On the LLR Metrics for DPSK Modulations Over Two-Symbol Observation Intervals for the Flat Rician Fading Channel

We consider the differential phase-shift keying modulation over the flat Rician fading channel with perfect channel model information (PCMI), and derive the two-symbol-observation-interval log-likelihood ratio (LLR) metric from first principles. The work generalizes the previous LLR derivations in [16]-[19], and demonstrates the necessity of the knowledge of the statistical channel model in the exact LLR computation. Assuming the channel is slowly time-varying, we also propose a simple approximate LLR metric based on the generalized likelihood ratio test (GLRT), which requires only the information of the noise spectral density. The advantage of the newly derived PCMI-LLR metric over the approximate metric and the other LLR metrics in the literature is explained from the information-theoretic perspective via the generalized mutual information. Computer simulations are also provided to demonstrate the superiority of the PCMI-LLR metric in both error rate performance and convergence speed for iterative decoding of turbo and low-density parity-check codes in practical systems. This paper aims to present the fundamental principles of LLR computation, emphasizing the importance of selecting the appropriate metric for iterative decoding over fading channels.