The use of the method of maximum likelihood in estimating continuous-modulated intelligence which has been corrupted by noise

A signal is received in the time interval (t ¿ T ¿ ¿ ¿ t). It is known that this signal is composed of noise plus intelligence a(t) which is statistical in nature and which has been modulated in some known way. Assuming that both intelligence and noise are Gaussian (although not necessarily stationary) time series, the analog of the classical maximum-likelihood estimate for ¿(t) is derived. The advantage of this approach is that it can handle arbitrary types of modulation. For unmodulated stationary intelligence and stationary noise, the solution reduces to that of Zadeh and Ragazzini. In the general case, the optimum estimate is given as the solution of a pair of integral equations. The amplitude-modulated case is treated in some detail. The application of the maximum-likelihood technique to problems involving arbitrary modulation was first suggested, as far as the author is aware, by F. W. Lehan and R. J. Parks of the Jet Propulsion Laboratory.