The Limiting Poisson Law of Massive MIMO Detection with Box Relaxation

TL;DR

This paper analyzes the asymptotic behavior of the box-relaxation decoder in massive MIMO systems, revealing a limiting Poisson distribution for decoding errors and a phase transition in perfect recovery probability.

Contribution

It establishes the limiting Poisson law for the number of decoding errors and characterizes the phase transition threshold for perfect recovery in massive MIMO detection.

Findings

Decoding errors follow a Poisson distribution asymptotically.

A phase transition in perfect recovery probability is identified.

Numerical results match theoretical predictions even at moderate system sizes.

Abstract

Estimating a binary vector from noisy linear measurements is a prototypical problem for MIMO systems. A popular algorithm, called the box-relaxation decoder, estimates the target signal by solving a least squares problem with convex constraints. This paper shows that the performance of the algorithm, measured by the number of incorrectly-decoded bits, has a limiting Poisson law. This occurs when the sampling ratio and noise variance, two key parameters of the problem, follow certain scalings as the system dimension grows. Moreover, at a well-defined threshold, the probability of perfect recovery is shown to undergo a phase transition that can be characterized by the Gumbel distribution. Numerical simulations corroborate these theoretical predictions, showing that they match the actual performance of the algorithm even in moderate system dimensions.