Effects of Some Lattice Reductions on the Success Probability of the Zero-Forcing Decoder

TL;DR

This paper investigates how different lattice reduction techniques, including LLL, SQRD, and V-BLAST, affect the success probability of the zero-forcing decoder in various dimensions, revealing that LLL can both improve and decrease success probability depending on the case.

Contribution

It provides a rigorous analysis of the effects of SQRD, V-BLAST, and LLL reductions on the zero-forcing decoder's success probability, including conditions for improvement and deterioration.

Findings

SQRD and V-BLAST do not affect success probability

LLL reduction improves success probability for n=2

LLL reduction can decrease success probability for n≥3

Abstract

Zero-forcing (ZF) decoder is a commonly used approximation solution of the integer least squares problem which arises in communications and many other applications. Numerically simulations have shown that the LLL reduction can usually improve the success probability $P_{ZF}$ of the ZF decoder. In this paper, we first rigorously show that both SQRD and V-BLAST, two commonly used lattice reductions, have no effect on $P_{ZF}$. Then, we show that LLL reduction can improve $P_{ZF}$ when $n=2$, we also analyze how the parameter $\delta$ in the LLL reduction affects the enhancement of $P_{ZF}$. Finally, an example is given which shows that the LLL reduction decrease $P_{ZF}$ when $n\geq3$.