Coulomb Gas Analogy: A Statistical Physics Approach to Performance Analysis of MIMO Systems

In this correspondence, by adopting the Coulomb gas analogy that is an analytical approach from statistical physics, we derive a closed-form approximation of the cumulative distribution function (CDF) of the largest eigenvalue of a random Gram matrix <inline-formula><tex-math notation="LaTeX">$ \boldsymbol {H}^{H} \boldsymbol {H}$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\boldsymbol {H}$</tex-math></inline-formula> denotes the channel matrix of a multiple-input multiple-output (MIMO) system. The expression of approximation has a simple structure in terms of several elementary functions, which is derived by assuming that the entries of <inline-formula><tex-math notation="LaTeX">$\boldsymbol {H}$</tex-math></inline-formula> have Nakagami-<inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> distributed amplitude with an independent phase. This assumption is made because it applies in two common cases of the Nakagami-<inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> distributions, i.e., with a uniformly distributed phase and a complex signal model. Simulation results indicate that the theoretical expression also provides a good approximation to the CDF for Rice and Hoyt distributions when the CDF is within a <italic>near-one</italic> range. This range takes the form of <inline-formula><tex-math notation="LaTeX">$(P_{no}, 1)$</tex-math></inline-formula>, where we assume <inline-formula><tex-math notation="LaTeX">$P_{no} = 10^{-3}$</tex-math></inline-formula> by default in this study. The derived approximation can find many uses for the performance analysis of MIMO beamforming and singular value decomposition MIMO systems, e.g., evaluating the outage probability for these systems.