Semi-Infinite Programming in Orthogonal Wavelet Filter Design

Abstract

Quadrature mirror filters can be introduced by studying algebraic properties of discrete input signals viewed as square summable infinite sequences together with linear operators, termed filters, which act linearly through convolution on the input signals. The algebra is readily revealed through operations on complex variable transfer functions which are formally associated with the filters. By using the space of transfer functions, one can define how a signal can be decomposed into two subsidiary signals which can then be combined, using only certain linear mappings all along throughout the entire process. Ideally, the original signal may be recovered, and in this case the term perfect reconstruction is used. In addition to ideal recovery, it is desired to impose orthogonality conditions on the subsidiary signals. These constraints imply that one need only concentrate on the optimality of one of the subsidiary signals to guarantee the optimality of the other. The orthogonality properties are associated with the term quadrature mirror filters.

We describe what optimality can mean within this special structure by making a transition to a description of statistical properties which are reasonably representative of the actual transmission of discrete signals and their recovery after transmission. The concept of coding gain is reviewed, and we show how this objective function may be combined with constraining relations on the coefficients of the (chosen) primary filter necessary to formally guarantee perfect reconstruction. The constraints lead to a nonlinear transformation of the original filter coefficients to variables that appear in an equivalent linear semi-infinite programming problem developed by the second author. We show how an optimal solution in the original filter-variables may be obtained from an LSIP optimal solution by spectral decomposition. Finally, we review some elementary duality-based sensitivity analysis and present some previously published numerical results (by us with other co-authors).