Globally optimal two channel FIR orthonormal filter banks adapted to the input signal statistics

We introduce a new approach to adapt a 2-channel FIR orthonormal filter bank to the input second order statistics. The problem is equivalent to optimizing the magnitude squared response F(e/sup jw/)=|H(e/sup jw/)|/sup 2/ of one of the subband filters for maximum energy compaction under the constraint that F(e/sup jw/) is Nyquist(2). The novel algorithm enjoys important advantages that are not present in previous work. First, we can ensure the positivity of F(e/sup jw/) over all frequencies simultaneously with the Nyquist constraint. Second, for a fixed input power spectrum, the resulting filter F/sub opt/(z) is guaranteed to be a global optimum due to the convexity of the new formulation. The optimization problem is expressed as a multi-objective semi-definite programming problem which can be solved efficiently and with great accuracy using interior point methods. Third, the new algorithm is extremely general in the sense that it works for any arbitrary filter order N and any given input power spectrum. Finally, obtaining H/sub opt/(e/sup jw/) from F/sub opt/(e/sup jw/) does not require an additional spectral factorization step.