Source-linked AI summary
On Capacity of Large-Scale MIMO Multiple Access Channels with Distributed Sets of Correlated Antennas
Jun Zhang, Chao-Kai Wen, Shi Jin, Xiqi Gao, Kai-Kit Wong
TL;DR
The paper studies how to analyze and optimize uplink large-scale MIMO multiple access channels with distributed antenna sets, spatial correlation, and LOS components. It derives deterministic equivalents using large-dimensional RMT and uses them to design capacity-achieving input covariances. The resulting approximations are reported to remain accurate for realistic system dimensions.
Problem
Existing deterministic-equivalent results do not support the general setting with multiple spatially correlated antennas at users and distributed base-station antenna sets.
Method
The paper derives ergodic-sum-rate deterministic equivalents with large-dimensional RMT and uses them to formulate an iterative water-filling covariance-design algorithm.
Results
The deterministic equivalent of ergodic sum rate is reported to be very close to the corresponding finite-dimensional result, even for realistic system dimensions.
Takeaways & Limitations
The results provide a computational basis for evaluating sum rate and designing capacity-achieving input covariance matrices in the considered large-scale MIMO MAC.
Takeaways & Limitations
Prior related extensions require diagonal transmitter-side correlation matrices, limiting support for multiple spatially correlated antennas.
Abstract
from arXiv · showhide
In this paper, a deterministic equivalent of ergodic sum rate and an algorithm for evaluating the capacity-achieving input covariance matrices for the uplink large-scale multiple-input multiple-output (MIMO) antenna channels are proposed. We consider a large-scale MIMO system consisting of multiple users and one base station with several distributed antenna sets. Each link between a user and an antenna set forms a two-sided spatially correlated MIMO channel with line-of-sight (LOS) components. Our derivations are based on novel techniques from large dimensional random matrix theory (RMT) under the assumption that the numbers of antennas at the terminals approach to infinity with a fixed ratio. The deterministic equivalent results (the deterministic equivalent of ergodic sum rate and the capacity-achieving input covariance matrices) are easy to compute and shown to be accurate for realistic system dimensions. In addition, they are shown to be invariant to several types of fading distribution.
1 Introduction
The paper addresses performance analysis and covariance design for large-scale uplink MIMO systems with distributed antenna sets, multiple antennas, spatial correlation, and LOS components. It develops deterministic equivalents using large-dimensional RMT to make ergodic sum-rate evaluation and optimization tractable.
- System model and motivation: Exact performance analysis becomes too complex for large systems, motivating large-dimensional random matrix theory as an alternative.The paper applies RMT to derive information-theoretic results for the considered uplink system.
- System model and motivation: Large-scale MIMO systems use distributed antenna sets and multiple-antenna users, but their realistic correlation and LOS effects complicate analysis.These systems may involve tens of antenna sets and hundreds of users, making direct computer simulations challenging.
- Paper contributions: The paper derives a deterministic equivalent of the ergodic sum rate for the distributed uplink MIMO multiple access channel.The approximation is intended to enable efficient and accurate computation of system sum rate.
- Limitations of prior work: Existing results generally restrict users and antenna sets to single antennas or diagonal spatial-correlation matrices, excluding multiple spatially correlated antennas.These restrictions do not capture the general model studied here, which includes multiple antennas at both terminals.
- Paper contributions: The analysis combines Gaussian and large-dimensional RMT techniques, includes LOS components, and extends deterministic-equivalent results from Gaussian to non-Gaussian entries.The resulting framework is presented as applicable to more complex channel models than prior work.
- Paper contributions: Simulation results show that the deterministic approximation closely estimates Monte Carlo ergodic sum rates for realistic system dimensions.The approximation is also applied to design input covariances and formulate an iterative water-filling optimization algorithm.
2 Channel Model and Problem Statement
The paper models an uplink large-scale MIMO multiple-access channel with multiple users and geographically distributed antenna sets, allowing two-sided spatial correlation and LOS components. It studies ergodic sum capacity using large-dimensional random matrix theory under a fixed-ratio antenna-growth regime.
- The system has K users transmitting simultaneously to a base station with L distributed antenna sets containing multiple antennas.
- Each user-to-antenna-set link uses a Kronecker model that separates receive-side and transmit-side spatial correlation.The receive and transmit correlation matrices are represented by R_l,k and T_l,k, respectively.
- The channel includes i.i.d. random components and deterministic LOS components, with the random entries not necessarily Gaussian.The model also permits some link pairs to have absent LOS components.
- The central problem is to understand the ergodic sum capacity of this MIMO multiple-access channel using large-dimensional random matrix theory.The analysis keeps L and K fixed while all antenna numbers grow with fixed ratios.
- Different antenna sets may have different spatial correlations, preventing the general channel matrix from being represented by a separable correlation model.This nonseparability is identified as the main obstacle addressed by the paper’s random-matrix analysis.
- The resulting deterministic approximation is intended to compute the system sum rate efficiently and accurately, including total ergodic sum rate and rate per antenna.The Shannon transform provides the relevant mutual-information functional used in this formulation.
3 Deterministic Equivalents and Ergodic Capacity
The paper derives deterministic equivalents for the ergodic sum rate in a distributed large-scale MIMO channel and uses them to design capacity-achieving input covariances. The results are rigorous under the stated asymptotic assumptions, extend beyond Gaussian fading, and lead to an iterative waterfilling algorithm.
- Deterministic Equivalents: The deterministic system of L × K equations has a unique solution for ω ∈ R+ under the stated model assumptions.The proof combines Gaussian random-matrix techniques, a Lindeberg argument for non-Gaussian entries, and an existence-and-uniqueness analysis.
- Deterministic Equivalents: Theorem 2 extends the deterministic-equivalent analysis to non-Gaussian fading distributions when the random entries have finite sixth-order moments.Nakagami and log-normal fading are identified as distributions to which the theorem applies under this condition.
- Deterministic Equivalents: Theorem 2 provides a deterministic equivalent of the ergodic sum rate in the large-dimensional regime.The approximation covers both the rate normalized per antenna and the total ergodic sum rate.
- Deterministic Equivalents: The result is more general than earlier cases because it permits nonzero LOS components and multiple spatially correlated antennas at both users and antenna sets.
- Applications: The deterministic equivalent provides a foundation for system-optimization applications, including deriving deterministic equivalents for MMSE-receiver SINR.Some such applications are left outside the paper because of space limitations.
- Ergodic Capacity: The paper proposes an approximate covariance-design approach whose optimal covariances converge to ergodic capacity and are structurally equivalent to iterative waterfilling over a deterministic channel.The deterministic-equivalent objective is strictly concave in the input covariance matrices.
- Ergodic Capacity: The proposed iterative waterfilling algorithm finds capacity-achieving input covariance matrices whose structure reflects antenna correlations and LOS components.The asymptotic optimal covariances are also invariant to the type of fading distribution.
4 Simulation Results
Simulations evaluate the deterministic-equivalent approximation for ergodic sum rate and its covariance-design algorithm under finite dimensions, multiple fading distributions, and distributed-antenna settings. The results show accurate rate approximation, efficient computation, and receiver-correlation-dependent transmit directions.
- Ergodic sum-rate approximation: With N1 = N2 = n1 = n2 = 8, the simulated curves tend to overlap regardless of the fading distribution.Even with N1 = N2 = n1 = n2 = 2, the approximation is evaluated for practical finite dimensions.
- Ergodic sum-rate approximation: The deterministic equivalent provides a very good approximation for the ergodic sum rate in finite-dimensional systems.The simulations compare the approximation with Monte-Carlo estimates under several fading distributions.
- Computational efficiency: For typical systems with twenties of distributed antenna sets and forties of users, Monte-Carlo simulations become prohibitive and rule out other optimization designs such as scheduling.The deterministic equivalent is described as much more efficient and as a foundation for further system-optimization applications.
- Input covariance design: The deterministic-equivalent covariance design produces indistinguishable ergodic sum rates from the optimal covariance solutions under Rayleigh and log-normal fading.The same experiments also report that Algorithm 1 is computationally much more efficient than the Vu-Paulraj algorithm.
- Input covariance design: When the receiver-side correlation has a broader beamwidth, the optimal covariance feeds the signal largely according to T2,1.This demonstrates that receiver correlation can change the optimal transmit directions.
5 Conclusion
The paper derives deterministic equivalents for large-scale MIMO MACs with general spatial correlation, LOS components, and non-Gaussian channel entries, then uses them to design capacity-achieving covariance matrices. Simulations report accurate practical-size approximations, computational efficiency, and close agreement with finite-dimensional optimization.
- The model covers large-scale MIMO MACs with general spatial correlation, LOS components, and non-Gaussian channel entries.
- The paper derives a deterministic equivalent for the ergodic sum rate and an iterative waterfilling algorithm for capacity-achieving input covariance matrices.
- The deterministic equivalent provides a very good approximation even when antenna numbers are of practical size.
- The deterministic-equivalent calculation is more efficient than Monte Carlo for large systems and is relevant to complex optimization problems.
- The predicted optimal covariance matrices are remarkably close to those from finite-dimensional optimization, but are obtained more efficiently.
- Future work includes a central limit theorem for the sum rate and applying deterministic equivalents to system-level designs.
in Theorem 1
The proof reformulates the channel model and uses resolvent properties, Gaussian integration by parts, and the Poincaré-Nash inequality. It establishes convergence through intermediate quantities and a two-step argument.
- The proof begins by reformulating the channel model for systematic analysis and defining resolvents of channel-related matrices.
- The channel construction embeds link matrices into the full channel, with mutually independent random components satisfying the stated assumptions.
- The proof invokes Gaussian integration by parts and the Poincaré-Nash inequality for complex Gaussian vectors and polynomially bounded functions.
- For LOS components, the proof requires additional manipulations beyond the standard no-LOS MIMO procedure and is split into two convergence steps.
- An intermediate quantity is introduced because direct proof of tr(E{S} − Ψ) → 0 is difficult, after which propositions establish the required convergence.
A.1 Proof of Proposition 3
The proof of Proposition 3 develops trace and resolvent estimates using integration by parts, concentration inequalities, and auxiliary lemmas. It then establishes convergence of the relevant scalar quantities and matrix terms.
- The proposition is completed by combining the established intermediate identities and convergence results.
- The derivation repeatedly applies Gaussian integration by parts to obtain identities for resolvent-related quantities.
- The proof reduces Proposition 3 to showing that scalar differences such as ˜α_l,k − ˜τ_l,k and tr(ΔΞ) converge to zero.
- Auxiliary lemmas bound trace expressions and error terms for uniformly bounded matrices, using the Poincaré-Nash inequality and Cauchy-Schwarz inequality.
- For sufficiently large ω, strict diagonal dominance makes Γ invertible with eigenvalues bounded away from zero.
- Analytic continuation and Stieltjes-transform properties extend the convergence argument to 0 < ω ≤ ω0.
A.2 Proof of Proposition 4
The proof of Proposition 4 transfers the preceding convergence approach to additional scalar and matrix quantities. It concludes that the corresponding differences converge to zero and completes the proposition.
- The proof uses the same approach as Proposition 3 to show that α_l,k − β_l,k e_l,k and ˜α_l,k − ˜e_l,k converge to zero.
- These convergence relations complete the proof of Proposition 4.
A.3 Proof of Proposition 5
The appendix proves the stated relations by rewriting the governing equations in matrix form, establishing positivity and spectral-radius conditions, and showing the relevant matrices are invertible. These steps complete the proof of (52a)–(52d).
- Matrix reformulation: The proof rewrites the defining equations in matrix form and introduces Γ′′ by replacing the expectation terms in Γ′.The resulting block matrices and entrywise definitions are used to analyze the system.
- Matrix identities: The matrix-inversion lemma yields ˜Φ ¯H H Ψ = ˜Ψ ¯H H Φ, which supports the required equivalence relations.This identity is obtained while comparing the auxiliary matrix expressions.
- Invertibility: Positivity of the relevant entries and spectral-radius bounds establish invertibility of I − Γ′ for sufficiently large N.The argument uses positivity conditions and Lemma 15 to control the spectral radius.
in Theorem 1
The proof uses a generalized Lindeberg principle to compare random-matrix quantities under finite sixth-order moment assumptions. Bounding the resulting derivative terms establishes the claimed convergence relation (132).
- Comparison principle: The proof applies the generalized Lindeberg principle to compare functions of random vectors with mutually independent components.The principle is stated for twice continuously differentiable functions and is applied to the matrix-valued setting.
- Complex case: Independence of the real and imaginary parts allows the real-case results to be applied directly to the complex case.The proof states this as the justification for proceeding without loss of generality.
- Moment assumption: Finite sixth-order moments of X(l,k)_ij support the derivative bounds required in the comparison argument.The proof explicitly invokes the finite 6-th order moment assumption.
- Derivative bounds: The derivative terms are bounded using norm inequalities and matrix estimates, with constants C1, C2, C3, and C4 appearing in the final bounds.The bounds combine triangle inequalities, Frobenius norms, and previously established lemmas.
- Conclusion: The resulting upper bound for the imaginary-part difference establishes (132).The proof states that the same bound applies to the relevant quantity and concludes that (132) is true.
C Existence and Uniqueness
The appendix establishes existence of the quantities (e_l,k, ˜e_l,k) for every user–antenna-set pair using Proposition 3 and an approach from prior work.
- Existence: The proof establishes the existence of (e_l,k, ˜e_l,k) for all l and k.The quantities are introduced as existing jointly over all indexed pairs.
- Proof basis: The existence argument follows and uses Proposition 3.The passage identifies both the prior approach and the proposition used.
- Scope: The result applies uniformly to every indexed pair (l,k).The notation ∀l,k specifies the scope of the existence claim.
C.2 Uniqueness
The uniqueness proof compares candidate solutions through matrix equations and proves that the associated spectral radius is below one. Positivity and standard matrix inequalities then force the candidate differences to vanish.
- Matrix formulation: The proof rewrites the defining equations in matrix form to analyze uniqueness through an auxiliary matrix Π.The matrix formulation combines equations (116) and (117).
- Block structure: The auxiliary matrices are built from block matrices and bounded using the dimensions and norms of the component matrices.The appendix states block-structure definitions and a bound on ∥AAH∥.
- Uniqueness: A contradiction with an eigenvalue equal to 1 implies that e_l,k − e°_l,k vanishes.The contradiction argument directly yields equality between the candidate quantities.
- Supporting tools: The proof relies on resolvent, matrix-inversion, trace, norm, and eigenvalue inequalities collected in the appendix.These tools include the resolvent identity, matrix inversion lemma, and standard matrix norm bounds.