Complexity-performance trade-off of algorithms for combined lattice reduction and QR decomposition

Over the last years, numerous equalization schemes for multiple-input/multiple-output channels have been studied in the literature. New low-complexity approaches based on lattice basis reduction are of special interest, since they achieve the optimum diversity behavior. Although the per-symbol equalization complexity of these schemes is very low, the initial calculation of the required matrices may impose an enormous burden in arithmetic complexity. In this paper, we give a tutorial overview and assess algorithms, which, given the channel matrix, result in the feedforward, feedback, and unimodular matrix required in lattice-reduction-aided decision-feedback equalization or precoding. To this end, via a unified exposition of the Lenstra–Lenstra–Lovász (LLL) algorithm, the LLL with deep insertions, and the reversed Siegel approach similarities and differences of these approaches are enlightened. A modification of the LLL swapping criterion, better matched to the equalization setting, is discussed. It is shown that using lattice-reduction-aided equalization/precoding better performance can be achieved at lower complexity compared to classical equalization or precoding approaches.