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Statistical Eigenmode Transmission over Jointly-Correlated MIMO Channels

Xiqi Gao, Bin Jiang, Xiao Li, Alex B. Gershman, Matthew R. McKay

arXiv:0903.1952v1cs.ITmath.IT

TL;DR

Statistical-CSI capacity analysis and transmitter design depend on the channel model, motivating treatment of general jointly-correlated MIMO channels. The paper derives a permanent-based capacity upper bound and a convergent power-allocation algorithm; simulations report a tight bound and near-negligible performance loss.

  • Problem

    Statistical-CSI capacity analysis and transceiver design depend on channel modeling, including for general jointly-correlated MIMO channels.

    Source: §I · p. 2

  • Method

    The paper derives a tight closed-form ergodic-capacity upper bound involving matrix permanents, then uses it to develop an efficient iterative water-filling algorithm with guaranteed convergence.

    Source: §VI · p. 23

  • Results

    Simulations confirm the bound’s tightness and show near-negligible loss for the low-complexity optimization approach relative to the jointly-correlated channel’s ergodic capacity.

    Source: §VI · pp. 23–24

  • Takeaways & Limitations

    The proposed approach provides low-complexity transmitter optimization for the general jointly-correlated channel while retaining near-ergodic-capacity performance in simulations.

    Source: §VI · pp. 23–24

Abstract

from arXiv · show

We investigate MIMO eigenmode transmission using statistical channel state information at the transmitter. We consider a general jointly-correlated MIMO channel model, which does not require separable spatial correlations at the transmitter and receiver. For this model, we first derive a closed-form tight upper bound for the ergodic capacity, which reveals a simple and interesting relationship in terms of the matrix permanent of the eigenmode channel coupling matrix and embraces many existing results in the literature as special cases. Based on this closed-form and tractable upper bound expression, we then employ convex optimization techniques to develop low-complexity power allocation solutions involving only the channel statistics. Necessary and sufficient optimality conditions are derived, from which we develop an iterative water-filling algorithm with guaranteed convergence. Simulations demonstrate the tightness of the capacity upper bound and the near-optimal performance of the proposed low-complexity transmitter optimization approach.

I. INTRODUCTION

The paper develops statistical eigenmode transmission for general jointly-correlated MIMO channels, addressing the difficulty of transmitter optimization when instantaneous CSI is hard to obtain and exact-capacity optimization is computationally involved. It derives a permanent-based capacity bound and statistics-only power allocation, with simulations showing a tight bound and near-negligible performance loss.

  • Jointly-correlated channel models capture mutual dependence between link ends, addressing deficiencies reported for the analytically friendly but separable Kronecker model.The paper considers statistical eigenmode transmission over a general jointly-correlated MIMO channel.Source: §I · p. 3
  • The closed-form tight ergodic-capacity upper bound for general jointly-correlated MIMO channels reveals a relationship to matrix permanents and includes existing results as special cases.The stated special cases include Kronecker channels.Source: §I · p. 3
  • Convex optimization of the bound yields low-complexity power allocation using only channel statistics, with necessary and sufficient optimality conditions and a water-filling algorithm that converges within a few iterations.Source: §I · pp. 3–4
  • Simulations show the proposed low-complexity transmitter optimization suffers near-negligible loss relative to the jointly-correlated channel’s ergodic capacity.The simulations also demonstrate the tightness of the capacity upper bound.Source: §I · p. 4

A. Notation

This section establishes the paper’s matrix and vector notation and specifies a jointly-correlated MIMO channel model. The model represents average coupling between transmit and receive eigenmodes and encompasses several established channel models.

A. Notation

  • Boldface denotes matrices or column vectors, while subscripts can clarify dimensionality; the notation also defines common matrix operations, including trace, determinant, and permanent.Source: A. Notation · p. 4

A. Channel Model

  • The channel model combines deterministic unitary matrices, a line-of-sight component, and weighted i.i.d. fading, without requiring Gaussian fading entries.Its coupling matrix records average transmit-receive eigenmode power, and its row and column sums give receive and transmit correlation eigenvalues.Source: A. Channel Model · pp. 5–6
  • The model encompasses Kronecker, virtual-channel, Weichselberger, and UIU models, and allows multiple line-of-sight components for links such as distributed radio networks.Source: A. Channel Model · p. 7

B. Statistical Eigenmode Transmission

The proposed statistical-CSI power-allocation approach is designed to avoid repeated channel-sample averaging while approximating the capacity-achieving solution. The section also establishes that optimal signaling directions follow the transmit-correlation eigenvectors.

B. Statistical Eigenmode Transmission

  • With receiver channel knowledge and transmitter access to channel statistics, finding the capacity-achieving covariance reduces to allocating power over the optimized transmit directions.The covariance eigenvalues determine the power allocation.Source: B. Statistical Eigenmode Transmission · pp. 7–8
  • The capacity-achieving input covariance uses transmit-correlation eigenvectors as its signaling directions, extending prior results to the more general channel model.The eigenvector matrix is U = Ut.Source: B. Statistical Eigenmode Transmission · p. 8
  • An exact closed-form power allocation is generally difficult because it requires evaluating an intractable expectation, and prior approaches used numerical optimization.The passage identifies this difficulty for both the jointly-correlated model and other scenarios, including the Kronecker model.Source: B. Statistical Eigenmode Transmission · p. 8
  • The paper derives a tight closed-form capacity upper bound and uses it to develop fast, low-complexity power allocations that approximate the capacity-achieving solution.The proposed computations use channel statistics and avoid random-matrix averaging, allowing computation at either the receiver or transmitter.Source: B. Statistical Eigenmode Transmission · p. 8
  • Unlike a prior iterative algorithm requiring numerical channel-sample averaging and instantaneous CSI, the proposed power-allocation algorithms use only channel statistics.The authors note that the prior algorithm’s computations must typically be performed at the receiver under statistical feedback.Source: B. Statistical Eigenmode Transmission · pp. 8–9

III. CLOSED-FORM CAPACITY UPPER BOUND USING PERMANENTS

This section develops a closed-form capacity upper bound using matrix permanents, extending the permanent framework to rectangular matrices. It also establishes supporting properties and identifies a concavity conjecture that remains unproved for general matrix dimensions.

III. CLOSED-FORM CAPACITY UPPER BOUND USING PERMANENTS

  • The section derives a closed-form capacity upper bound for the jointly-correlated MIMO channel and develops algorithms for its efficient computation.The derivation exploits linear-algebraic properties of matrix permanents.Source: §III · p. 9

A. Matrix Permanents: Definitions and Properties

  • The paper extends the permanent definition and its useful properties to arbitrary M × N matrices, as needed for the capacity-bound derivation.The permanent’s expansion resembles the determinant’s, but its minor terms carry positive signs.Source: A. Matrix Permanents: Definitions and… · pp. 9–10
  • The section introduces extended permanents through permanents of matrices augmented with identity blocks.These extended quantities are denoted as extended permanents of A.Source: A. Matrix Permanents: Definitions and… · p. 11
  • A lemma relates the expected permanent of an independent-entry random matrix to its mean and elementwise second moments, under a row-sparsity condition on its mean.The condition requires at most one nonzero element in each row of the mean matrix.Source: A. Matrix Permanents: Definitions and… · pp. 11–12
  • For independent, identically distributed zero-mean, unit-variance entries, the permanent of the all-ones matrix is N!.The passage notes agreement with prior results.Source: A. Matrix Permanents: Definitions and… · p. 12
  • The authors conjecture that log Per(Adiag(λ)) is concave for nonnegative λ when the permanent is positive.A formal proof for arbitrary M and N is unavailable; the paper provides proofs for several special cases.Source: A. Matrix Permanents: Definitions and… · p. 12

B. Capacity Upper Bound

The section derives a closed-form upper bound on ergodic capacity for jointly-correlated MIMO channels. The bound depends on average SNR and the eigenmode channel coupling matrix, and supports power allocation without Monte Carlo averaging over channel realizations.

B. Capacity Upper Bound

  • Theorem 2 gives an upper bound on ergodic capacity for the jointly-correlated channel using the permanent-based derivation.The proof uses determinant expansions and the Cauchy-Binet formula.Source: B. Capacity Upper Bound · pp. 12–13
  • The bound is completely determined by average SNR and the eigenmode channel coupling matrix Ω.This dependence supports optimizing the bound over the power-allocation vector without Monte Carlo averaging over random channel realizations.Source: B. Capacity Upper Bound · pp. 13–14
  • When the channel reduces to the Kronecker correlation model, the expression matches previously presented upper bounds for Kronecker-correlated channels.The passage also relates the equal-power special case to an earlier capacity upper bound.Source: B. Capacity Upper Bound · p. 14

C. Efficient Computation Algorithms

The section develops polynomial-based recursive algorithms for computing the extended permanent needed to evaluate the capacity bound. These methods reduce computational complexity relative to direct permanent calculations, with the most efficient polynomial method depending on antenna size.

C. Efficient Computation Algorithms

  • Computing permanents becomes increasingly complex as matrix dimensions grow, motivating more efficient extended-permanent algorithms.The section considers direct definitions, Laplace expansion, and Ryser’s method as existing approaches.Source: C. Efficient Computation Algorithms · pp. 14–15
  • The paper presents three polynomial expansions for the extended permanent and shows that each admits an efficient recursive algorithm for its coefficients.The resulting algorithms apply to arbitrary antenna dimensions.Source: C. Efficient Computation Algorithms · pp. 15–16
  • The coefficient recursion, combined with the key result, yields an efficient algorithm for computing the extended permanent.The section also states that related algorithms follow from the other two polynomial expressions.Source: C. Efficient Computation Algorithms · p. 16
  • For configurations N = Nt = Nr, polynomial-based algorithms significantly reduce computational complexity versus direct methods, with least complexity for N ≤ 5 in many cases; Ryser’s formula is most efficient for N > 5.Source: C. Efficient Computation Algorithms · pp. 16–17

IV. OPTIMAL POWER ALLOCATION WITH THE CAPACITY BOUND

The paper optimizes statistical-CSI power allocation using a tractable capacity upper bound, first checking its behavior at low and high SNR. The resulting asymptotic policies match capacity-achieving allocations in those regimes.

A. Asymptotic Optimality at Low and High SNR

  • The general allocation problem maximizes the bound using only the eigenmode coupling matrix Ω and transmit SNR ρ.The maximizing allocation serves as an approximation to the capacity-achieving allocation, and numerical results confirm its accuracy.Source: A. Asymptotic Optimality at Low and… · p. 17
  • At low SNR, beamforming on the strongest transmit eigenmodes specified by Ω is optimal.The allocation puts power on the l strongest modes and zero on the remaining modes.Source: A. Asymptotic Optimality at Low and… · pp. 17–18
  • At high SNR, equal power across transmit eigenmodes is optimal.The low- and high-SNR policies derived from the bound coincide with the capacity-achieving policies for those regimes.Source: A. Asymptotic Optimality at Low and… · p. 18

B. Optimality Conditions for Arbitrary SNRs

For arbitrary SNRs, the capacity-bound objective is concave in the power allocation, allowing necessary and sufficient optimality conditions. The general coupling case requires numerical optimization, while a square diagonal coupling matrix reduces to ordinary water-filling.

B. Optimality Conditions for Arbitrary SNRs

  • The KKT conditions give necessary and sufficient conditions for optimal power allocation under the nonnegativity and total-power constraints.The multiplier for the total-power constraint is chosen to satisfy the power constraint.Source: B. Optimality Conditions for Arbitrary… · pp. 18–20
  • The expected mutual-information upper bound is concave in λ, so the power-allocation problem is a concave optimization problem with a global optimum.Standard convex optimization algorithms, including interior-point methods, can evaluate the solution.Source: B. Optimality Conditions for Arbitrary… · pp. 18–19
  • For a square diagonal coupling matrix, the conditions reduce to the conventional water-filling formula.The passage identifies this as the same formula as water-filling with instantaneous CSI.Source: B. Optimality Conditions for Arbitrary… · p. 20
  • With an arbitrary eigenmode coupling matrix, the solution is not obtained as easily and requires numerical approaches.The paper derives derivatives of the bound using the Laplace expansion property of permanents.Source: B. Optimality Conditions for Arbitrary… · pp. 19–20

C. Iterative Water-Filling Algorithm

The paper proposes an iterative water-filling algorithm (IWFA) to evaluate optimal power allocation from the upper-bound conditions. Its updates are designed to increase the bound, and the algorithm converges to the capacity upper bound.

C. Iterative Water-Filling Algorithm

  • The IWFA typically converges after only a few iterations, with its first iteration achieving near-optimal performance.These simulation results establish the reported speed and performance of the algorithm.Source: C. Iterative Water-Filling Algorithm · p. 20
  • The allocation can be initialized with equal power or with the previous frame’s optimal allocation when channel statistics change smoothly.Using the previous frame’s allocation could speed convergence.Source: C. Iterative Water-Filling Algorithm · p. 21
  • The IWFA converges to the capacity upper bound.The proof uses concavity and an update step that does not decrease the bound.Source: C. Iterative Water-Filling Algorithm · p. 21
  • Each iteration uses a water-filling solution when it increases the bound; otherwise, the algorithm uses a different update to guarantee convergence.The algorithm checks whether the bound increases and uses the alternative update when it does not.Source: C. Iterative Water-Filling Algorithm · p. 22

V. SIMULATIONS

Simulations show that the proposed statistical eigenmode transmission performs near-optimally in jointly-correlated and Kronecker channel scenarios, and that the capacity upper bound is tight, especially at low to moderate SNR. The conclusion highlights a permanent-based bound and a convergent low-complexity allocation approach.

V. SIMULATIONS

  • The upper bound is rather tight for both channel models, especially at low to moderate SNRs below 8 dB.The Kronecker-model bound is slightly tighter than the jointly-correlated-model bound.Source: §V · p. 22
  • The proposed SET approach suffers almost negligible loss compared with true channel capacity in both jointly-correlated and Kronecker channel scenarios.Equal power allocation and beamforming are also optimal in the high- and low-SNR regimes, respectively.Source: §V · p. 23
  • The IWFA converges after only a few iterations, with the first iteration achieving near-optimal performance in all simulated cases.Source: §V · p. 23

VI. CONCLUSIONS

  • The paper derives a tight closed-form ergodic-capacity upper bound for jointly-correlated MIMO channels, with a relationship expressed through matrix permanents.The bound embraces many existing results as special cases.Source: §VI · p. 23
  • The paper develops low-complexity power allocation with necessary and sufficient optimality conditions and an iterative water-filling algorithm with guaranteed convergence.Simulations confirm the bound’s tightness and the optimization approach’s performance.Source: §VI · pp. 23–24

APPENDIX I

The appendices provide algebraic proof steps and establish concavity of the permanent-based function used in the paper. They rely on permanent and determinant expansions, with concavity shown across several cases.

APPENDIX I

  • The appendix tracks permanent-expansion terms through index and complementary-sequence relations to complete its proof.Source: Appendix I · pp. 24–25

APPENDIX II

  • The determinant proof uses permutation signs and row independence, then substitutes element-wise expressions to complete the derivation.Source: APPENDIX II · p. 25

APPENDIX III PROOF OF THE CONCAVITY OF f(λ) = log Per(Adiag(λ)) IN SEVERAL CASES

APPENDIX IV

This appendix proves a permanent identity by expanding the permanent and relating the terms. It treats the case Nr ≤ Nt, noting that the proof for Nr > Nt is similar.

APPENDIX IV

  • The proof considers Nr ≤ Nt; the authors state that the Nr > Nt case is similar.Source: APPENDIX IV · p. 27
  • The derivation starts from the definition of permanents and applies a relation to each product term in the expansion.Source: APPENDIX IV · p. 27
  • The appendix completes the proof of (93) by expanding the permanent and substituting a relation between its terms.Source: APPENDIX IV · p. 27

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