Message Passing Algorithms for Phase Noise Tracking Using Tikhonov Mixtures

Phase noise poses a serious challenge for high-speed digital communications systems mainly when going to higher and higher carrier frequencies, such as in satellite communications. Traditionally, phase noise estimation was performed separately from the decoding task and it was shown, recently, that there is much to be gained from joint estimation and decoding, particularly when using LDPC (low-density parity check)/turbo codes. However, jointly estimating phase noise and decoding is a very complex and computationally demanding task. In this paper, we propose several algorithms based on the sum and product algorithm (SPA) for low complexity joint decoding and estimation of coded information in strong phase noise channels. These algorithms are based on a novel approximation of SPA messages as Tikhonov mixtures of a given order. Since mixture-based Bayesian inference such as SPA, creates an exponential increase in mixture order for consecutive messages, a mixture reduction scheme is a must. Therefore, in this paper, we propose a low complexity mixture reduction algorithm, which provably satisfies an upper bound on the Kullback Leibler (KL) divergence between the mixture and the reduced mixture. We then reduce the complexity even further, including limiting the model order and reducing the clustering effort to simple component selection. As an extreme case, it is even possible to reduce the number of modes to one. We show the relation between the simplified algorithm to the phase locked loop (PLL). Finally, we show simulation results and complexity analysis for the proposed algorithms, which show superior performance over other state of the art low complexity algorithms.