A general likelihood-ratio formula for random signals in Gaussian noise

It is shown that the likelihood ratio for the detection of a random, not necessarily Gaussian, signal in additive white Gaussian noise has the same form as that for a known signal in white Gaussian noise. The role of the known signal is played by the casual least-squares estimate of the signal from the observations. However, the "correlation" integral has to be interpreted in a special sense as an Itô stochastic integral. It will be shown that the formula includes all known explicit formulas for signals in white Gaussian noise. However, and more important, the formula suggests an "estimator-correlator" philosophy for engineering approximation of the optimum receiver. Some extensions of the above result are also discussed, e.g., additive finite-variance, not necessarily Gaussian, noise plus a white Gaussian noise component. Purely colored Gaussian noise can be treated if whitening filters can be specified. The analog implementation of Itô integrals is briefly discussed. The proofs of the formulas are based on the concept of an innovation process, which has been useful in certain related problems of linear and nonlinear least-squares estimation, and on the concept of covariance factorization.