BER Analysis of the box relaxation for BPSK Signal Recovery

TL;DR

This paper derives an exact error probability expression for the box relaxation method in BPSK signal recovery, showing its performance approaches the matched filter bound at high SNR and large system sizes.

Contribution

It provides a precise analysis of the box relaxation technique's error probability in high-dimensional BPSK recovery, including asymptotic independence of bit errors.

Findings

Error probability closely approaches the matched filter bound at high SNR.

Error events of different bits become independent as system size grows.

Performance improves with increasing measurement ratio m/n.

Abstract

We study the problem of recovering an $n$-dimensional vector of $\{\pm1\}^n$ (BPSK) signals from $m$ noise corrupted measurements $\mathbf{y}=\mathbf{A}\mathbf{x}_0+\mathbf{z}$. In particular, we consider the box relaxation method which relaxes the discrete set $\{\pm1\}^n$ to the convex set $[-1,1]^n$ to obtain a convex optimization algorithm followed by hard thresholding. When the noise $\mathbf{z}$ and measurement matrix $\mathbf{A}$ have iid standard normal entries, we obtain an exact expression for the bit-wise probability of error $P_e$ in the limit of $n$ and $m$ growing and $\frac{m}{n}$ fixed. At high SNR our result shows that the $P_e$ of box relaxation is within 3dB of the matched filter bound MFB for square systems, and that it approaches MFB as $m $ grows large compared to $n$. Our results also indicates that as $m,n\rightarrow\infty$, for any fixed set of size $k$, the error events of the corresponding $k$ bits in the box relaxation method are independent.