The Analysis of Certain Nonlinear Feedback Systems with Random Inputs

A method is developed for the determination of the probability density function of the output of a nonlinear feedback system whose input is a random voltage of known statistical properties. The method of analysis is based upon the establishment of a mathematical model of the feedback system in such a way that the output is a Markov process. The transition probabilities of the Markov process are determined from the open-loop nonlinear characteristics of the system. From this model, the closed-loop output probability density function can be determined by the solution of an integral equation or, equivalently, by the solution of a set of simultaneous linear equations. As a consequence of the properties of stationary Markov chains, the same result can also be obtained by a process of successive matrix squaring operations. The method is then applied to a complex nonlinear feedback system, a frequency tracking loop whose function is to follow the center frequency of a narrow-band random signal in the presence of wide-band noise. In addition to the study of this system with a stationary input, a simple extension of the method is made which allows the effect of a particular time-variation of the input statistical properties to be studied. The results of a digital computer study of this system are presented and discussed.