Widely Linear Sphere Decoder in MIMO Systems by Exploiting the Conjugate Symmetry of Linearly Modulated Signals

Abstract

This paper investigates the widely linear processing (WLP) for the detection of circular signals, such as M-ary phase-shift keying (MPSK) signals and M-ary quadrature amplitude modulation (MQAM) signals. First, a unified mathematical model is derived to describe the conjugate symmetry of general MPSK/MQAM signals. In the unified model, a phase-rotation matrix (PRM) is introduced to partition the constellation of multiple-input multiple-output (MIMO) signals into subsets. Signals in a subset share the same PRM. Second, a widely linear receiver is proposed in each subset for MIMO detection. To avoid repetitive WLP in each subset, a widely linear sphere decoder (WLSD) is further proposed for MIMO systems. WLSD transforms the traditional sphere decoder (SD) searching for a true transmitted vector into a shrunk one by searching for the corresponding phase-rotation vector. Finally, the diversity order of WLSD is proven to be more than $N_R-\frac{N_T-1}{2}$ and less than $N_R$ , where $N_T$ (or $N_R$ ) denotes the number of transmitting (or receiving) antennas. Additional performance analysis is also conducted to quantify the signal-to-noise ratio improvement. The complexity analysis reveals that the candidate phase-rotation vectors of WLSD are no more than $(\frac{1}{2})^{N_T}$ of the SD candidates. Simulation results show that the proposed WLSD can achieve quasi-optimal bit error rate performance, while the computational complexity is reduced by more than a half compared with the Schnorr–Euchner sphere decoder.