A Novel Reduced-complexity Map Equalizer Using Soft-statistics for Decision-feedback Isi Cancellation

A novel version of the reduced-state Bayesian Maximum A Posteriori Probability /Decision-Feedback (MAP/DF) equalizer for ISI channels with long impulse responses is presented. The main feature of the proposed equalizer is that the soft-statistics generated by the MAP receiver are employed to recursively compute a suitable index of the actual reliability of the (soft) decisions feeding the DF filter. Therefore, in the presented equalizer the usual (over-optimistic) assumption of error-free decisions at the input of the feedback filter is relaxed and numerical results supporting the actual effectiveness of the proposed MAP/DF equalizer are provided for the so-called High-BitRate Digital Subscriber Line (HDSL) test-loop # 4. I. MOTIVATIONS OF THE WORK High-bit-rate data transmissions over waveform channels with large delay-spreads as those typically encountered in HDSL environments [8,9] are subject to severe ISI impairments which, in principle, could be well compensated by resorting to receivers based on Maximum-Likelihood Sequence Estimators (MLSEs) or Bayesian MAP Symbol Detectors (MAPSDs). However, the computational complexity of these receivers grows (at least) exponentially with the channel memory length, so that considerable efforts have been undertaken to develop reducedcomplexity versions of the MLSE and MAP equalizers suitable for the application on transmission channels with large delay-spreads [2,3,4,7]. Various versions of reduced-state MLSEs which employ decision-feedback to “short” the channel memory and then lower the resulting receiver complexity are described, for example, in [1,2,3] and references therein. More recently, in [4,7] the decision-feedback approach has been also applied to reduce the complexity of the standard MAPSDs [5 and references therein]; the resulting MAP/DF equalizer reveals a structure similar to the DFE-MLSE of [3], the main difference being that a single feedback-filter is present in the MAP/DF receiver of [4,7] whereas in the DFE-MLSE of [3] each state of the trellis requires a feedback-filter. All the above mentioned reduced-state DF-based equalizers share the common feature to attempt to cancel the tails of the channel ISI via hard decisions and furthermore have been derived under the overoptimistic assumption of error-free decisions feeding the feedback filter; so, they can incur errorpropagation phenomena, specially over channels with non-minimum-phase long impulse responses as those characterizing some typical HDSLs [9]. Starting from the system-modelling of Sect.II and then following a Bayesian approach, in Sect.III of this contribution we present a novel version of the above mentioned reduced-state MAP Hard-DecisionFeedback (MAP/HDF) equalizer of [4,7] where the soft-statistics delivered by the MAPSD are used to enhance the suppression of the ISI tails. More in detail, the distinguishing features of the proposed MAP/Soft-Decision-Feedback (MAP/SDF) equalizer are twofold. First of all, the A Posteriori Probabilities (APPs) of the reduced-state of the ISI channel generated by the MAP receiver are employed to deliver low-delayed “soft” decisions which are utilized in the feedback-branch of the proposed equalizer for a more reliable cancellation of the ISI tails. Secondly, a suitable observation-depending index of the actual reliability of the decisions feeding the feedback-filter is also computed and then used for updating on a per-step basis the branch-metrics of the MAP/SDF equalizer. Therefore, since the proposed equalizer operates without relying on the usual assumption of error-free decisions at the input of the DF filter, its performance tends to be less impaired by error propagation phenomena, specially in data applications at medium SNRs; the numerical results reported in the final Sect.IV for some HDSLtype applications support this conclusion. II. THE SYSTEM MODEL The ISI-impaired noisy sequence {r(n)∈R, n≥1} observed at the output of a baud-rate-sampled whitened matched receiving filter can be modelled by the usual relationship −