Theory and design of optimum FIR compaction filters

The problem of optimum FIR energy compaction filter design for a given number of channels M and a filter order N is considered. The special cases where N<M and N=∞ have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix, respectively. We deal with the more difficult case of M<N<∞. For the two-channel case and for a restricted but important class of random processes, we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle. This also corresponds to the optimal two-channel FIR filter bank that maximizes the coding gain under the traditional quantization noise assumptions. With a minor extension, this can also be used to generate optimal wavelets. For the arbitrary M-channel case, we provide a very efficient suboptimal design method called the window method. The method involves two stages that are associated with the above two special cases. As the order increases, the suboptimality becomes negligible, and the filter converges to the ideal optimal solution. We compare the window method with a previously introduced technique based on linear programming