Evaluation of likelihood functions for Gaussian signals

State variable techniques are used to derive new expressions for the likelihood function for Gaussian signals corrupted by additive Gaussian noise. The continuous time case is obtained as a limit of the discrete time case. The likelihood function is expressed in terms of the conditional expectation of the signal given only past and present observations, multipliers, and integrators (adders). Thus, the likelihood function can be generated in real time using a physically realizable system. Time-varying finite-dimensional Markov models are also discussed as they lead to a direct mechanization for the required conditional expectation. A simple example of a multipath communication system is discussed and an explicit mechanization indicated.