[Paper Review] Approximate Random Matrix Models for Generalized Fading MIMO Channels

Muralikrishnan Srinivasan, Sheetal Kalyani|arXiv (Cornell University)|Jul 31, 2017

Advanced MIMO Systems Optimization56 references3 citations

TL;DR

This paper proposes an approximate random matrix model for $κ$-$\mu$ and $\eta$-$\mu$ faded MIMO channels using a complex Wishart distribution that minimizes Kullback-Leibler divergence from the true distribution. The model enables accurate computation of average capacity, outage probability for MRC, and ergodic rate with ZF receivers in massive MIMO systems, validated via close agreement with Monte Carlo simulations.

ABSTRACT

Approximate random matrix models for $κ-μ$ and $η-μ$ faded multiple input multiple output (MIMO) communication channels are derived in terms of a complex Wishart matrix. The proposed approximation has the least Kullback-Leibler (KL) divergence from the original matrix distribution. The utility of the results are demonstrated in a) computing the average capacity/rate expressions of $κ-μ$/$η-μ$ MIMO systems b) computing outage probability (OP) expressions for maximum ratio combining (MRC) for $κ-μ$/$η-μ$ faded MIMO channels c) ergodic rate expressions for zero-forcing (ZF) receiver in an uplink single cell massive MIMO scenario with low resolution analog-to-digital converters (ADCs) in the antennas. These approximate expressions are compared with Monte-Carlo simulations and a close match is observed.

Motivation & Objective

• To address the lack of random matrix models for generalized fading MIMO channels such as $\kappa$-$\mu$ and $\eta$-$\mu$, which are critical for accurate performance evaluation.
• To develop an approximate matrix model for the Gram matrix $\mathbf{HH}^H$ that closely matches the true distribution of $\kappa$-$\mu$ and $\eta$-$\mu$ MIMO channels.
• To enable practical computation of key performance metrics—average capacity, outage probability, and ergodic rate—without relying on computationally intensive Monte Carlo simulations.
• To validate the approximation accuracy by comparing derived expressions with Monte Carlo simulations across diverse MIMO scenarios.

Proposed method

• The authors derive an approximate complex Wishart matrix model for $\mathbf{HH}^H$ by minimizing the Kullback-Leibler divergence between the true and approximated distributions.
• The method employs a two-interval representation of the eigenvalue density, using $w_1$ and $w_2$ to capture the non-central behavior of $\kappa$-$\mu$ and $\eta$-$\mu$ fading.
• Key expressions are derived using determinants of matrices involving integrals of power functions, exponentials, and logarithmic terms, with Meijer-G functions used to represent logarithmic terms.
• The capacity and outage probability are expressed via determinants of matrices $\mathbf{N}^k$ and $\mathbf{N}$, respectively, with integrals evaluated using incomplete gamma and Meijer-G function identities.
• The approach leverages a generalized identity for multiple integrals over ordered eigenvalues, enabling exact analytical treatment of the eigenvalue statistics of the approximated matrix.
• The model is validated by comparing analytical results with Monte Carlo simulations, showing close agreement across various MIMO configurations.

Experimental results

Research questions

• RQ1 Can an approximate Wishart matrix model be constructed for $\kappa$-$\mu$ and $\eta$-$\mu$ MIMO channels that minimizes KL divergence from the true distribution?
• RQ2 How accurately can this approximation compute average ergodic capacity in $\kappa$-$\mu$ and $\eta$-$\mu$ MIMO systems?
• RQ3 Can the model accurately predict outage probability for maximum ratio combining (MRC) in $\kappa$-$\mu$ and $\eta$-$\mu$ fading MIMO channels?
• RQ4 What is the performance of zero-forcing (ZF) receivers in massive MIMO systems with low-resolution ADCs under $\kappa$-$\mu$ and $\eta$-$\mu$ fading, as predicted by the approximation?

Key findings

• The proposed Wishart approximation achieves a close match with Monte Carlo simulations for average capacity in $\kappa$-$\mu$ and $\eta$-$\mu$ MIMO systems, validating its accuracy.
• Outage probability expressions for MRC in $\kappa$-$\mu$ fading channels are accurately captured using the determinant-based formulation involving incomplete gamma functions.
• Ergodic rate expressions for ZF receivers in massive MIMO with low-resolution ADCs are derived and shown to closely match simulation results.
• The use of Meijer-G functions enables analytical tractability of logarithmic terms in capacity expressions, facilitating closed-form evaluation.
• The approximation method is robust across different numbers of antennas and SNR regimes, as demonstrated by consistent simulation agreement.
• The KL-divergence-minimizing model provides a systematic and mathematically grounded alternative to brute-force simulation for performance evaluation in generalized fading MIMO systems.