On the Sphere Decoding Complexity of STBCs for Asymmetric MIMO Systems

TL;DR

This paper investigates the sphere decoding complexity of space-time block codes in asymmetric MIMO systems, revealing conditions under which the decoding matrix is rank-deficient and analyzing how this affects decoding complexity.

Contribution

It clarifies when the R matrix in sphere decoding is full-ranked, identifies rank deficiency in high-rate codes for asymmetric MIMO, and derives complexity bounds based on antenna configurations.

Findings

Full-rank R matrix only when receive antennas ≥ transmit antennas.

High-rate codes often have rank-deficient R in asymmetric MIMO.

Decoding complexity decreases as the number of receive antennas increases.

Abstract

In the landmark paper by Hassibi and Hochwald, it is claimed without proof that the upper triangular matrix R encountered during the sphere decoding of any linear dispersion code is full-ranked whenever the rate of the code is less than the minimum of the number of transmit and receive antennas. In this paper, we show that this claim is true only when the number of receive antennas is at least as much as the number of transmit antennas. We also show that all known families of high rate (rate greater than 1 complex symbol per channel use) multigroup ML decodable codes have rank-deficient R matrix even when the criterion on rate is satisfied, and that this rank-deficiency problem arises only in asymmetric MIMO with number of receive antennas less than the number of transmit antennas. Unlike the codes with full-rank R matrix, the average sphere decoding complexity of the STBCs whose R matrix is rank-deficient is polynomial in the constellation size. We derive the sphere decoding complexity of most of the known high rate multigroup ML decodable codes, and show that for each code, the complexity is a decreasing function of the number of receive antennas.