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Energy Efficiency Optimization in Relay-Assisted MIMO Systems with Perfect and Statistical CSI

Alessio Zappone, Pan Cao, Eduard A. Jorswieck

arXiv:1310.7794v3cs.ITmath.ITmath.OC

TL;DR

Bit/Joule energy-efficiency optimization remains largely unexplored for relay-assisted MIMO, especially beyond perfect CSI. This paper develops resource allocation for perfect and statistical CSI cases, and reports better performance with perfect CSI, with perfect knowledge of H outperforming perfect knowledge of G.

  • Problem

    No results were available for bit/Joule energy-efficiency optimization in relay-assisted MIMO systems, and even the perfect-CSI case was almost unexplored.

    Source: §I · p. 3

  • Method

    The paper determines optimal source and relay eigenvector matrices and develops energy-efficient resource allocation for perfect and statistical CSI cases.

    Source: §VIII · p. 25

  • Results

    Perfect CSI grants better performance, and perfect knowledge of H performs better than perfect knowledge of G.

    Source: §VII · p. 24

  • Takeaways & Limitations

    Diagonalizing the correlation matrices of statistically available channels is optimal in the considered scenario.

    Source: §VIII · p. 25

    Algorithm 3 has a small probability of converging to a different fixed point.

    Source: §VII · p. 24

Abstract

from arXiv · show

A framework for energy-efficient resource allocation in a single-user, amplify-and-forward relay-assisted MIMO system is devised in this paper. Previous results in this area have focused on rate maximization or sum power minimization problems, whereas fewer results are available when bits/Joule energy efficiency (EE) optimization is the goal. The performance metric to optimize is the ratio between the system's achievable rate and the total consumed power. The optimization is carried out with respect to the source and relay precoding matrices, subject to QoS and power constraints. Such a challenging non-convex problem is tackled by means of fractional programming and and alternating maximization algorithms, for various CSI assumptions at the source and relay. In particular the scenarios of perfect CSI and those of statistical CSI for either the source-relay or the relay-destination channel are addressed. Moreover, sufficient conditions for beamforming optimality are derived, which is useful in simplifying the system design. Numerical results are provided to corroborate the validity of the theoretical findings.

I. INTRODUCTION

The paper targets the underexplored problem of bit/Joule energy-efficiency optimization in relay-assisted MIMO systems. It develops resource allocation for several CSI settings and derives conditions for source beamforming optimality.

I. INTRODUCTION

  • Bit/Joule energy-efficiency optimization had no available results for relay-assisted MIMO systems, and the simpler perfect-CSI case was nearly unexplored.The paper frames this gap against prior work that largely optimized traditional measures or assumed perfect CSI.Source: §I · p. 3
  • The work jointly allocates source and relay precoding matrices to maximize achievable rate over consumed power, subject to QoS and power constraints.Consumed power includes both transmit power and circuit power dissipated in device electronics.Source: §I · p. 3
  • The analysis covers perfect CSI for both channels and partial-CSI cases where either the source-relay or relay-destination channel is statistically known.The destination is not specified in these introductory scenario descriptions.Source: §I · pp. 3–4
  • A two-step solution determines optimal transmit directions in closed form, then uses fractional programming and alternating maximization for power allocation.After fixing the directions, the resulting problem is separately pseudo-concave in the source and relay power-allocation vectors.Source: §I · p. 4
  • For the partial-CSI scenarios, sufficient conditions for source beamforming optimality identify when the source should put all available power on one stream.The paper presents this condition as a way to reduce resource-allocation complexity.Source: §I · p. 4

II. SYSTEM MODEL AND PROBLEM STATEMENT

The system model is a half-duplex amplify-and-forward relay link with source, relay, and destination precoding and power constraints. The objective is global energy efficiency, defined using achievable rate and total consumed power, under a minimum-rate requirement.

II. SYSTEM MODEL AND PROBLEM STATEMENT

  • The modeled link has one source, one half-duplex amplify-and-forward relay, and one destination, with source-relay channel H, relay-destination channel G, and relay matrix A.The source, relay, and destination have NS, NR, and ND antennas, respectively.Source: §II · p. 5
  • Global energy efficiency is the achievable rate divided by consumed energy, balancing high rate against low energy use.The rate is expressed in bits/s/Hz, and consumed power accounts for circuit consumption as well as transmission.Source: §II · p. 5
  • The resource-allocation problem maximizes global energy efficiency subject to source and relay power limits and a minimum acceptable achievable rate.The maximum feasible transmit powers at the source and relay are denoted by their respective power limits.Source: §II · pp. 5–6
  • The channel-information cases are perfect CSI for both links, perfect CSI for G with statistical CSI for H, or perfect CSI for H with statistical CSI for G.The destination is assumed to know both channels perfectly in all three cases.Source: §II · p. 6

III. GEE MAXIMIZATION WITH PERFECT CSI

With perfect CSI on both channels, the paper formulates a non-jointly-convex fractional energy-efficiency problem. It establishes that the optimal source covariance and relay matrix align with the channel singular-vector bases.

III. GEE MAXIMIZATION WITH PERFECT CSI

  • The perfect-CSI problem is a complex fractional optimization that is not jointly convex in the source covariance Q and relay matrix A.The numerator and denominator favor opposite eigenvalue orderings, so diagonalization is not immediate from optimizing either term alone.Source: §III · p. 7
  • The optimal source covariance and relay matrix use aligned singular-vector bases: U_Q = V_H, U_A = V_G, and V_A = U_H.This is the stated optimal structure for the perfect-CSI problem.Source: §III · p. 7

DIND + GA

The proof selects matrix bases that simultaneously maximize the energy-efficiency numerator and minimize its denominator. The selected bases remain feasible because those terms correspond to the QoS and power constraints.

DIND + GA

  • The selected bases U_y = U_H, U_x = U_G, and V_x = U_y maximize the numerator while minimizing the denominator of the reformulated objective.The denominator summands are minimized by U_y = U_H and U_x = U_G; the numerator is maximized when V_x = U_y = U_H.Source: DIND + GA · p. 7
  • These basis choices are feasible because the objective numerator is also the QoS constraint's left-hand side, while the denominator summands correspond to the power constraints.The proof uses this correspondence to support the simultaneous numerator and denominator optimization.Source: DIND + GA · p. 7

RINR)1/2U H

The section reduces the precoding design to a vector-valued power-allocation problem and develops an alternating fractional-programming approach. It also states the scope of the structural result and the limits of the algorithm’s global-optimality guarantees.

RINR)1/2U H

  • To achieve the stated optimal eigenvector alignments, the AF matrix must satisfy the corresponding relations between its left and right eigenvector matrices and those of the channels.Source: RINR)1/2U H · p. 7

H,NRU H

  • The structural result extends beyond tall, full-rank channel matrices; the tall, full-rank assumption is made for notational ease, and the method also applies to rank-deficient matrices.Source: H,NRU H · p. 8
  • Components outside the reduced covariance and AF-matrix blocks can be set to zero, since they do not affect the numerator and only increase consumed power.Source: H,NRU H · p. 8
  • The optimal directions yield a vector-valued power-allocation problem that remains non-convex but can be tackled with fractional programming and alternating maximization.Each power vector is optimized while holding the other fixed.Source: H,NRU H · p. 9
  • Algorithm 1 converges because each iteration does not decrease the objective and the objective is upper-bounded.Source: H,NRU H · p. 9
  • The overall alternating algorithm is not guaranteed to reach the global GEE optimum because the objective is not jointly pseudo-concave in both power vectors.Each subproblem is solved globally using Dinkelbach’s algorithm, but the vectors are optimized in alternation.Source: H,NRU H · p. 10
  • Algorithm 1 can be implemented centrally at the relay or in parallel at the source and relay, where each learns its respective precoding matrix.Source: H,NRU H · p. 10

IV. GEE MAXIMIZATION WITH PARTIAL CSI ON H

This section addresses GEE maximization with partial CSI on the source-relay channel and a known relay-destination channel, defining GEE as ergodic achievable rate over average consumed energy and deriving optimal precoder structures with an alternating-maximization solution. It also presents Jensen’s inequality for a suboptimal resource-allocation algorithm that avoids evaluating statistical expectations, and notes WMMSE as an approach to computing them.

IV. GEE MAXIMIZATION WITH PARTIAL CSI ON H

  • With statistical CSI for H, GEE is ergodic achievable rate divided by average consumed energy, not the average of instantaneous GEE.The latter is rejected because it is not the ratio between the system benefit and the cost incurred to achieve it.Source: §IV · p. 11
  • For any AF matrix A, the optimal source covariance eigenvectors satisfy U_Q = U_t,H.This structure optimizes the source eigenvector matrix under the partial-CSI problem.Source: §IV · p. 11
  • When either Λ_G or Λ_r,H is scaled, the optimal relay precoder satisfies U_A = V_G and V_A = U_r,H.This result assumes the source covariance has the optimal structure U_Q = U_t,H.Source: §IV · p. 12
  • The method uses alternating maximization because the reduced objective is separately pseudo-concave, though not jointly concave or pseudo-concave, in its two variable blocks.The algorithm is guaranteed to converge because GEE is upper-bounded, and its subproblems are pseudo-concave.Source: §IV · pp. 14–15
  • The algorithm requires statistical-expectation evaluations, which can be approximated by a deterministic function when computational complexity is critical.It can be implemented centrally or in a distributed way.Source: §IV · p. 15
  • WMMSE can replace the rate numerator with weighted traces to enable expectation computation, while Jensen’s inequality is another approximation approach briefly outlined here.Detailed discussion of approximation techniques is not the work’s main purpose.Source: §IV · p. 15
  • Jensen’s inequality lower-bounds the objective numerator, yielding a sub-optimal allocation approach that requires no statistical-expectation evaluation.The resulting approximation is used in place of the numerator of Problem (25).Source: §IV · pp. 15–16
  • The deterministic approximation is concave in either optimization block when the other is fixed, enabling alternating maximization without evaluating statistical expectations.The algorithm follows Algorithm 2’s pseudo-code with the objective numerator replaced by approximation (26).Source: §IV · pp. 15–16

V. GEE MAXIMIZATION WITH PARTIAL CSI ON G

This section treats the case where the relay-to-destination channel is statistically known under a Kronecker model while the source-to-relay channel is perfectly known. It derives the optimal matrix structure and reduces the design to power control.

V. GEE MAXIMIZATION WITH PARTIAL CSI ON G

  • The setting assumes statistical knowledge of the relay-to-destination channel under a Kronecker model, with the source-to-relay channel perfectly known.The scenario is relevant when the source-to-relay channel varies slowly and the relay-to-destination channel varies rapidly, as in a downlink with a mobile destination.Source: §V · p. 16
  • The optimal source covariance and AF matrix have eigenvector alignments specified by Proposition 5.Source: §V · p. 16
  • With these alignments, the design reduces to power control, and the resulting choices are feasible for the QoS and power constraints.Source: §V · p. 17

RINR)1/2U H

After fixing the optimal directions, the remaining problem is power control over the source and relay allocation vectors. The objective is separately pseudo-concave in each vector, enabling an alternating maximization algorithm.

RINR)1/2U H

  • The remaining power-control problem is not jointly concave or jointly pseudo-concave in the source and relay allocation vectors.Source: RINR)1/2U H · p. 17
  • For either allocation vector held fixed, the objective is separately pseudo-concave in the other vector.For the source vector, the numerator is concave and denominator linear; for the relay vector, the denominator is linear and the numerator is shown concave.Source: RINR)1/2U H · p. 17
  • Algorithm 3 alternately optimizes the relay and source power vectors until convergence.Source: RINR)1/2U H · p. 18

VI. OPTIMALITY OF FULL-POWER BEAMFORMING.

The section derives sufficient conditions for source beamforming to be optimal in energy-efficiency maximization, distinguishing full-power from non-full-power beamforming. For full-power beamforming, those conditions can identify a single-stream, full-power solution without solving the source optimization problem.

VI. OPTIMALITY OF FULL-POWER BEAMFORMING.

  • The section derives conditions for single-stream source transmission under both statistical-CSI scenarios, with the QoS constraint relaxed during their derivation.For the original problem, a beamforming solution to the relaxed problem is optimal if it satisfies the QoS constraint.Source: §VI · p. 18, A. FP beamforming with partial CSI on H · p. 19
  • Full-power beamforming means allocating all feasible source power to one stream, while non-full-power beamforming requires only a rank-one source covariance.The full-power conditions are also sufficient for non-full-power beamforming.Source: §VI · p. 19
  • When the full-power condition holds, the optimal source allocation is (P, 0, …, 0), so the source optimization can be bypassed.The condition is checked before solving the source allocation problem.Source: §VI · pp. 18–19, A. FP beamforming with partial CSI on H · p. 21
  • Proposition 6 supplies a condition to check whether single-stream transmission is optimal, and its C2,H < 0 case addresses whether FP beamforming is optimal under an if-and-only-if criterion.Source: A. FP beamforming with partial CSI on H · pp. 20–21
  • For statistical CSI on the relay-destination channel, Proposition 7 gives full-power beamforming conditions for the cases C2,G ≥ 0 and C2,G ≤ 0.These conditions are derived for the source allocation problem with the QoS constraint relaxed.Source: B. FP beamforming with partial CSI on G · p. 22

VII. NUMERICAL RESULTS

The numerical results compare the proposed algorithms across CSI assumptions and test beamforming conditions. They report convergence behavior, performance differences between CSI scenarios, and agreement between a derived condition and the simulated allocation.

VII. NUMERICAL RESULTS

  • For both statistical-CSI scenarios, the algorithms converge after a few iterations; at high SNR they reach a unique fixed point regardless of initialization.At low SNR, initialization affects the fixed point.Source: §VII · pp. 23–24
  • Perfect CSI performs better than statistical CSI, and perfect knowledge of the source-relay channel H performs better than perfect knowledge of relay-destination channel G.The paper explains that H affects both the GEE numerator and denominator, whereas G affects only the numerator.Source: §VII · p. 24
  • With correlation indices ρ = 0.1 and ρ = 0.9, perfect knowledge of H yields higher instantaneous GEE than perfect knowledge of G, and the gap to perfect CSI is larger at ρ = 0.1.The paper identifies ρ as a measure of how much channel information is available.Source: §VII · p. 24
  • The simulated source allocation enters the beamforming-optimality region at the limit marked by the Proposition 6 condition.Beyond that limit, some power is allocated to the second transmission stream.Source: §VII · pp. 24–25

VIII. CONCLUSIONS

The paper formulates relay-assisted MIMO energy-efficiency optimization for perfect and selected statistical CSI settings, then derives optimal eigenvector structures and solves the resulting power allocation problem. It also identifies a broader statistical-CSI case whose power-control structure leaves low-complexity algorithm design open.

VIII. CONCLUSIONS

  • The paper optimizes GEE, defined as achievable rate divided by consumed power, for relay-assisted systems with perfect or statistical CSI.The considered statistical-CSI cases include one channel known statistically while the other is perfectly known.Source: §VIII · p. 25
  • The optimal source and relay eigenvector matrices diagonalize both the numerator and denominator of GEE.The resulting power-allocation problem is separately pseudo-concave in the source and relay allocation vectors and is addressed using fractional programming and alternating maximization.Source: §VIII · p. 25
  • For the considered statistical-CSI scenarios, the paper derives conditions for beamforming optimality and numerically contrasts the scenarios.Source: §VIII · p. 25
  • With both channels known only statistically, diagonalizing their correlation matrices is optimal, but separate pseudo-concavity of the resulting power-control problem cannot be shown.The paper identifies low-complexity power-control design in this setting as requiring further investigation.Source: §VIII · p. 25

APPENDIX

The appendix collects matrix results used in the analysis, including eigenvalue-ordering criteria for trace and log-determinant expressions and diagonal structure results for matrix functions. It also records concavity properties for matrix-valued functions.

APPENDIX

  • The trace of a product of Hermitian matrices is minimized when the matrices commute and their eigenvalues are arranged in opposite order.Source: APPENDIX · p. 25
  • The log-determinant expression log|I + T^-1R| is minimized when T and R commute and their eigenvalues are arranged in the same order.Source: APPENDIX · p. 25
  • For concave or convex g, the appendix states that g is maximized or minimized, respectively, when X is diagonal.For linear g, setting X's eigenvector matrix to the identity leaves the function unchanged.Source: APPENDIX · p. 26
  • The matrix-valued functions (I + MX^-1M^H)^-1 and Λ(νI + LΛ)^-1 are stated to be matrix-concave under the listed conditions.Source: APPENDIX · p. 26
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