design of principal component filter banks (PCFBs ), The newly proposed method finds the optimum product filter F-opt(z) = H- opt(z)H-opt(z(-1)) corresponding to the compaction filter H-opt(z). By expr essing F(z) in the form D(z) + D(z(-1)), we show that the compaction proble m can be completely parameterized in terms of the state-space realization o f the causal function D(z). For a given input power spectrum, the resulting filter F-opt(z) is guaranteed to be a global optimum solution due to the c onvexity of the new formulation. The new-algorithm is universal in the sens e that it works for any nir, arbitrary filter length N, and any given input power spectrum. Furthermore, additional linear constraints such as wavelet s regularity constraints can be incorporated into the design problem. Final ly, obtaining H-opt(z) from F-opt(z) does not require an additional spectra l factorization step. The minimum-phase spectral factor H-min(a) can be obt ained automatically by relating the state space realization of D-opt(z) to that of H-opt(z).