Soft-output detection of differentially encoded M-PSK over channels with phase noise

We consider a differentially encoded M-PSK signal transmitted over a channel affected by phase noise. For this problem, we derive the exact maximum a posteriori (MAP) symbol detection algorithm. By analyzing its properties, we demonstrate that it can be implemented by a forward-backward estimator of the phase probability density function, followed by a symbol-by-symbol completion to produce the a posteriori probabilities of the information symbols. To practically implement the forward-backward phase estimator, we propose a couple of schemes with different complexity. The resulting algorithms exhibit an excellent performance and, in one case, only a slight complexity increase with respect to the algorithm which perfectly knows the channel phase. The application of the proposed algorithms to repeat and accumulate codes is assessed in the numerical results.