Limits on discrete modulated signals

We develop theorems of a general nature that apply to the analysis of AM-FM signals of the form a(m) exp [j (m)] or a(m) cos [ (m)] and to their behavior both in linear systems and in simple nonlinear systems comprised of products of linear elements. Such product-systems include interesting nonlinear demodulation operators, such as the Teager–Kaiser operator. Expressions for the approximate system responses to AM-FM signals are derived by making an analogy to the eigenfunction interpretation of sinusoids in linear systems; for the case of sinusoidal signals, the approximations are exact. These expressions are collectively called quasieigenfunction approximations (QEA’s). For nonsinusoidal AM-FM signals, the approximations have errors that are tightly bounded by functionals that express the smoothness of the AM and FM information signals and the durations of the involved system impulse responses. The bounds are independent of the bandwidths of the AM and FM functions. Two general applications are considered. First, the approximations are found to be useful for analyzing discretetime nonlinear energy operators, including the Teager–Kaiser operator. Next, the approximation theorems lead to the selection of an optimal class of bandpass filters for use in a discrete multiband AM-FM demodulation system. The filter class selected is optimal in the sense of achieving the lower bound of a novel discrete uncertainty principle.