Tightness of a New and Enhanced Semidefinite Relaxation for MIMO Detection

Abstract

In this paper, we consider a fundamental problem in modern digital communications known as multiple-input multiple-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite relaxation (SDR) based algorithms have been proposed to solve the problem in the literature. In this paper, we first show that the conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So in [35]. Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap.

On the Equivalence of Semidifinite Relaxations for MIMO Detection with General Constellations

“…Two remarks on Theorem 1 are in order. First, combining Theorem 1 and [24,Theorem 4.4], we can immediately obtain the following tightness result of (ERSDR2).…”

“…Based on (RSDR), an enhanced SDR for (P) has recently been proposed in [24]. Define the following 3 × 3 matrices By the definition of y in (2), ideally each Yi in (3) must be one of matrices Pj with j = 1, 2, .…”

“…By relaxing the above combinatorial constraints and dropping some redundant constraints, reference [24] proposes the following enhanced SDR for (P):…”

An Enhanced SDR Based Global Algorithm for Nonconvex Complex Quadratic Programs With Signal Processing Applications

“…, n} such that r i > |x i |. This special case has been studied in [15], [23]. This paper studies a more general problem (CQP) (with interval modulus constraints) and can be regarded as a nontrivial extension from the unit-modular case in [15], [23].…”

“…Most of existing algorithms for solving problem (CQP) are approximation algorithms, local optimization algorithms, or other heuristics (e.g., [3], [4], [5], [8], [10], [11], [13], [14], [19], [20], [21]). These algorithms generally cannot guarantee to find the global solutions of problem (CQP), except only for some special cases [9], [10], [22], [23]. A straightforward way of globally solving problem (CQP) is to first reformulate the problem as an equivalent real QP by representing the complex variables by their real and imaginary components and then apply the existing general-purpose global algorithms (e.g., algorithms proposed in [24], [25]) for solving the equivalent real reformulation.…”