BER analysis of regularized least squares for BPSK recovery

This paper investigates the problem of recovering an n-dimensional BPSK signal x_0 ∈ {-1, 1}^n from m-dimensional measurement vector y = Ax+z, where A and z are assumed to be Gaussian with iid entries. We consider two variants of decoders based on the regularized least squares followed by hard-thresholding: the case where the convex relaxation is from {-1, 1}^n to ℝ^n and the box constrained case where the relaxation is to [-1, 1]^n. For both cases, we derive an exact expression of the bit error probability when n and m grow simultaneously large at a fixed ratio. For the box constrained case, we show that there exists a critical value of the SNR, above which the optimal regularizer is zero. On the other side, the regularization can further improve the performance of the box relaxation at low to moderate SNR regimes. We also prove that the optimal regularizer in the bit error rate sense for the unboxed case is nothing but the MMSE detector

The BOX-LASSO with application to GSSK modulation in massive MIMO systems

The BOX-LASSO is a variant of the popular LASSO that includes an additional box-constraint. We propose its use as a decoder in modern Multiple Input Multiple Output (MIMO) communication systems with modulation methods such as the Generalized Space Shift Keying (GSSK) modulation, which produces constellation vectors that are inherently sparse and with bounded elements. In that direction, we prove novel explicit asymptotic characterizations of the squared-error and of the per-element error rate of the BOX-LASSO, under iid Gaussian measurements. In particular, the theoretical predictions can be used to quantify the improved performance of the BOX-LASSO, when compared to the previously used standard LASSO. We include simulation results that validate both these premises and our theoretical predictions