Free Deterministic Equivalents for the Analysis of MIMO Multiple Access Channel

Abstract

Abstract-In this paper, a free deterministic equivalent is proposed for the capacity analysis of the multi-input multi-output (MIMO) multiple access channel (MAC) with a more general channel model compared to previous works. Specifically, a MIMO MAC with one base station (BS) equipped with several distributed antenna sets is considered. Each link between a user and a BS antenna set forms a jointly correlated Rician fading channel. The analysis is based on operator-valued free probability theory, which broadens the range of applicability of free probability techniques tremendously. By replacing independent Gaussian random matrices with operator-valued random variables satisfying certain operator-valued freeness relations, the free deterministic equivalent of the considered channel Gram matrix is obtained. The Shannon transform of the free deterministic equivalent is derived, which provides an approximate expression for the ergodic input-output mutual information of the channel. The sum-rate capacity achieving input covariance matrices are also derived based on the approximate ergodic input-output mutual information. The free deterministic equivalent results are easy to compute, and simulation results show that these approximations are numerically accurate and computationally efficient.Index Terms-Operator-valued free probability, deterministic equivalent, massive multi-input multi-output (MIMO), multiple access channel (MAC).

Spectral Efficiency and Energy Efficiency Tradeoff in Massive MIMO Downlink Transmission With Statistical CSIT

“…Calculate the derivative D Calculate the derivative dη (ℓ) RE (PT(t)) dPT by (39). 15: Update P T (t + 1) by (41) and set t = t + 1.…”

MIXANDMIX: numerical techniques for the computation of empirical spectral distributions of population mixtures

“…Our tool has focused on the model described in Wagner et al (2012) but it could straightforwardly be adapted and optimized to address several analogous models in the literature, such as those mentioned in § 1. Nevertheless, we have not contemplated even more general models such as the Kronecker model (Zhang et al, 2013), with a recent contribution for the separable case in Leeb (2019), or more general couplings between the matrix elements (Wen et al, 2011;Lu et al, 2016). However, these models generally involve the solution of more intricate systems of nonlinear equations with more auxiliary functions, but with strong functional resemblances to the system we have studied here, so there is potential to reuse or adapt our tools to tackle them.…”

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“…According to the relationship in (36), if the solution to P1 (ℓ+1) is not feasible for F (ℓ) 5 , we then need to solve P2 (ℓ+1) . Employing a similar procedure for solving (37), we have the proposition on the solution to the sum-rate maximization problem P2 (ℓ+1) in the ℓth iteration of MM method as follows. ) by (41) and (42).…”

“…in each iteration, i.e., the system sum-rate, manipulating the expectation operation through Monte-Carlo methods is quite computationally cumbersome. To avoid this, we use the random matrix theory[36],[37] to replace the rate expression by its deterministic equivalent. More specifically, the DE of R + k (Λ) is computed byR + k (Λ) = log det (I M + Γ k Λ k ) + log det Γ k + K k (Λ) − tr I N k − Φ −1 k(21)whereΓ k = Ξ k Φ −1 k K k (Λ) −1(22)Γ k = Π k Φ −1 k Λ k(23)…”