Finite-Length MMSE Tomlinson–Harashima Precoding for Frequency Selective Vector Channels

We design Tomlinson-Harashima precoding for decentralized receivers and frequency-selective channels based on the minimum mean square error criterion, where the feedforward filter is restricted to have finite length. Contrary to most other publications on Tomlinson-Harashima precoding which rely on solutions for decision feedback equalization to find the corresponding precoding filters in a heuristic manner, we deduce the optimization for Tomlinson-Harashima precoding from the optimization for the linear minimum mean square error transmit filter. Thereby, we include the precoding order explicitly in the problem formulation and thus obtain the precoding filter solutions, together with the algorithms to compute the latency time, i.e., the time difference between application of the precoder at the transmitter and detection at the receiver and the precoding order from a single optimization. Since the algorithm for THP filter computation resulting from the optimization has a high computational complexity, we present an alternative algorithm to compute the Tomlinson-Harashima precoding filters based on a Cholesky factorization with symmetric permutation, resulting in an order of complexity that is the same as for the computation of the linear transmit filters. The simulations reveal that the latency time optimization can be omitted without performance degradation for most practical channel models, i.e., the latency time can be chosen to be the order of the feedforward filter