A generalized BCJR algorithm and its use in turbo synchronization

A generalization of the BCJR algorithm is derived to compute joint posterior probabilities of arbitrary sets of symbols given received data. A special case of the new algorithm that computes pairwise joint probabilities is used to evaluate the second order statistics needed for EM estimation of carrier phase and symbol timing offsets in a digitally modulated waveform. This EM estimator is integrated into a turbo synchronization loop in which soft information is exchanged between the generalized BCJR algorithm (the synchronizer) and an LDPC decoder.