The Design of Optimal Convolutional Filters via Linear Programming

Computational algorithms are given for the design of optimal, finite-length, convolutional filters with finite-length input sequences. Design techniques are developed for minimum-weighted-mean-square-error filters (MWMSE), for minimum-weighted-absolute-error filters (MWAE), and for filters which minimize the maximum output error (minimax). It is shown that the coefficients of the MWAE and minimax filters can be obtained by using standard linear programming methods. Next, the problem of developing a filter whose function is to "sharpen" a particular input waveform is considered. The filter input sequence is assumed to be derived from a Ricker wavelet of the velocity type and the desired output is the Dirac delta function. Convolutional filters are developed for this problem using each of the three performance criteria described above. The output sequences of each of the three optimal filters are discussed. It is shown that the minimax filter gives significantly better discrimination than can be obtained from either the MWAE or MWMSE filters.