Symbol Error Rate Performance of Box-relaxation Decoders in Massive MIMO

The maximum-likelihood (ML) decoder for symbol detection in large multiple-input multiple-output wireless communication systems is typically computationally prohibitive. In this paper, we study a popular and practical alternative, namely the Box-relaxation optimization (BRO) decoder, which is a natural convex relaxation of the ML. For iid real Gaussian channels with additive Gaussian noise, we obtain exact asymptotic expressions for the symbol error rate (SER) of the BRO. The formulas are particularly simple, they yield useful insights, and they allow accurate comparisons to the matched-filter bound (MFB) and to the zero-forcing decoder. For BPSK signals the SER performance of the BRO is within 3dB of the MFB for square systems, and it approaches the MFB as the number of receive antennas grows large compared to the number of transmit antennas. Our analysis further characterizes the empirical density function of the solution of the BRO, and shows that error events for any fixed number of symbols are asymptotically independent. The fundamental tool behind the analysis is the convex Gaussian min-max theorem.