Optimal reception of signals having a p.n. subcarrier to phase modulate a transmitted carrier

A method of synthesis for optimum receiver for signals having a pseudorandom noise (p.n.) sequence subcarrier to phase modulate a transmitted carrier is proposed. The p.n. sequence, unknown at the receiving end, is assumed to be represented by a discrete (binary) Markov process while the phase of the modulated signal is modelled as a continuous Markov process. Optimum receiver structure and corresponding general measures of system performance are derived under the Bayes criterion of minimum average risk by reducing the problem to that of estimation under uncertainty as to the binary levels of the p.n. sequence and then probabilisticallyweighting each component of the estimate by the a posteriori probability densities of the p.n. sequence given all past observations. It has been shown that implementation of such a receiver involves a computer with ever growing memory to store all past statistics concerning the conditional means and conditional covariances of the estimate as well as the a posteriori probability densities of the p.n. sequence at the expense of any threshold detector which is used in conventional receivers.