The autocorrelation function of the output of a nonlinear angle modulator

The calculation of the autocorrelation function of the output of a randomly frequency-modulated oscillator with a non-linear tuning characteristic is similar to the calculation of the characteristic function of a functional of the form
\int_{I}h(t', \tau) V(y(t')) dt'
. This functional has arisen in a number of other stochastic problems, and so the techniques which have been applied to these problems may be used to obtain results for the nonlinear frequency-modulator. For example, if as a first approach to a nonlinear frequency-modulation a linear plus square-law modulator characteristic and a Gaussian modulating signal are assumed, a technique similar to that used in the analysis of the square-law detector by Kac and Siegert, and Emerson, may be applied. Then an expression for the autocorrelation function of the oscillator output is obtained in terms of two finite Wiener-Hopf integral equations. For an important class of kernels, these equations may be solved and the autocorrelation function evaluated. As an example, the calculation is carried out for a low-pass modulating signal. Results for phase modulation are included also.