In this chapter, we introduce the maximum a posteriori probability (MAP) symbol detection over static ISI channels. The MAP symbol detector finds the symbols that maximize the a posteriori probabilities. A computationally efficient algorithm, called the BCJR (acronym from the authors names: Bahl, Cocke, Jelinek, and Raviv) algorithm (Bahl et al., 1974), is introduced for the MAP detection.

In the MAP detection, the CIR should be known. There are various approaches suitable for finding the channel estimation. An approach based on the ML criterion is introduced in this chapter with the expectation and maximization (EM) algorithm (Dempster, Laird, and Rubin, 1977). The EM algorithm is an iterative numerical method employed to solve the ML problem. The EM-based channel estimation is closely related to the MAP symbol detection. The resulting receiver employing the EM-based channel estimation becomes an iterative receiver in which the channel estimation and detection are carried out iteratively. This iterative receiver is also studied in this chapter.

5.1 MAP Symbol Detection Over ISI Channels

In this section, we study MAP symbol detection and the BCJR algorithm used to perform the MAP symbol detection.

5.1.1 MAP symbol detection

Recall the received signal model over a static ISI channel. The received signal at time l is given by

where { h _p} is the CIR of length P