Proof of the optimum running approximation of FIR filter banks without using the mini-max theorem in functional analysis

摘要

We show a running approximation of discrete signals {f{nT)} = {f(n/R_0)} {R_0 = a natural number, n = an integer) by a FIR filter bank that minimizes various worst-case measures of error, at the same time. We assume that {f(nr)} is a sampled version of an unknown band-limited original signal f(t) having a main lobe and small side-lobes. The output signal {g(nt)} of the filter bank is a final approximation signal.