[Paper Review] Achievable Sum Rate of MIMO MMSE Recievers: A General Analytic Framework

Matthew R. McKay, Iain B. Collings+1 more|ArXiv.org|Mar 4, 2009

Advanced MIMO Systems Optimization41 references4 citations

TL;DR

This paper introduces a general analytic framework that links the achievable sum rate of MIMO linear MMSE receivers to the ergodic mutual information of optimal receivers, enabling the direct application of existing MIMO mutual information results to MMSE systems. The key contribution is a closed-form expression for the sum rate in various fading scenarios, including Rayleigh and Rician channels, with exact and asymptotic expressions derived for high and low SNR regimes.

ABSTRACT

This paper investigates the achievable sum rate of multiple-input multiple-output (MIMO) wireless systems employing linear minimum mean-squared error (MMSE) receivers. We present a new analytic framework which unveils an interesting connection between the achievable sum rate with MMSE receivers and the ergodic mutual information achieved with optimal receivers. This simple but powerful result enables the vast prior literature on ergodic MIMO mutual information to be directly applied to the analysis of MMSE receivers. The framework is particularized to various Rayleigh and Rician channel scenarios to yield new exact closed-form expressions for the achievable sum rate, as well as simplified expressions in the asymptotic regimes of high and low signal to noise ratios. These expressions lead to the discovery of key insights into the performance of MIMO MMSE receivers under practical channel conditions.

Motivation & Objective

• Address the lack of closed-form analytical results for achievable sum rates in MIMO systems with linear MMSE receivers, especially for finite-dimensional systems.
• Overcome the limitations of prior asymptotic analyses based on large-dimensional random matrix theory, which are less accurate for practical MIMO system dimensions.
• Provide a general method applicable to a broad class of MIMO channel models, including correlated Rayleigh and Rician fading, beyond the uncorrelated Rayleigh case previously studied.
• Enable the use of the extensive body of literature on ergodic MIMO mutual information to analyze MMSE receiver performance without explicitly characterizing the complex SINR distribution at the receiver output.

Proposed method

• Establish a mathematical relationship between the achievable sum rate of linear MMSE receivers and the ergodic mutual information of optimal receivers via algebraic manipulations of matrix determinants.
• Leverage the identity involving the inverse of a matrix and its minors to express the MMSE sum rate as a difference of expected log-determinants of channel Gram matrices.
• Utilize known results on the probability density function of eigenvalues of $oldsymbol{H}^ opoldsymbol{H}$ for arbitrary $N_t$ and $N_r$, particularly the unified expression from [46] for the eigenvalue PDF.
• Apply the trace and second derivative of the log-determinant of $oldsymbol{I} + xoldsymbol{A}$ at $x=0$ to derive low-SNR asymptotic expansions.
• Use the channel power normalization condition to simplify expectations of trace terms, enabling closed-form low-SNR sum rate expressions.
• Derive high-SNR approximations by analyzing the behavior of the sum rate expression as SNR increases, based on the asymptotic properties of the mutual information.

Experimental results

Research questions

• RQ1 How can the achievable sum rate of MIMO MMSE receivers be analytically characterized in finite-dimensional systems without relying on large-system asymptotics?
• RQ2 What is the fundamental relationship between the sum rate of MMSE receivers and the ergodic mutual information of optimal receivers?
• RQ3 Can existing closed-form results on ergodic MIMO mutual information be repurposed to analyze MMSE receiver performance in non-i.i.d. fading channels?
• RQ4 What are the exact and asymptotic sum rate expressions for MIMO MMSE receivers in correlated Rayleigh and Rician fading channels?
• RQ5 How does the sum rate scale at low and high SNR in practical MIMO systems with realistic channel statistics?

Key findings

• The achievable sum rate of MIMO MMSE receivers is exactly expressible as the difference between the expected log-determinant of the full channel Gram matrix and the expected log-determinant of its minor matrices, enabling direct use of existing mutual information results.
• For uncorrelated Rayleigh fading, the exact sum rate expression derived via the proposed framework matches previously published results, validating the approach.
• In the low-SNR regime, the sum rate scales as $ rac{N_r}{ ext{ln } 2} $ bits/s/Hz, which corresponds to the capacity pre-log factor, confirming the optimality of the pre-log in this regime.
• The high-SNR asymptotic sum rate is shown to be proportional to $ N_t imes ext{log}_2( ext{SNR}) $, with the multiplexing gain matching that of the optimal receiver.
• The framework enables exact closed-form expressions for sum rate in Rician fading and correlated Rayleigh fading channels, which were previously intractable using SINR-based methods.
• The method is extendable to multi-user MIMO systems, including multiple access channels with MMSE receivers and MIMO broadcast channels with MMSE precoding or reception, demonstrating broad applicability beyond single-user scenarios.