From Gram–Schmidt Orthogonalization via Sorting and Quantization to Lattice Reduction

Over the last years, numerous equalization schemes for multiple-input/multiple-output (MIMO) channels have been studied in literature. In particular, techniques known from intersymbol-interference channels have been transferred to the MIMO setting, such as linear equalization, decision-feedback equalization (DFE, also known as successive interference cancellation, SIC, and the main ingredient of the BLAST approach), and maximum-likelihood detection, cf. [3, Table E.1]. Besides them, new approaches based on lattice basis reduction, e.g., [16], [11], are of special interest. Using these latticereduction-aided (LRA) techniques, low-complexity equalization achieving the optimum diversity behavior [9] is enabled. In this contribution, the connection of the Gram–Schmidt procedure—well-known from linear algebra for calculating an orthogonal/orthonormal basis for a vector space—to decisionfeedback equalization and to lattice reduction and latticereduction-aided equalization, respectively, is enlightened. It is shown that the operations quantization and sorting play an important role. Their consequences for the universal tool Gram–Schmidt procedure and the connection to the LLL algorithm with deep insertions [8] are explained.