The optimum discrete scan-type approximation of low-pass type signals bounded by a measure like Kullback-Leibler divergence

Abstract

We present a running approximation of discrete signals by a FIR filter bank that minimizes various worst-case measures of error, simultaneously. We assume that the discrete signal is a sampled version of unknown original band-limited signal that has a main lobe and small side-lobes. To restrict frequency characteristics of signals in this discussion, we impose two restrictions to a set of signals. One is a restriction to a weighted-energy of the Fourier transform of the discrete signal treated actually in the approximation and another is a restriction to a measure like Kullback-Leibler divergence between the initial analog signal and the final discrete approximation signal. Firstly, we show interpolation functions that have an extended band-width and satisfy condition called discrete orthogonality. Secondly, we present a set of signals and a running approximation satisfying all of these conditions of the optimum approximation.