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Energy Efficiency and Spectral Efficiency Tradeoff in RIS-Aided Multiuser MIMO Uplink Transmission

Li You, Jiayuan Xiong, Derrick Wing Kwan Ng, Chau Yuen, Wenjin Wang, Xiqi Gao

arXiv:2011.09724v1cs.IT

TL;DR

The paper addresses EE-SE balancing in RIS-aided multi-user MIMO uplinks with partial CSI and practical RIS phase constraints. It jointly optimizes user precoding and RIS phases through RE maximization, yielding tunable tradeoffs under high power budgets and convergent designs.

  • Problem

    RIS transmission design must balance EE and SE while adapting practical continuous or discrete RIS phase shifters under partial CSI.

  • Method

    The framework jointly optimizes user transmit precoding and RIS phase shifts for RE using alternating optimization, closed-form transmit subspaces, asymptotic power allocation, and iterative RIS updates.

  • Results

    At high power budgets, RE maximization provides different EE-SE tradeoffs according to the weighting factor β, while tradeoffs are nearly identical below 25 dBm.

  • Takeaways & Limitations

    The proposed framework supports EE-SE tradeoff design for RIS-aided multi-user MIMO uplinks with both continuous- and discrete-phase RIS configurations.

Abstract

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The emergence of reconfigurable intelligent surfaces (RISs) enables us to establish programmable radio wave propagation that caters for wireless communications, via employing low-cost passive reflecting units. This work studies the non-trivial tradeoff between energy efficiency (EE) and spectral efficiency (SE) in multiuser multiple-input multiple-output (MIMO) uplink communications aided by a RIS equipped with discrete phase shifters. For reducing the required signaling overhead and energy consumption, our transmission strategy design is based on the partial channel state information (CSI), including the statistical CSI between the RIS and user terminals (UTs) and the instantaneous CSI between the RIS and the base station. To investigate the EE-SE tradeoff, we develop a framework for the joint optimization of UTs' transmit precoding and RIS reflective beamforming to maximize a performance metric called resource efficiency (RE). For the design of UT's precoding, it is simplified into the design of UTs' transmit powers with the aid of the closed-form solutions of UTs' optimal transmit directions. To avoid the high complexity in computing the nested integrals involved in the expectations, we derive an asymptotic deterministic objective expression. For the design of the RIS phases, an iterative mean-square error minimization approach is proposed via capitalizing on the homotopy, accelerated projected gradient, and majorization-minimization methods. Numerical results illustrate the effectiveness and rapid convergence rate of our proposed optimization framework.

I. INTRODUCTION

This paper addresses the EE-SE tradeoff in RIS-aided multi-user MIMO uplink systems under partial CSI and continuous or discrete RIS phase shifts. It develops a joint RE-maximization framework for UT transmission and RIS phase design.

  • Motivation: EE and SE maximization can conflict, motivating an optimization framework for their tradeoff in RIS-aided multi-user MIMO uplinks.The paper identifies this as the first framework targeting the EE-SE tradeoff in this setting.
  • Optimization objective: The framework jointly optimizes UT transmit covariance matrices and RIS phase shifts to maximize resource efficiency, while also supporting EE or SE maximization.Resource efficiency provides the selected balance between energy and spectral efficiency.
  • System setting: The transmission strategy uses partial CSI: instantaneous RIS-to-BS CSI and statistical UT-to-RIS CSI.The framework considers both continuous and discrete RIS phase shifts.
  • UT precoding: Closed-form UT transmit directions and an asymptotic SE expression simplify transmit covariance optimization and reduce the complexity of nested-integral evaluation.The resulting design separates transmit subspaces from power allocation.
  • RIS phase design: RIS phase optimization is formulated through MSE minimization and handled using homotopy, projected-gradient, and majorization-minimization techniques for continuous and discrete phases.The overall framework uses alternating optimization to iteratively update UT covariances and RIS phases.

A. System Model

The system comprises multiple-antenna UTs transmitting to an M-antenna BS through an RIS under a jointly correlated Rayleigh UT-to-RIS channel model. Its SE and EE metrics account for partial CSI, bandwidth, transmit power, circuit power, and RIS hardware consumption.

  • Channel model: The RIS-to-BS channel is H_1, while each UT-to-RIS channel H_2,k follows a jointly correlated Rayleigh fading model.The model captures spatial correlations through deterministic eigenvector matrices and a random Gaussian component.
  • CSI assumptions: Only partial CSI is assumed: H_1 is instantaneously known, whereas fast-varying UT-to-RIS channel statistics are available.This setting reflects the paper’s reduced-CSI transmission design.
  • RIS model: Each RIS coefficient has unit magnitude, with adjustable phase; CPS permits arbitrary phases, whereas DPS restricts phases to a finite set.The RIS is modeled as ideally passive and does not change reflected-signal amplitude.
  • Performance metrics: Average system EE is defined from ergodic system SE and total energy consumption over transmission bandwidth W.The ergodic SE averages over the UT-to-RIS channel realizations.
  • Energy model: System power consumption includes UT transmit power, static UT and BS circuit power, and RIS hardware power proportional to the number of reflecting units.The per-unit RIS power depends on phase-shifter resolution, with τ = 2^b discrete levels.

C. Problem Formulation

The paper formulates resource-efficiency maximization to balance EE and SE in RIS-aided multiuser MIMO uplinks, jointly designing UT transmit covariances and RIS phases under practical constraints. It then decomposes the difficult joint problem into sequential covariance and phase-shift updates.

  • C. Problem Formulation: Resource efficiency (RE) provides a unit-consistent metric for balancing energy efficiency and spectral efficiency.The weighting factor β controls the tradeoff, with small β emphasizing EE and large β emphasizing SE.
  • C. Problem Formulation: The design jointly optimizes UT transmit covariance matrices Q_k and the RIS phase-shift matrix Φ to improve system RE.The RIS may use continuous or discrete phase-shift constraints.
  • C. Problem Formulation: The optimization is challenging because Q and Φ are tightly coupled, ergodic SE requires high-dimensional expectations, and RE inherits fractional non-convexity.These difficulties become more significant as the number of RIS elements grows.
  • III. OPTIMIZATION FRAMEWORK FOR RE MAXIMIZATION: Alternating optimization decouples Q and Φ, updating UT covariances and RIS phases separately and sequentially.The covariance block is optimized with Φ fixed, followed by phase optimization with Q fixed.
  • A. Optimization of UTs’ Transmit Covariance Matrices: The covariance subproblem is subject to per-UT transmit-power limits tr{Q_k} ≤ P_max,k and positive-semidefinite constraints.These constraints define feasible UT transmit covariance matrices.
  • A. Optimization of UTs’ Transmit Covariance Matrices: Each covariance matrix is decomposed as Q_k = V_kΛ_kV_k^H, separating UT transmit directions from power allocation.V_k contains eigenvectors defining the transmit subspace, while Λ_k contains the allocated powers.

1) Optimal Transmit Directions at UTs:

The optimal UT transmit directions have a closed-form characterization, reducing precoding design to power allocation. A deterministic-equivalent approximation then replaces costly channel expectations using statistical channel information and iterative auxiliary updates.

  • 1) Optimal Transmit Directions at UTs:: The optimal eigenmatrix V_k equals the unitary matrix V_2,k from the UT-to-RIS channel decomposition.Thus, the optimal transmit directions lie in the signal space spanned by the channel transmit-correlation eigenvectors.
  • 1) Optimal Transmit Directions at UTs:: With V_k fixed to V_2,k, the UT precoding problem becomes a power-allocation problem over Λ_k.This substantially reduces the number of optimization variables.
  • 1) Optimal Transmit Directions at UTs:: The deterministic equivalent avoids directly computing high-dimensional expectations required by the ergodic SE objective.It uses statistical knowledge of the random UT-to-RIS channel components rather than their actual realizations.
  • 1) Optimal Transmit Directions at UTs:: The DE formulation represents the effective channel using D, G, and auxiliary quantities derived from Λ_k, RIS phases, and channel statistics.The auxiliary vectors γ_k and ψ_k are obtained from coupled fixed-point equations.
  • 1) Optimal Transmit Directions at UTs:: The auxiliary quantities γ and ψ are computed by cyclic iterative updates from initialized values.These updates generate the fixed-point solutions used in the DE objective.
  • 1) Optimal Transmit Directions at UTs:: The asymptotic SE approximation is almost surely accurate as the UT-to-RIS channel matrix dimensions grow.The paper also states that the approximation remains sufficiently accurate for small-scale MIMO systems.

3) Quadratic Transformation:

The quadratic transformation removes the fractional structure in the RE objective by introducing an auxiliary variable and iteratively optimizing it with the primal variables. The resulting fixed-variable subproblem is concave and efficiently solvable.

  • 3) Quadratic Transformation:: The RE subproblem remains generally non-concave because its objective contains a fractional term.Although the SE component is strictly concave, the fractional objective prevents direct convex optimization.
  • 3) Quadratic Transformation:: Introducing an auxiliary variable y decouples the numerator and denominator through the quadratic transformation.This converts P3 into an equivalent non-fractional problem P4.
  • 3) Quadratic Transformation:: For fixed Λ, the optimal y is directly calculated, while Λ is optimized with y held fixed.The two variable blocks are updated in an alternating iterative procedure.
  • 3) Quadratic Transformation:: For fixed y, P4 is concave in Λ because the SE term and −P(Λ) are concave.Classical convex optimization can therefore solve the Λ subproblem numerically and efficiently.
  • 3) Quadratic Transformation:: Algorithm 2 generates a convergent, non-decreasing sequence of objective values, with its final Λ* output being a stationary point of P3.The algorithm summarizes the quadratic-transformation updates for power allocation.

B. Adjustment of RIS Phase Shifters

With UT transmit covariances fixed, RIS phase optimization can target SE because RIS operation consumes no transmit power. The resulting problem remains difficult because its objective and continuous or discrete phase constraints are non-convex.

  • B. Adjustment of RIS Phase Shifters: When Q is fixed, optimizing RIS phases Φ for RE is equivalent to maximizing system SE.The total energy consumption is independent of Φ because the passive RIS consumes no transmit power.
  • B. Adjustment of RIS Phase Shifters: The RIS phase problem is non-convex in Φ and difficult under both continuous and discrete phase-shift constraints.Continuous phase shifters impose unit-modulus manifold constraints, whereas discrete phase shifters produce a mixed-integer program.

1) Weighted Minimum MSE (WMMSE) Method:

The WMMSE method reformulates the SE maximization subproblem as an equivalent MSE minimization problem, then updates auxiliary and beamforming variables blockwise. RIS phase optimization remains difficult because phase-shift constraints make the problem non-convex.

  • WMMSE reformulation: WMMSE equivalently converts the challenging SE maximization problem into a more tractable MSE minimization problem.The transformed objective is convex in each variable when the other variables are fixed.
  • Block-coordinate updates: BCD decouples the transformed problem by separately updating the auxiliary variable, receiving matrix, and RIS phase matrix.The auxiliary-variable and receiving-matrix updates have explicit solutions, while RIS phases require a separate procedure.
  • RIS phase update: The RIS phase update is isolated as the remaining issue in a BCD iteration after the WMMSE reformulation.The phase adjustment is formulated through an equivalent optimization expression involving the phase vector and matrix.
  • Algorithm: Algorithm 3 presents the iterative WMMSE procedure, which alternates variable updates until a stopping criterion is satisfied.The resulting phase matrix is returned as the solution to the SE subproblem.
  • Non-convex constraints: Non-convexity in the phase update arises from the RIS phase-shift constraints, despite convexity of the objective function.This motivates the penalty-based treatment developed in the subsequent method.

3) Penalty Method:

The penalty method handles non-convex RIS phase constraints by replacing them with a penalized problem over a convex hull. An adequately large penalty parameter preserves equivalence with the original formulation while enabling a more manageable optimization problem.

  • Penalty reformulation: The NSP method addresses non-convex phase-shift constraints by imposing a negative square penalty on the objective.The homotopy approach approximates the difficult constrained problem with an easier penalized one.
  • Equivalence: With a sufficiently large penalty parameter, the original and penalized problems have equivalent global optima.The required threshold depends on the Lipschitz constant and phase-shift setting.
  • Convex relaxation: The penalized formulation uses the convex hull of the feasible phase-shift set, making the resulting feasible region convex for continuous and discrete phases.The convex hull is a unit circle for continuous phase shifts and a regular polygon for discrete phase shifts.
  • Method choice: The penalty function is quadratic and independent of the phase-vector dimension, unlike higher-order polynomial alternatives.This motivates selecting NSP for the phase-update problem.
  • Parameter selection: An adequately large penalty parameter is sufficient for equivalence, so finely adjusting it is unnecessary.The method therefore need not identify an exact penalty value.

4) Gradient Extrapolated MM Method:

The gradient extrapolated MM method solves the penalized RIS phase problem by majorizing its difference-of-convex objective and applying projected gradient updates. Using one accelerated projected-gradient step per MM iteration reduces computational burden while retaining stationary convergence.

  • Majorization-minimization: MM handles the penalized phase problem because its objective has a difference-of-convex form.The convex quadratic terms and concave negative penalty term permit construction of a majorizer.
  • Convex subproblem: The majorizer linearizes the concave negative penalty term, producing a convex smooth subproblem over the convex feasible set.Projected-gradient methods are therefore applicable to each MM subproblem.
  • Accelerated updates: APG is selected for the smooth convex subproblems because it has a faster convergence rate than ordinary projected gradient.The projection operators have closed-form element-wise expressions for both phase-shift settings.
  • GEMM acceleration: GEMM performs only one APG iteration per MM update instead of solving each MM subproblem exactly.This inexact update is designed to reduce computational complexity.
  • Convergence: Despite inexact MM updates, GEMM retains the same stationary convergence guarantee as exact MM and converges much faster in the reported comparison.The faster convergence is attributed to limiting APG work to one iteration per MM update.
  • Penalty schedule: The NSP-based GEMM procedure gradually increases the penalty parameter until the equivalence condition is met.It is initialized relatively small to help avoid an ill-posed problem.

C. Overall Algorithm and Complexity Analysis

The overall AO-based algorithm alternates optimization of user transmit covariances and RIS phases to maximize RE under partial CSI. Its weighting also specializes the framework to EE or SE maximization, while the phase-update complexity is dominated by GEMM iterations.

  • Overall algorithm: Algorithm 5 combines the transmit-covariance and RIS-phase procedures into a complete AO-based RE maximization approach.The method is designed for RIS-aided multi-user MIMO uplink transmission with partial CSI.
  • EE-SE tradeoff: Different values of β generate different EE-SE tradeoff schemes, and the framework can specialize to either EE or SE maximization.Setting β = 0 yields EE maximization, while ξ_k = 0 enables the SE-maximization special case.
  • AO iterations: Algorithm 5 alternates updates of transmit variables and RIS phases, recalculating asymptotic SE and auxiliary parameters during the iterations.The procedure repeats until a stopping criterion is satisfied and outputs the transmit covariance matrices and RIS phase matrix.
  • Phase optimization: The phase-optimization component uses the NSP-based GEMM method within the overall alternating framework.Algorithm 4 supplies the RIS phase update used by Algorithm 5.
  • Complexity analysis: The overall complexity depends on AO, quadratic-transformation, BCD, and GEMM iteration counts, with phase optimization contributing approximately O(N_R^3) per BCD iteration.The gradient computation in GEMM is approximately O(N_R^2), and the stated phase complexity assumes N_R > M.
  • CSI scope: The framework can extend to instantaneous CSI by replacing deterministic-equivalent objectives with instantaneous expressions and removing expectation operations.Only slight modification is required in that CSI setting.

IV. NUMERICAL RESULTS

The numerical evaluation uses a suburban macro scenario with 3GPP-based statistical channel parameters and reports the simulation setup.

  • Simulation setup: Simulations model small-scale fading in a suburban macro scenario using primary statistical channel parameters from the 3GPP spatial channel model.The channel statistics Ω_k are obtained using existing methods.
  • Simulation setup: The assumed large-scale fading includes a path loss of −120 dB for all end-to-end composite channels.
  • Simulation setup: Table II lists the simulation setup parameters used to evaluate the RIS-aided multi-user MIMO uplink system.

A. Convergence Performances

The proposed algorithms converge rapidly and support SE, EE, and adjustable EE-SE tradeoff evaluations across RIS phase-shifter resolutions and power budgets.

  • Convergence Performances: All evaluated algorithms exhibit fast convergence in typical power-budget regions.Algorithm 2 usually converges after one step, while Algorithm 3 does so for small Pmax.
  • SE Maximization Design: Optimizing both transmit power allocation and RIS phase shifts provides remarkable SE gains, including with the lowest-resolution discrete phase shifters.The comparison includes fixed-RIS and equal-power baselines, as well as a full-instantaneous-CSI benchmark.
  • EE Maximization Design: Under EE maximization, EE rises only below a transmit-power threshold and then saturates, whereas SE maximization eventually causes EE to decline at large Pmax.The EE-maximizing transmit power remains constant once the maximum EE is attained.
  • EE Maximization Design: Higher-resolution phase shifters can improve SE but substantially increase static hardware power, potentially making discrete phase shifters more energy efficient.
  • EE-SE Tradeoff: When Pmax is below 25 dBm, weighting factors produce nearly identical EE-SE tradeoffs; at higher budgets, increasing β yields higher EE but lower SE.The extreme cases β/Ptot = 0.01 and β/Ptot = 100 essentially perform EE and SE maximization, respectively.
  • EE-SE Tradeoff: GEMM achieves performance nearly identical to exact MM while requiring nearly half the runtime for updating Φ.The comparison uses K = 4, N_k = 2, M = 32, and β/Ptot = 0.01.
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