  In order to effetively suppress intersymbol interference (ISI) at low
complexity, we propose in this paper an approximate maximum likelihood (ML)
decision feedback block equalizer (A-ML-DFBE) for doubly selective
(frequency-selective, time-selective) fading channels. The proposed equalizer
design makes efficient use of the special time-domain representation of the
multipath channels through a matched filter, a sliding window, a Gaussian
approximation, and a decision feedback. The A-ML-DFBE has the following
features: 1) It achieves performance close to maximum likelihood sequence
estimation (MLSE), and significantly outperforms the minimum mean square error
(MMSE) based detectors; 2) It has substantially lower complexity than the
conventional equalizers; 3) It easily realizes the complexity and performance
tradeoff by adjusting the length of the sliding window; 4) It has a simple and
fixed-length feedback filter. The symbol error rate (SER) is derived to
characterize the behaviour of the A-ML-DFBE, and it can also be used to find
the key parameters of the proposed equalizer. In addition, we further prove
that the A-ML-DFBE obtains full multipath diversity.
