Nontrivial analytic signals with positive instantaneous frequency and band-limited amplitude

Abstract

Questions have previously been raised about the existence of an analytic signal with positive instantaneous frequency when the form of the analytic signal is prescribed. Here, it is shown that the complex function a(t)exp[j(/spl omega//sub 0/t+m(t))] is an analytic signal when m(t) is a real periodic function and a(t) is a band-limited real function with the maximum bandwidth depending on /spl omega//sub 0/ and the fundamental frequency of m(t). That implies as a special case m(t) which is simultaneously a periodic and piecewise polynomial. Positivity of the instantaneous frequency is simply obtained by requiring that the absolute value of the first derivative of m(t) is smaller than /spl omega//sub 0/.