Precoding for blocking signalling and shaped signal sets

Even on channels with severe intersymbol interference, the decision feedback equalizer (DFE) performs within 9 dB of channel capacity (at error rates of 10^-6) if one assumes no errors in the feedback path (ideal DFE). Using coset codes, with gains in the range of 5 dB, the ideal DFE promises transmission at rates within 4 dB of channel capacity. But this again is subject to the assumption of no errors in the feedback path of the DFE. In any actual implementation of the DFE, errors do occur in the feedback path, and these errors tend to propagate. Forney and Calderbank (1989) suggest the use of a modulo arithmetic precoder (Tomlinson precoder) at the transmitter to replace the DFE at the receiver and avoid DFE error propagation. Their precoder avoids error propagation, but it increases the average power of any shaped signal constellation fed into it. This increase in power negates the shaping gain that the constellation has at the input to the precoder. The precoder is extended to multidimensional systems, and how it can be designed to preserve shaping gains of shaped signal constellations is shown