A nonlinear optimum-detection problem. I. Theory

An approximate log-likelihood ratio for detecting a deterministic signal in linear and nonlinear Gaussian noise is derived. The new aspect of this detection problem is the inclusion of a nonlinear form of Gaussian noise and a certain interaction between the signal and the linear noise, such as modulation of the signal by the noise. The derived log-likelihood ratio consists of a modified version of the classical correlation detector and a new processor where the latter is an order of magnitude smaller than the former. In the absence of the signal-noise interaction, the modification consists of subtracting the estimate of the nonlinear noise from the input (to the correlation detector); the nonlinear processor is a quadratic correlation processor where the correlation is not against the (transformed) signal but against the difference of two estimates of the nonlinear noise under the two hypotheses (signal-presence and signal-absence). Presence of the signal-noise interaction introduces further modification and complication on both processors