On the amplitude- and frequency-modulation decomposition of signals

Abstract

In general, the problem of determining the amplitude and frequency modulations (AM and FM) of a signal is ill posed because there is an unlimited number of combinations of AM and FM that will generate a given signal. Although Gabor proposed a method for uniquely defining the AM and FM of a signal, namely via the analytic signal, the results obtained are sometimes physically paradoxical. In this paper, four reasonable physical conditions that the calculated AM and FM of a signal should satisfy are proposed. The analytic signal method generally fails to satisfy two of the four conditions. A method utilizing the positive (Cohen–Posch) time-frequency distribution and time-varying coherent demodulation of the signal is given for obtaining an AM and FM that satisfy the four proposed conditions. Contrary to the accepted definition, the instantaneous frequency (i.e., the FM) that satisfies these conditions is generally not the derivative of the phase of the signal. Rather, the phase is separated into two parts, one which gives the instantaneous frequency via differentiation, and the other which can be interpreted either as phase modulation or quadrature amplitude modulation of the signal. Examples are given for synthetic signals and speech, with comparisons to the analytic signal method.