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Beampattern Synthesis for Discrete Phase RIS in Communication and Sensing Systems

Xiao Cai, Hei Victor Cheng, Daniel E. Lucani

TL;DR

This paper tackles wide-beam beampattern design for RIS under discrete phase constraints to aid direction estimation and sensing when target directions are unknown. It introduces a penalty-based convex-hull relaxation and a Minorization-Maximization (MM) algorithm to solve a max-min optimization over a region of interest, achieving wide beams with discrete levels close to continuous, per-element power-constrained performance. The approach is validated for both Uniform Linear Arrays and Uniform Planar Arrays, and is extended to temporal-domain MUSIC for AOA estimation and to GLRT/energy detectors for target detection, with results showing substantial gains over beam sweeping and favorable scalability with RIS size. The work provides a practical framework for JCAS applications by enabling wide angular coverage with a single pilot and reduced training overhead.

Abstract

Extensive research on Reconfigurable Intelligent Surfaces (RIS) has primarily focused on optimizing reflective coefficients for passive beamforming in specific target directions. This optimization typically assumes prior knowledge of the target direction, which is unavailable before the target is detected. To enhance direction estimation, it is critical to develop array pattern synthesis techniques that yield a wider beam by maximizing the received power over the entire target area. Although this challenge has been addressed with active antennas, RIS systems pose a unique challenge due to their inherent phase constraints, which can be continuous or discrete. This work addresses this challenge through a novel array pattern synthesis method tailored for discrete phase constraints in RIS. We introduce a penalty method that pushes these constraints to the boundary of the convex hull. Then, the Minorization-Maximization (MM) method is utilized to reformulate the problem into a convex one. Our numerical results show that our algorithm can generate a wide beam pattern comparable to that achievable with per-power constraints, with both the amplitudes and phases being adjustable. We compare our method with a traditional beam sweeping technique, showing a) several orders of magnitude reduction of the MSE of Angle of Arrival (AOA) at low to medium Signal-to-Noise Ratio (SNR)s; and b) $8$~dB SNR reduction to achieve a high probability of detection.

Beampattern Synthesis for Discrete Phase RIS in Communication and Sensing Systems

TL;DR

This paper tackles wide-beam beampattern design for RIS under discrete phase constraints to aid direction estimation and sensing when target directions are unknown. It introduces a penalty-based convex-hull relaxation and a Minorization-Maximization (MM) algorithm to solve a max-min optimization over a region of interest, achieving wide beams with discrete levels close to continuous, per-element power-constrained performance. The approach is validated for both Uniform Linear Arrays and Uniform Planar Arrays, and is extended to temporal-domain MUSIC for AOA estimation and to GLRT/energy detectors for target detection, with results showing substantial gains over beam sweeping and favorable scalability with RIS size. The work provides a practical framework for JCAS applications by enabling wide angular coverage with a single pilot and reduced training overhead.

Abstract

Extensive research on Reconfigurable Intelligent Surfaces (RIS) has primarily focused on optimizing reflective coefficients for passive beamforming in specific target directions. This optimization typically assumes prior knowledge of the target direction, which is unavailable before the target is detected. To enhance direction estimation, it is critical to develop array pattern synthesis techniques that yield a wider beam by maximizing the received power over the entire target area. Although this challenge has been addressed with active antennas, RIS systems pose a unique challenge due to their inherent phase constraints, which can be continuous or discrete. This work addresses this challenge through a novel array pattern synthesis method tailored for discrete phase constraints in RIS. We introduce a penalty method that pushes these constraints to the boundary of the convex hull. Then, the Minorization-Maximization (MM) method is utilized to reformulate the problem into a convex one. Our numerical results show that our algorithm can generate a wide beam pattern comparable to that achievable with per-power constraints, with both the amplitudes and phases being adjustable. We compare our method with a traditional beam sweeping technique, showing a) several orders of magnitude reduction of the MSE of Angle of Arrival (AOA) at low to medium Signal-to-Noise Ratio (SNR)s; and b) ~dB SNR reduction to achieve a high probability of detection.

Paper Structure

This paper contains 35 sections, 2 theorems, 60 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Main Contribution
  4. Organization & Notations
  5. system model
  6. Uniform Linear Array (ULA)
  7. Uniform Planar Array (UPA)
  8. Problem formulation
  9. ULA-based Problem formulation
  10. UPA-based Problem formulation
  11. Extension to multiple time slots
  12. Proposed Method
  13. $\Delta \theta$: Discretization of $\theta$
  14. Handling Discrete Constraints via Convex Hull Relaxation
  15. Solving the Nonconvex Problem via the MM Algorithm
  16. ...and 20 more sections

Key Result

Proposition 1

Let $\mathcal{S} = \{ s_1, \dots, s_L \} \subset \mathbb{C}$ be a discrete set of complex numbers, $|s_\ell| = \rho, \ \forall\ell$ and $\rho > 0$. Define the feasible set and let its convex hull be Let $f:\mathbb{C}^N\to\mathbb{R}$ be convex and $\lambda\ge0$. Then

Figures (10)

  • Figure 1: System setting
  • Figure 2: Illustration of constraints. (a) Per-element power constraint, (b) Constant modulus constraint (CMC), (c) Discrete phase constraint, (d) Convex hull of the discrete set.
  • Figure 3: Beam pattern for the target at different locations. Parameters: $N =256$, $L = 4$.
  • Figure 4: The effect of different parameters. Target region at $[-30^\circ, 30^\circ]$.
  • Figure 5: Beam pattern for a UPA with $N = 64\times64$, $L =4$.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proof 1
  • Proof 2