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Reconfigurable Intelligent Surfaces Relying on Non-Diagonal Phase Shift Matrices

Qingchao Li, Mohammed El-Hajjar, Ibrahim Hemadeh, Arman Shojaeifard, Alain A. M. Mourad, Bruno Clerckx, Lajos Hanzo

arXiv:2203.08184v1cs.ITeess.SPmath.IT

TL;DR

Conventional RIS designs reflect each element’s signal independently, restricting the phase-shift matrix to a diagonal form. This paper proposes a non-diagonal architecture with inter-element reflection and jointly optimizes beamforming and phase shifts. The proposed design improves on conventional RIS and, as the number of elements increases, outperforms group-connected RIS while approaching fully-connected RIS performance.

  • Problem

    Conventional RIS reflects each element’s incident signal from that same element, producing a diagonal phase-shift matrix, while cooperative fully- and group-connected designs incur increased optimization complexity.

    Source: §I · p. 2

  • Method

    The paper introduces non-diagonal RIS phase shifts that allow inter-element reflection and jointly optimizes transmit beamforming and the phase-shift matrix using alternating optimization and SDR.

    Source: §I · p. 2

  • Results

    The proposed RIS outperforms conventional RIS in channel gain, outage probability, average BER, and achievable rate, and as the number of elements increases it exceeds group-connected RIS while approaching fully-connected RIS performance.

    Source: §I · p. 2

  • Takeaways & Limitations

    The proposed architecture combines performance approaching fully-connected RIS with reduced complexity and less BS–RIS control-link information exchange.

    Source: §I · p. 2

Abstract

from arXiv · show

Reconfigurable intelligent surfaces (RIS) have been actively researched as a potential technique for future wireless communications, which intelligently ameliorate the signal propagation environment. In the conventional design, each RIS element configures and reflects its received signal independently of all other RIS elements, which results in a diagonal phase shift matrix. By contrast, we propose a novel RIS architecture, where the incident signal impinging on one element can be reflected from another element after an appropriate phase shift adjustment, which increases the flexibility in the design of RIS phase shifts, hence, potentially improving the system performance. The resultant RIS phase shift matrix also has off-diagonal elements, as opposed to the pure diagonal structure of the conventional design. Compared to the state-of-art fully-connected/group-connected RIS structures, our proposed RIS architecture has lower complexity, while attaining a higher channel gain than the group-connected RIS structure, and approaching that of the fully-connected RIS structure. We formulate and solve the problem of maximizing the achievable rate of our proposed RIS architecture by jointly optimizing the transmit beamforming and the non-diagonal phase shift matrix based on alternating optimization and semi-define relaxation (SDR) methods. Moreover, the closed-form expressions of the channel gain, the outage probability and bit error ratio (BER) are derived. Simulation results demonstrate that our proposed RIS architecture results in an improved performance in terms of the achievable rate compared to the conventional architecture, both in single-user as well as in multi-user scenarios.

I. INTRODUCTION

Conventional RIS designs keep each element’s reflection independent, producing a diagonal phase-shift matrix; the paper proposes controlled inter-element reflection to increase design flexibility. The proposed architecture is evaluated against conventional, group-connected, and fully-connected designs, and the channel model specifies the RIS-assisted system and its assumptions.

I. INTRODUCTION

  • Conventional RIS reflects each element’s incident signal from that same element, yielding a diagonal phase-shift matrix that does not exploit controlled relationships among RIS elements.Earlier RIS structures – used this same-element assumption; fully- and group-connected designs were proposed in but at increased optimization complexity.Source: §I · p. 2
  • The proposed non-diagonal architecture permits an incident signal at one element to be reflected from another, and jointly optimizes transmit beamforming and phase shifts using alternating optimization and SDR.The paper formulates achievable-rate maximization for this design.Source: §I · p. 2
  • The proposed architecture performs better than conventional RIS in channel gain, outage probability, average BER, and achievable rate, while approaching fully-connected performance and exceeding group-connected performance as RIS elements increase.It uses fewer optimized phase-shift entries and less BS–RIS control-link information than the fully-connected architecture.Source: §I · p. 2

A. Channel Model

  • The channel model has an M-antenna base station, K single-antenna users, and an N-element RIS, with the direct BS–user link blocked and communication routed through the RIS.The model assumes instantaneous CSI at the BS, far-field RIS channels, distance-dependent path loss, and Rician small-scale fading.Source: A. Channel Model · p. 3, Figure 1 · p. 4

1) Conventional RIS Architecture:

Conventional RIS elements reflect their own incident signals independently, producing a diagonal phase-shift matrix. The proposed architecture routes each signal to another element, retaining only N nonzero matrix entries while adding routing and phase-design flexibility.

1) Conventional RIS Architecture:

  • Conventional RIS elements independently phase-shift and reflect their own incident signals, so the phase-shift matrix is diagonal.The proposed architecture instead permits a signal incident on one element to be reflected from another element.Source: 1) Conventional RIS Architecture · p. 4, 2) The proposed RIS Architecture · p. 4
  • The fully connected design uses N × N matrix entries and extra controller-link information; the group-connected design lowers complexity with modest performance loss.The proposed architecture is described as approaching fully connected performance at significantly reduced complexity.Source: 1) Conventional RIS Architecture · p. 4

2) The proposed RIS Architecture:

  • The proposed non-diagonal matrix has one nonzero entry per row and column, with N entries and their positions recorded each coherence time.The paper says this modestly increases optimization complexity and controller-link information relative to the conventional architecture.Source: 2) The proposed RIS Architecture · p. 5
  • 20.2 versus 12.4: the four-element example’s non-diagonal design achieves higher channel gain than the conventional design.The non-diagonal and conventional gains are reported for the same example with coherently designed phase shifts.Source: 2) The proposed RIS Architecture · p. 5

C. Implementation Circuits

  • A potential N-element implementation uses an N × N switch array, with switch states set by the positions of nonzero phase-matrix entries.The transmission-line example uses reconfigurable impedance and variable capacitance to control phase shifts.Source: C. Implementation Circuits · pp. 5–6

III. BEAMFORMING DESIGN

The beamforming design first derives a closed-form SISO solution, then extends joint rate optimization to single-user MISO and multi-user MIMO systems. For SISO, element permutations pair channel amplitudes before phase shifts are set for coherent combination.

A. Beamforming Design for SISO Systems

1) Conventional RIS Architecture:

  • In the conventional SISO design, each element’s phase is chosen to align its reflected signal for coherent combination.The conventional phase-shift matrix is diagonal.Source: 1) Conventional RIS Architecture · p. 6

2) The proposed RIS Architecture:

  • For the proposed SISO design, the maximum channel gain is proportional to the square of the sum of paired amplitudes a(i)b(i) when N is large.The channel vectors are permuted by ascending element amplitude, then phase shifts are selected to combine them according to the MRC criterion.Source: 2) The proposed RIS Architecture · pp. 6–7

B. Beamforming Design for Single-user MISO Systems

For single-user MISO, the paper jointly optimizes the BS transmit beamforming vector and RIS phase-shift matrix to maximize achievable rate. It solves the non-convex problem through alternating optimization.

B. Beamforming Design for Single-user MISO Systems

  • The single-user MISO objective is to maximize achievable rate by jointly choosing the phase-shift matrix and transmit beamforming vector.The section frames the optimization as maximizing channel gain.Source: B. Beamforming Design for Single-user… · p. 7
  • Alternating optimization iterates between RIS phase design and BS transmit beamforming for the single-user MISO problem.With the beamforming vector fixed, the RIS is designed as in the SISO case; with the RIS fixed, the beamforming vector uses maximum ratio transmission.Source: B. Beamforming Design for Single-user… · p. 7

C. Beamforming Design for Multi-user MIMO Systems

For multi-user MIMO, the paper uses a two-stage design to maximize achievable rate. It first optimizes the RIS phase matrix with SDR, then designs BS beamforming and power allocation for the resulting equivalent channel.

C. Beamforming Design for Multi-user MIMO Systems

  • The two-stage algorithm decouples joint beamforming design, reducing computational complexity with slight performance erosion versus alternating optimization.This is the stated reason for selecting the two-stage method in the multi-user MIMO case.Source: C. Beamforming Design for Multi-user… · p. 8
  • Stage I maximizes the users’ combined channel gain and applies SDR after relaxing the rank-one constraint.The solution uses the phase vector q and the positive-semidefinite matrix Q = qq^H.Source: C. Beamforming Design for Multi-user… · p. 8
  • Stage II uses SVD to design the transmit beamforming matrix and water-filling to allocate power over the equivalent channel.The beamforming matrix uses the first K columns of the right singular matrix.Source: C. Beamforming Design for Multi-user… · p. 8

IV. THEORETICAL ANALYSIS

This section analyzes channel gain for the proposed RIS architecture and compares it theoretically with the conventional diagonal design. It also outlines analysis of outage probability, BER, and complexity.

IV. THEORETICAL ANALYSIS

  • The theoretical analysis covers channel gain, outage probability, average BER, and complexity for the proposed architecture.The channel-gain comparison is framed for SISO systems and Rician channels.Source: §IV · p. 9

A. Channel Gain Analysis

  • The channel-gain analysis quantifies the conventional RIS architecture’s average gain as a function of the number of reflecting elements N.The conventional architecture uses a diagonal phase shift matrix.Source: A. Channel Gain Analysis · p. 9, 1) Average channel gain in the… · p. 9
  • The SISO channel model describes the BS–RIS and RIS–user channel vectors, with their amplitudes characterized using Rician distributions and moments.The Rician factors κg and κh correspond to the BS–RIS and RIS–user paths, respectively.Source: A. Channel Gain Analysis · p. 9

2) Average channel gain analysis in the proposed RIS architecture:

The proposed architecture’s average channel gain is expressed using ordered channel-amplitude sequences. The analysis also examines how Rician factors affect channel gain, including a fully LoS-dominated limit.

2) Average channel gain analysis in the proposed RIS architecture:

  • The proposed architecture’s average channel gain is expressed using ascendingly ordered channel-amplitude sequences a(i) and b(i).The derivation uses order-statistic distributions and their first and second moments.Source: 2) Average channel gain analysis in… · pp. 9–10

B. Effect of the Value of the Rician Factor on the Channel Gain

Case I, κg →∞(or κh

The analysis contrasts ideal LoS and Rayleigh-fading conditions for the proposed non-diagonal RIS. In Rayleigh fading, its channel gain is related to maximum-ratio rather than equal-gain combination.

Case I, κg →∞(or κh

  • Under simultaneous fully LoS-dominated BS–RIS and RIS–user channels, both architectures attain the upper bound ϱtϱrN^2.The stated equality is gdiag = gnond = ϱtϱrN^2.Source: Case I, κg →∞(or κh · p. 10

Case II, κg = 0 and κh = 0:

  • As N increases in Rayleigh fading, the proposed architecture outperforms the group-connected design and approaches the fully-connected design, though it is worse than the latter for small N.Fig. 5 compares normalized channel gain against N across RIS architectures.Source: Case II, κg = 0 and κh = 0 · p. 11, Figure 5 · p. 11
  • In Rayleigh fading, the proposed architecture outperforms the conventional architecture regardless of N, with its large-N gain associated with maximum-ratio combination.The conventional gain is proportional to an equal-gain combination, whereas the proposed gain is proportional to a maximum-ratio combination.Source: Case II, κg = 0 and κh = 0 · p. 11

C. Channel Gain Analysis when N →∞

As N grows, the proposed architecture retains normalized channel gain 1 across Rician-factor ranges, unlike the conventional design. The section also derives received-SNR, outage-probability, and BER expressions for Rayleigh fading.

C. Channel Gain Analysis when N →∞

  • As N →∞, the proposed architecture’s normalized channel gain remains 1 across Rician-factor ranges, while the conventional gain degrades as the factors decrease.The proposed architecture’s gain reaches ϱtϱrN^2, equivalent to the fully-connected RIS architecture’s channel-gain performance.Source: C. Channel Gain Analysis when N →∞ · p. 11

D. Outage Probability and BER Performance Analysis

2) Average BER:

  • The average BER is derived from the received-SNR CDF, with parameters p and q depending on the modulation scheme.For BPSK, p = 1/2 and q = 1.Source: 2) Average BER · p. 12

E. Complexity Analysis

The proposed RIS architecture uses non-diagonal phase shifts and is evaluated for complexity and performance against conventional and connected RIS designs. The reported results show gains over the conventional architecture and, as the number of elements grows, performance between the group-connected and fully-connected designs at reduced complexity.

E. Complexity Analysis

  • The proposed architecture requires N configurable impedances, matching conventional RIS and fewer than fully-connected or group-connected RIS.The complexity analysis also considers BS-RIS control-link load, but the supplied passages do not provide its complete comparison.Source: E. Complexity Analysis · p. 12

VI. CONCLUSIONS

  • The proposed architecture improves channel gain, outage probability, average BER, and achievable rate over conventional RIS, while approaching fully-connected performance and exceeding group-connected performance as N grows.The comparison with connected designs is reported alongside reduced complexity relative to the fully-connected architecture.Source: §V · p. 12, §VI · p. 15

V. PERFORMANCE RESULTS AND ANALYSIS

  • At an SNR threshold of 25 dB, the proposed architecture’s outage-probability lower bound is tight; its BPSK average BER matches theory and approaches LoS performance as N increases.The proposed architecture outperforms conventional RIS in outage probability and average BER.Source: §V · p. 12, Figure 8 · p. 13
  • The proposed architecture achieves higher single-user achievable rate than conventional RIS across RIS-user distances, with especially strong gains at larger N and nearly the same optimization convergence speed.The reported rate gain with increasing N is attributed to the MRC criterion.Source: Figure 9 · p. 13, §V · p. 13
  • In multi-user MIMO, the proposed architecture provides better coverage at the same data-rate requirement, and achievable rate increases nearly linearly with N.The reported enhancement persists across user counts, path-loss exponents, and channel-estimation error variances.Source: §V · pp. 13–14, Figure 10 · p. 14
  • The proposed RIS architecture achieves better performance than conventional RIS; at adjacent-element spacing δ0 = 1/64λ, its achievable rate is almost the same as conventional RIS at δ0 = 1/4λ.At δ0 = 1/4λ, the proposed architecture’s achievable rate is even higher than conventional RIS at δ0 = 4λ.Source: §V · p. 14
  • As the BS-RIS Rician factor increases, both architectures improve, while the proposed architecture’s rate enhancement is significant in NLoS-dominated channels and less obvious at high Rician factors.Source: §V · p. 14

VI. CONCLUSIONS

  • The non-diagonal architecture uses cross-element reflection and, as N increases, exceeds group-connected performance while approaching fully-connected performance at considerably reduced complexity.Its phase-shift matrix can be non-symmetric, and the rate optimization jointly adjusts transmit beamforming and the phase-shift matrix.Source: §VI · p. 15

APPENDIX A PROOF OF FORMULA (65)

The appendix establishes limiting equivalence between diagonal and non-diagonal channel gains and develops a channel-gain result using distribution functions and an inequality argument.

APPENDIX A PROOF OF FORMULA (65)

  • As κg approaches infinity, ai equals 1; as κg or κh approaches infinity, the diagonal and non-diagonal channel gains are equal.Source: Appendix A · p. 15

APPENDIX B PROOF OF Theorem 1

  • The appendix denotes the CDF of a Rayleigh random variable with scale parameter σ as FX(x; σ), then gives the PDF of a(i).Source: Appendix B · p. 15
  • The Appendix B proof concludes after deriving the stated PDF.Source: Appendix B · p. 15

APPENDIX C PROOF OF FORMULA (79)

  • The Appendix C argument invokes the Cauchy-Schwarz inequality in the derivation.Source: Appendix C · p. 15
  • Equality in (88) holds only when a(i) equals b(i).Source: Appendix C · p. 15
  • As N approaches infinity, the proof uses the inverse CDFs of a(i) and b(i) to establish the proposed architecture’s channel gain.Source: Appendix C · p. 15
  • The resulting limiting channel gain is ϱtϱrN^2.Source: Appendix C · p. 15

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