A maximum a posteriori (MAP) equalizer is derived in this paper by formulating the forward/backward recursion MAP algorithm (i.e. the BCJR algorithm) on a parallel-trellis representation that was previously proposed for equalizing sparse multipath channels. Several enhancements are suggested to further improve the performance and/or reduce the complexity of the resultant MAP equalizer. Results show that the parallel-trellis MAP equalizers utilizing 2-state trellises are the predominant structures applicable to minimum-phase sparse multipath channels when binary phase shift keying (BPSK) modulation is assumed. However, for nonminimum-phase sparse multipath channels, prefilterng is generally indispensable and Is accomplished using the feedforward filter (FFF) of a nonuniformly-spaced decision feedback equalizer (NU-DFE) optimized under the minimum mean square error (MMSE) criterion, which also preserves the sparseness of the resultant minimum-phase impulse response. The simplicity of the 2-state, parallel-trellis structure leads to a low computational load that is comparable to those of uniformly-spaced and nonuniformly-spaced tapped-delay-line (TDL) equalizers, but attains much superior performance over the latter types of equalizers. Moreover, iterations can be added to improve the accuracy of the inter-trellis intersymbol interference (ISI) estimates and the residual ISI estimates due to channel truncation, and are especially effective in mitigating the precursor ISI terms due to non-ideal prefiltering.
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