Soft-Output Decoding of Rotationally Invariant Codes Over Channels with Phase Noise

We consider rotationally invariant (RI) trellis-coded modulations (TCMs) transmitted over channels affected by phase noise. For this problem, we use the framework based on factor graphs (FGs) and the sum-product algorithm (SPA) to derive the exact maximum a posteriori (MAP) symbol detection algorithm. We demonstrate that it can be implemented by a reduced number of forward-backward estimators of the phase probability density function, followed by a symbol-by-symbol completion to compute the a posteriori probabilities of the information symbols. To practically implement the forward-backward phase estimators, we propose a couple of schemes with different complexity. The resulting algorithms exhibit an excellent performance and a limited complexity. The proposed soft-output algorithms can be also used in iterative decoding schemes for concatenated codes employing RI inner components