A minimum mean-squared error interpretation of residual ISI channel shortening for discrete multitone transceivers

P.J.W. Melsa et al. (see IEEE Transactions on Communications, vol.44, p.1662-72, 1996) presented a channel shortening technique for discrete multitone transceivers that reduces intersymbol interference (ISI) by forcing the effective channel's impulse response to lie within a window of v+1 consecutive samples. G. Arslan et al. (see IEEE International Conference on Acoustic Speech and Signal Processing (ICASSP), p.2965-8, 2000) claim that although this method is intuitive, no previous study has been made on its optimality. They comment on its optimality by simulation. We demonstrate that Melsa's approach is in fact theoretically equivalent to a minimum mean-squared error (MMSE) solution to the channel-shortening problem. As a corollary to this we are afforded an insight into MMSE channel shortening as originally proposed by D.D. Falconer and F.R. Magee (see Bell Systems Technical Journal, p.1541-62, 1973). Previously, it has not been intuitive as to why the desired impulse response (DIR) should be made adaptive in this approach. Our result demonstrates that allowing DIR adaptation achieves a minimisation of the effective impulse response energy outside the desired window of v samples.