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Exploiting Amplitude Control in Intelligent Reflecting Surface Aided Wireless Communication with Imperfect CSI

Ming-Min Zhao, Qingqing Wu, Min-Jian Zhao, Rui Zhang

arXiv:2005.07002v3cs.ITmath.IT

TL;DR

The paper addresses whether IRS reflection-amplitude control can benefit multiuser wireless communication when CSI is imperfect. It jointly designs AP precoding and IRS reflection coefficients, and reports gains over full reflection, with amplitude control more cost-effective than phase-shift control in certain practical setups.

  • Problem

    Under imperfect CSI, it was unknown whether IRS amplitude control benefits performance, while prior robust designs mainly assumed full reflection and canonical CSI-error models.

    Source: §I · p. 2

  • Method

    The paper derives achievable rates using practical channel-estimation error statistics and jointly optimizes AP precoders and IRS reflection amplitudes and phases, using separate algorithms for single-user and multiuser cases.

    Source: §I · p. 2

  • Results

    Controlling IRS reflection amplitude under imperfect CSI achieves additional gains over conventional full-reflection schemes and can be more cost-effective than phase-shift control under certain practical setups.

    Source: §I · p. 2

  • Takeaways & Limitations

    Amplitude beamforming can be a low-cost alternative to conventional phase beamforming in practice, with comparable or more favorable performance when CSI errors are severe.

    Source: §VI · p. 13

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) is a promising new paradigm to achieve high spectral and energy efficiency for future wireless networks by reconfiguring the wireless signal propagation via passive reflection. To reap the potential gains of IRS, channel state information (CSI) is essential, whereas channel estimation errors are inevitable in practice due to limited channel training resources. In this paper, in order to optimize the performance of IRS-aided multiuser systems with imperfect CSI, we propose to jointly design the active transmit precoding at the access point (AP) and passive reflection coefficients of IRS, each consisting of not only the conventional phase shift and also the newly exploited amplitude variation. First, the achievable rate of each user is derived assuming a practical IRS channel estimation method, which shows that the interference due to CSI errors is intricately related to the AP transmit precoders, the channel training power and the IRS reflection coefficients during both channel training and data transmission. Then, for the single-user case, by combining the benefits of the penalty method, Dinkelbach method and block successive upper-bound minimization (BSUM) method, a new penalized Dinkelbach-BSUM algorithm is proposed to optimize the IRS reflection coefficients for maximizing the achievable data transmission rate subjected to CSI errors; while for the multiuser case, a new penalty dual decomposition (PDD)-based algorithm is proposed to maximize the users' weighted sum-rate. Simulation results are presented to validate the effectiveness of our proposed algorithms as compared to benchmark schemes. In particular, useful insights are drawn to characterize the effect of IRS reflection amplitude control (with/without the conventional phase shift) on the system performance under imperfect CSI.

I. INTRODUCTION

The paper studies joint active and passive beamforming for IRS-aided multiuser communication under imperfect CSI, where it investigates whether amplitude control can improve performance. Its system model allows IRS elements to vary both reflection amplitude and phase, rather than assuming full reflection.

I. INTRODUCTION

  • Whether IRS amplitude control is beneficial under imperfect CSI remains unknown in prior work, which mainly used canonical CSI-error models and assumed full reflection.Channel-estimation errors are inevitable with limited training resources, making them important to account for in reflection design.Source: §I · p. 2
  • Using practical IRS channel-estimation methods,, the paper derives achievable-rate lower bounds and relates CSI-error interference to precoders, training power, and IRS designs during training and data transmission.Source: §I · p. 2
  • The paper jointly optimizes AP precoders and IRS reflection coefficients, proposing a penalized Dinkelbach-BSUM algorithm for the single-user case and a PDD-based algorithm for the multiuser case.The algorithms can be adapted to continuous or discrete reflection-amplitude and phase-shift control.Source: §I · p. 2
  • Amplitude control under imperfect CSI achieves additional gains over full-reflection schemes and can be more cost-effective than phase-shift control under certain practical setups.The algorithms can handle continuous or discrete amplitude and phase-shift cases.Source: §I · p. 2

A. System Model

  • The downlink model has an M-antenna AP, an N-element IRS, and K single-antenna users, with each IRS coefficient parameterized by amplitude and phase.The amplitude lies in [0, 1], while the phase lies in [0, 2π).Source: A. System Model · p. 3
  • Independent amplitude and phase control is more flexible than a coupled reflection model, but requires more sophisticated reflecting-element design.Source: A. System Model · p. 3

B. Channel Estimation and Achievable Rate

The paper estimates channels from uplink pilots under reciprocity and uses a lower bound on mutual information as the achievable rate. Under this model, CSI-error interference depends on both training and data-transmission choices.

B. Channel Estimation and Achievable Rate

  • Uplink pilots are sent while IRS reflection patterns vary; least-squares estimation yields direct and cascaded channel estimates.The paper assumes users are trained consecutively and sets the IRS amplitudes to 1 during training to maximize signal reflection.Source: B. Channel Estimation and Achievable… · pp. 3–4
  • For discrete phase shifts, the estimated-channel error has zero mean, with its covariance determined by the training pattern and power.The continuous-phase case with Nr = N + 1 has i.i.d. CSCG channel errors.Source: B. Channel Estimation and Achievable… · p. 4
  • Because mutual information is difficult to derive in closed form, the paper uses a tractable lower bound as each user's achievable rate.The rate treats the estimated-channel desired signal separately from multiuser interference and CSI-error-induced interference.Source: B. Channel Estimation and Achievable… · p. 4
  • CSI-error interference depends on the training reflection pattern, uplink training powers, data reflection coefficients, and AP precoders.The paper assumes reciprocity and notes that its beamforming designs can also use other estimation methods when error statistics are available.Source: B. Channel Estimation and Achievable… · pp. 3–4

C. Problem Formulation

The formulation maximizes users' weighted sum-rate by jointly optimizing AP precoders and IRS reflection coefficients under imperfect CSI. The coupled non-convex objective and discrete coefficient constraints make the problem difficult to solve optimally.

C. Problem Formulation

  • The objective is the users' weighted sum of achievable rates, jointly optimized over AP precoders and IRS reflection coefficients under imperfect CSI.The constraints include the AP's total transmit power and the IRS amplitude and phase-shift limits.Source: C. Problem Formulation · p. 4
  • Imperfect-CSI interference depends on error variance, the sum of squared reflection amplitudes, and AP transmit precoders in the stated special case.The error variance depends on uplink training power and the number of reflecting elements.Source: C. Problem Formulation · p. 5
  • The problems are non-convex with nonlinearly coupled precoders and IRS coefficients; discrete coefficient constraints make one formulation a MINLP.The added amplitude variables jointly affect performance with phase shifts, and the paper therefore seeks efficient suboptimal algorithms.Source: C. Problem Formulation · p. 5

III. SINGLE-USER SYSTEM

The single-user analysis isolates the effect of IRS amplitude control under imperfect CSI. It finds that full reflection need not maximize rate, particularly when CSI errors are large, and proposes an algorithm for discrete amplitude and phase control.

III. SINGLE-USER SYSTEM

  • The single-user case removes multiuser interference to isolate the gain from adding amplitude control to phase-shift control.The paper drops the user index and multiuser interference terms in this case.Source: §III · p. 5
  • Full reflection is not necessarily rate-optimal, especially when CSI errors are large and the error-related terms dominate the corresponding signal terms.The stated conditions are Rnn∥w∥² ≫ Ann and en ≫ cn.Source: §III · p. 5
  • The penalized Dinkelbach-BSUM algorithm is reported to achieve better performance with lower complexity than SDR-BCD for the single-user problem.The comparison is to an SDR-BCD method adapted from a prior single-antenna, full-reflection formulation.Source: §III · p. 5

A. Joint Discrete Amplitude and Phase-Shift Control

For discrete IRS amplitudes and phase shifts, the paper uses a penalized Dinkelbach-BSUM procedure to optimize reflection coefficients. It alternates continuous and discrete auxiliary variables, penalizes their mismatch, and converges without claiming global optimality.

A. Joint Discrete Amplitude and Phase-Shift Control

  • An outer loop decreases the penalty for v ≠ u, while an inner Dinkelbach-BSUM loop alternately optimizes the two variable blocks.The auxiliary vector facilitates parallel updating of IRS coefficients.Source: A. Joint Discrete Amplitude and… · pp. 5–6
  • With a fixed penalty coefficient, the Dinkelbach-BSUM steps are monotonically convergent, and the paper concludes that Algorithm 1 converges.The final algorithm also applies BCD to refine IRS coefficients by one-dimensional search over the discrete set.Source: A. Joint Discrete Amplitude and… · pp. 6–7
  • The discrete-coefficient problem is NP-complete, so the paper presents a convergent solution at acceptable computational complexity rather than an optimal one.Further investigation of more advanced algorithms is left open; the stated complexity is driven mainly by the v update.Source: A. Joint Discrete Amplitude and… · p. 7
  • The method introduces an auxiliary copy u of the reflection vector v, keeps u discrete, and relaxes v to continuous values before penalizing their mismatch.This differs from the conventional relax-and-then-quantize scheme.Source: A. Joint Discrete Amplitude and… · p. 7

B. Joint Continuous Amplitude and Phase-Shift Control

For continuous joint amplitude and phase control, the paper adapts the penalized Dinkelbach-BSUM algorithm by changing its auxiliary-variable update. A Lagrange-duality-based BCD refinement yields semi-closed-form reflection coefficients, and the method extends to mixed continuous/discrete controls.

B. Joint Continuous Amplitude and Phase-Shift Control

  • The continuous-control algorithm modifies the penalized Dinkelbach-BSUM method only in its u-subproblem and updates reflection coefficients successively until the objective's fractional increase falls below a threshold.It initializes v using Algorithm 1 with a modified constraint.Source: B. Joint Continuous Amplitude and… · p. 7
  • The u-update is a decoupled projection problem, so its variables can be optimized independently.The text states that the resulting optimal solution is readily obtained.Source: B. Joint Continuous Amplitude and… · p. 7
  • Lagrange duality replaces inefficient one-dimensional searches over each element's amplitude with semi-closed-form reflection-coefficient updates in the BCD refinement.The coefficient of each element is optimized while the others are fixed.Source: B. Joint Continuous Amplitude and… · p. 7
  • Algorithm 2 has complexity O(IBCDN^3 + IPIDN^3 log(1/εbi)) and can be extended to continuous-amplitude/discrete-phase or discrete-amplitude/continuous-phase cases.The extension combines techniques from Algorithms 1 and 2.Source: B. Joint Continuous Amplitude and… · p. 7

IV. MULTIUSER SYSTEM

For the multiuser system, the paper uses WMMSE and PDD to optimize active precoders and IRS reflections under imperfect CSI. The approach supports continuous and discrete amplitude/phase controls, and the paper expects amplitude control to offer larger gains than in the single-user case because multiuser interference is more severe.

IV. MULTIUSER SYSTEM

  • The multiuser method transforms the weighted sum-rate problem with WMMSE, then uses a double-loop PDD algorithm with block minimization inside and dual-variable and penalty updates outside.The inner augmented-Lagrangian problem is solved until convergence criteria are met.Source: §IV · p. 7
  • Each block, including the active precoders and IRS reflection coefficients, can be updated in closed form or by simple bisection, and the algorithm accommodates continuous or discrete amplitude and phase controls.Source: §IV · p. 8
  • Amplitude-control gains are expected to be larger in multiuser than single-user systems because multiuser interference is more severe.The paper attributes this expectation to the quadratic dependence of each element's interference-plus-noise power on its amplitude and the similar form of each user's achievable rate.Source: §IV · p. 8

A. Joint Discrete Amplitude and Phase-Shift Optimization

The multiuser optimization is recast as a WMMSE problem and solved by a PDD-based block algorithm. Its updates include receive coefficients, weights, transmit precoders, and IRS-related variables; the method is guaranteed to converge.

A. Joint Discrete Amplitude and Phase-Shift Optimization

  • WMMSE gives problem (27) the same globally optimal solution as its equivalent WMMSE formulation, using per-user receive coefficients and weighting factors.The receive coefficient and weight are optimized from the mean squared error.Source: A. Joint Discrete Amplitude and… · p. 8
  • An auxiliary variable u separates the optimization of v and enables its parallel updating in the PDD algorithm.Source: A. Joint Discrete Amplitude and… · p. 8
  • The transmit-precoder update uses a QCQP whose decoupled Lagrangian permits a first-order solution, with bisection used to find the optimal dual variable when the initial precoders exceed the power constraint.Source: 3) The optimization of wk · pp. 8–9
  • The inner block-coordinate loop successively updates receive coefficients, weights, precoders, v, and u; the outer loop updates the dual variables and reduces the penalty coefficient until the constraint violation is sufficiently small.Source: 4) The optimization of v · p. 9
  • The proposed PDD algorithm is guaranteed to converge, with complexity O(IoIi(KN^3 log(1/εbi) + N^3)) dominated by matrix inversions for the transmit precoders and v.Introducing the dual variable μ improves convergence behavior in both objective value and constraint violation compared with setting μ = 0.Source: 4) The optimization of v · p. 9

B. Joint Continuous Amplitude and Phase-Shift Optimization

The continuous-amplitude/phase multiuser PDD algorithm retains nearly all updates from the discrete case, changing only the auxiliary-variable update. For the single-user case, the paper favors its separate algorithms because they avoid WMMSE and use fewer auxiliary variables or inner-loop blocks.

B. Joint Continuous Amplitude and Phase-Shift Optimization

  • In the continuous counterpart, updates for the receive variables, precoders, and v are almost identical to the discrete case; only the u optimization differs.The u update decomposes into N subproblems.Source: B. Joint Continuous Amplitude and… · p. 9
  • The continuous-case u subproblems are solved in parallel, and the algorithm's complexity is similar to the discrete case because precoder and v optimization dominate.The same u-update distinction applies to DACP and CADP cases.Source: B. Joint Continuous Amplitude and… · p. 9
  • For single-user optimization, Algorithms 1 and 2 are preferred over adapting the PDD algorithm because they avoid WMMSE and use fewer auxiliary variables or inner-loop blocks, leading to faster convergence and more efficient solutions.Source: B. Joint Continuous Amplitude and… · p. 10

V. SIMULATION RESULTS

The simulations evaluate channel estimation and achievable-rate performance for an IRS-aided multiuser system under imperfect CSI. They find that amplitude control can improve rate over full reflection, with its gains depending on training resources, phase resolution, and system size.

V. SIMULATION RESULTS

  • The simulation places the AP at (2 m, 0, 0) and the IRS at (0, 45 m, 2 m), with users randomly located in a cluster.The AP uses a ULA, the IRS a UPA, and the direct AP-user link is modeled with a larger path-loss exponent than the AP-IRS and IRS-user links.Source: §V · p. 10, Figure 2 · p. 10
  • Channel-estimation performance improves with uplink training power and phase resolution; larger element counts generally reduce normalized MSE through increased aperture gain.For Qθ = 1, N = 60 has better estimation performance than N = 80 because the truncated 61 × 61 Hadamard patterns are more near-orthogonal than the 81 × 81 patterns.Source: §V · p. 10, Figure 3 · p. 10

1) Performance Comparison with Existing Algorithms:

  • The proposed algorithm achieves the best rate among the compared algorithms, especially when uplink training power is at most 18 dBm, and uses significantly less computation time than SDR-BCD.It directly handles discrete reflection constraints; its computational time may exceed the channel coherence time, though hardware customization could reduce it.Source: 1) Performance Comparison with… · p. 11, Table I · p. 11
  • Rate gains from amplitude control over full reflection are more significant at lower training power, when more IRS elements are turned off to reduce CSI-error interference.At high training power, phase beamforming performs better; at low training power, amplitude-only beamforming approaches its performance and is generally cheaper to implement.Source: 2) Impact of Uplink Training Power, pu · p. 11, Figure 5 · p. 11
  • The proposed scheme outperforms both the no-IRS and nonrobust schemes, while the nonrobust scheme falls below no IRS when training power is below about 20 dBm.The passage attributes this to stronger CSI-error impact on the AP-IRS-user link than on the direct AP-user link.Source: 2) Impact of Uplink Training Power, pu · p. 12, Figure 6 · p. 12
  • Training duration trades estimation quality and reflection-design gain against the symbols remaining for data, and the optimal training duration decreases as phase resolution Qθ increases.Amplitude-control gains also decrease as training duration or Qθ increases.Source: 3) Impact of Uplink Training Duration,… · p. 12, Figure 7 · p. 12
  • Amplitude-control gains are not sensitive to element count N; IRS elements-off percentage decreases as N grows, while amplitude beamforming performs better than phase beamforming at small N.The passage explains that CSI-error interference does not scale with N and that larger arrays allow finer phase beamforming.Source: 4) Impact of Number of Reflecting… · p. 12, Figure 8 · p. 12
  • As maximum downlink power P increases, amplitude-control gains and IRS elements-off percentage increase, and amplitude beamforming improves faster than phase beamforming.The passage attributes this trend to greater CSI-error interference at higher transmit power.Source: 5) Impact of Maximum Downlink Transmit… · pp. 12–13, Figure 9 · p. 13

B. Multiuser System

The proposed PDD-based algorithm shows that amplitude control can improve multiuser sum-rate under imperfect CSI, with its gain becoming more pronounced as the number of users increases. The conclusion identifies amplitude control as especially advantageous when channel-training resources are limited or AP transmit power is high.

B. Multiuser System

  • With more users, amplitude-control gains become more pronounced as imperfect-CSI multiuser interference grows, and amplitude beamforming outperforms phase beamforming.The comparison uses a multiuser system with K ≥2, M = 6 AP antennas, pu = 18 dBm, and N = 60.Source: B. Multiuser System · p. 13

VI. CONCLUSIONS

  • A PDD-based algorithm is proposed for the multiuser case, alongside a penalized Dinkelbach-BSUM algorithm for the single-user case.The algorithms optimize IRS reflection coefficients and AP transmit precoders under imperfect CSI.Source: §VI · p. 13
  • The conclusion reports additional gains from reflection-amplitude control over full reflection or phase-shift control, especially with limited training resources or high AP transmit power.It also identifies IRS amplitude beamforming as a low-cost alternative that can match or outperform conventional phase beamforming when CSI errors are severe.Source: §VI · p. 13

APPENDIX

The appendix proves a mutual-information lower bound by upper-bounding conditional entropy, then establishes convergence of the penalized Dinkelbach-BSUM algorithm through monotonicity of its Dinkelbach variable.

A. Proof of Proposition 1

  • The proof of Proposition 1 uses the mutual-information entropy identity and bounds conditional entropy to obtain the lower bound in (4).It invokes the Gaussian maximum-entropy property and an MMSE estimate in developing the bound.Source: A. Proof of Proposition 1 · pp. 13–14

B. Proof of Proposition 2

  • The proof of Proposition 2 shows that the Dinkelbach variable is non-decreasing and upper-bounded, guaranteeing convergence for any given β.The monotonicity follows from the algorithm's updates and the global upper-bound property of its approximation.Source: B. Proof of Proposition 2 · p. 14

C. BCD Algorithm for Solving Problem (26)

The BCD procedure optimizes each reflecting element in turn by considering cases based on the magnitude of its coefficient and comparing their objective values.

C. BCD Algorithm for Solving Problem (26)

  • The proposed BCD algorithm successively optimizes each reflection coefficient while holding the others fixed.Its per-element optimization is based on the simplified form of problem (26).Source: C. BCD Algorithm for Solving Problem… · p. 14
  • The derivation considers three cases based on |vn|; when |vn| < 1, complementary slackness requires λ = 0, and comparing the three cases’ objective values yields the optimal solution of problem (26).For |vn| = 0, vn = 0.Source: C. BCD Algorithm for Solving Problem… · p. 14
  • The |vn| < 1 case applies a first-order optimality condition to obtain a quadratic equation whose roots can be evaluated.For the phase variable, differentiating yields a cosine condition for θn.Source: C. BCD Algorithm for Solving Problem… · p. 14

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