Abstract

The Paley-Wiener criterion is used to define the universe of functions in the frequency domain for which the impulse responses are causal, i.e., physically realizable in the time domain. By methods similar to those of Paley and Wiener, we prove a kindred result. Specifically, we find a family of functions K_k[ω, α(ω)] so that given a transfer function amplitude αω, if ${\displaystyle {\int }_{-\infty }^{\infty }{K}_k}d\omega$ is finite for some k, there is a .(highly nonunique) transfer phase θ_k(ω) for which the impulse response I_k(t) corresponding to α exp(θ_k), is of the order of exp(kt). A striking corollary is that, if transfer amplitude decays as exp(−|ω|), for any finite γ > 0, then, for any k> 0, there are phase choices for which the corresponding impulse responses are of the order of exp(kt). After normalizing and for large k, these I_k(t) will for practical purposes vanish on the negative taxis.

© 1988 Optical Society of America