Reliability Distributions of Truncated Max-log-map (MLM) Detectors Applied to ISI Channels

The max-log-map (MLM) receiver is an approximated version of the well-known, Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. The MLM algorithm is attractive due to its implementation simplicity. In practice, sliding-window implementations are preferred; these practical implementations consider truncated signaling neighborhoods around each transmission time instant. In this paper, we consider the binary signaling case. We consider sliding-window MLM receivers, where for any integer m, the MLM detector is truncated to a length-m signaling neighborhood. For any number n of chosen times instants, we derive exact expressions for both i) the joint distribution of the MLM symbol reliabilities, and ii) the joint probability of the erroneous MLM symbol detections. We show that the obtained expressions can be efficiently evaluated using Monte-Carlo techniques. Our proposed method is efficient; the most computationally expensive operation (in each Monte-Carlo trial) is an eigenvalue decomposition of a size 2mn by 2mn matrix. Finally, our proposed method handles various scenarios such as correlated noise distributions, modulation coding, etc.