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Exact SINR Analysis of Matched-filter Precoder

Hui Zhao, Dirk Slock, Petros Elia

TL;DR

This work derives the exact SINR distribution for matched-filter precoding in downlink MU-MIMO under Rayleigh fading, yielding explicit CDF/PDF expressions and showing SINR convergence in both high-SNR and massive-MIMO regimes. It introduces a Beta-based interference model to simplify general-case analysis and develops an ergodic-rate approximation that remains accurate across SNRs, plus a rigorous result showing the exact rate converges to the known massive-MIMO asymptote. The combination of exact analysis, limiting results, and practical approximations provides deep insights into MF performance and enables robust performance evaluation without excessive computation. Numerical results corroborate the analytical models and demonstrate improved accuracy over existing approximations, enhancing design and evaluation of MF-based systems.

Abstract

This paper answers a fundamental question about the exact distribution of the signal-to-interference-plus-noise ratio (SINR) under matched-filter (MF) precoding. Specifically, we derive the exact expressions for the cumulative distribution function (CDF) and the probability density function (PDF) of SINR under MF precoding over Rayleigh fading channels. Based on the exact analysis, we then rigorously prove that the SINR converges to some specific distributions separately in high SNR and in massive MIMO. To simplify the exact result in general cases, we develop a good approximation by modelling the interference as a Beta distribution. We then shift to the exact analysis of the transmit rate, and answer the fundamental question: How does the exact rate converge to the well-known asymptotic rate in massive MIMO? After that, we propose a novel approximation for the ergodic rate, which performs better than various existing approximations. Finally, we present some numerical results to demonstrate the accuracy of the derived analytical models.

Exact SINR Analysis of Matched-filter Precoder

TL;DR

This work derives the exact SINR distribution for matched-filter precoding in downlink MU-MIMO under Rayleigh fading, yielding explicit CDF/PDF expressions and showing SINR convergence in both high-SNR and massive-MIMO regimes. It introduces a Beta-based interference model to simplify general-case analysis and develops an ergodic-rate approximation that remains accurate across SNRs, plus a rigorous result showing the exact rate converges to the known massive-MIMO asymptote. The combination of exact analysis, limiting results, and practical approximations provides deep insights into MF performance and enables robust performance evaluation without excessive computation. Numerical results corroborate the analytical models and demonstrate improved accuracy over existing approximations, enhancing design and evaluation of MF-based systems.

Abstract

This paper answers a fundamental question about the exact distribution of the signal-to-interference-plus-noise ratio (SINR) under matched-filter (MF) precoding. Specifically, we derive the exact expressions for the cumulative distribution function (CDF) and the probability density function (PDF) of SINR under MF precoding over Rayleigh fading channels. Based on the exact analysis, we then rigorously prove that the SINR converges to some specific distributions separately in high SNR and in massive MIMO. To simplify the exact result in general cases, we develop a good approximation by modelling the interference as a Beta distribution. We then shift to the exact analysis of the transmit rate, and answer the fundamental question: How does the exact rate converge to the well-known asymptotic rate in massive MIMO? After that, we propose a novel approximation for the ergodic rate, which performs better than various existing approximations. Finally, we present some numerical results to demonstrate the accuracy of the derived analytical models.
Paper Structure (11 sections, 7 theorems, 41 equations, 3 figures)

This paper contains 11 sections, 7 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model and Exact SINR Analysis
  3. Simplified Results of SINR Distribution
  4. High SNR Regime
  5. Massive MIMO Regime
  6. Beta Approximation
  7. Ergodic Rate Analysis
  8. Numerical Results
  9. Conclusion
  10. Proof of Lemma \ref{['Xi_Distri_Lem']}
  11. Proof of Theorem \ref{['SINR_MF_dist_thm']}

Key Result

Lemma 1

The random variable $X_i\triangleq \frac{ 1}{|| {\bf h}_i^* ||^2} | {\bf u}_k^T {\bf h}_i^* |^2 \in [0,1]$ has a Beta distribution with the first shape parameter $1$ and the second shape parameter $L-1$, denoted by $X_i \sim \text{Beta}(1,L-1)$. The CDF and PDF of $X_i$ are respectively Furthermore, $X_i$ is independent of ${\bf u}_k$ (or equivalently, ${\bf h}_k$), and $\{X_i\}_{i=1,i\neq k}^

Figures (3)

  • Figure 1: Outage probability versus $P_t$ for $\gamma =0.8$, $K=4$ and $\sigma_k^2=1$.
  • Figure 2: CDFs of $\frac{1}{(K-1) \text{SINR}_k}$ and $\frac{\text{SINR}_k}{L}$ for $K=4$ and $\sigma_k^2=1$.
  • Figure 3: Ergodic rate versus $P_t$ for $K=6$ and $\sigma_k^2=1$.

Theorems & Definitions (10)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 1
  • Remark 2
  • Lemma 5
  • Remark 3
  • Lemma 6