Spectral Density of a Nonlinear Function of a Gaussian Process

A band-limited gaussian process having a rectangular power spectrum is assumed for some purposes to be an adequate approximation for several classes of modulating signals encountered in phase modulation (PM) systems.1 However, in an actual implementation of a PM system, the modulating signal passes through circuitry which saturates when the signal rises above a fixed level. This clipping (i.e., limiting) level is usually adjusted fairly high (nominally at four times the rms of the modulating signal) and then ignored in any subsequent analysis of the system. Consequently, the objective of this study is to determine the qualitative effect of hard-limiting the modulating signal in a PM system. From a mathematical viewpoint, we can obtain an understanding of the preceding question by investigating the following problem: Find the spectral density of a sinusoidal wave which is phase modulated by a function G(XT) of the stationary gaussian process XT. Of course, this version of the problem can also be viewed as finding the spectral density of a (composite) nonlinear function of a gaussian process; a problem originally studied by S. 0. Rice,2 D. Middleton 3 and W. R. Bennett. 4 In fact, we do use their approach (representing the nonlinearity in terms of a transform) to derive an expression which is essentially the starting point of our analysis. However, using our relation avoids some of the complexity associated with the trans1025