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Beamforming Design for Integrated Sensing and Communications Using Uplink-Downlink Duality

Kareem M. Attiah, Wei Yu

TL;DR

The paper tackles joint sensing and communications beamforming by minimizing the Bayesian CRB, $\mu(\mathbf{V})=\mathrm{Tr}(\mathbf{J}^{-1}_{\mathbf{V}})$, under downlink SINR constraints. It shows this nonconvex objective can be recast as a downlink power-minimization problem with an uplink-downlink duality, obviating semidefinite relaxation lifting. An efficient algorithm solves an uplink problem to obtain beamforming directions and maps them to the downlink, with a dual-parameter outer loop $(\lambda,\boldsymbol{\beta})$ updated via a subgradient method in an admissible set $\mathcal{A}$. Numerical results for angle-of-arrival sensing demonstrate that the proposed method reproduces SDR-like beam patterns but at substantially lower computational cost, validating the approach for ISAC scenarios.

Abstract

This paper presents a novel optimization framework for beamforming design in integrated sensing and communication systems where a base station seeks to minimize the Bayesian Cramér-Rao bound of a sensing problem while satisfying quality of service constraints for the communication users. Prior approaches formulate the design problem as a semidefinite program for which acquiring a beamforming solution is computationally expensive. In this work, we show that the computational burden can be considerably alleviated. To achieve this, we transform the design problem to a tractable form that not only provides a new understanding of Cramér-Rao bound optimization, but also allows for an uplink-downlink duality relation to be developed. Such a duality result gives rise to an efficient algorithm that enables the beamforming design problem to be solved at a much lower complexity as compared to the-state-of-the-art methods.

Beamforming Design for Integrated Sensing and Communications Using Uplink-Downlink Duality

TL;DR

The paper tackles joint sensing and communications beamforming by minimizing the Bayesian CRB, , under downlink SINR constraints. It shows this nonconvex objective can be recast as a downlink power-minimization problem with an uplink-downlink duality, obviating semidefinite relaxation lifting. An efficient algorithm solves an uplink problem to obtain beamforming directions and maps them to the downlink, with a dual-parameter outer loop updated via a subgradient method in an admissible set . Numerical results for angle-of-arrival sensing demonstrate that the proposed method reproduces SDR-like beam patterns but at substantially lower computational cost, validating the approach for ISAC scenarios.

Abstract

This paper presents a novel optimization framework for beamforming design in integrated sensing and communication systems where a base station seeks to minimize the Bayesian Cramér-Rao bound of a sensing problem while satisfying quality of service constraints for the communication users. Prior approaches formulate the design problem as a semidefinite program for which acquiring a beamforming solution is computationally expensive. In this work, we show that the computational burden can be considerably alleviated. To achieve this, we transform the design problem to a tractable form that not only provides a new understanding of Cramér-Rao bound optimization, but also allows for an uplink-downlink duality relation to be developed. Such a duality result gives rise to an efficient algorithm that enables the beamforming design problem to be solved at a much lower complexity as compared to the-state-of-the-art methods.
Paper Structure (9 sections, 3 theorems, 27 equations, 2 figures)

This paper contains 9 sections, 3 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. ISAC Model and Performance Metrics
  4. Problem Formulation and Solution Strategy
  5. Problem Reformulation as Downlink Power Maximization Problem
  6. Uplink-Downlink Duality for ISAC
  7. Proposed Algorithm
  8. Simulations
  9. Conclusion

Key Result

Theorem 1

If problem prob:generalCase is feasible, it is equivalent to where $\boldsymbol{\beta} \triangleq [\boldsymbol{\boldsymbol{\beta}}_1, \ldots, \boldsymbol{\boldsymbol{\beta}}_L] \in \mathbb{R}^{L \times L}$, $\mathbf{e}_\ell$ is the $\ell$-th column of $\mathbf{I}$, and $\mathcal{V}$ denotes the constraints eq:qcqp_constraints-eq:qcqp_power_constraint. Further where $\mathbf{Q}_{\boldsymbol{\beta}

Figures (2)

  • Figure 1: The ISAC system considered in this work. The BS seeks to serve $K$ communication users and learn some unknown vector parameter $\boldsymbol{\eta}$.
  • Figure 2: Beam pattern of the proposed solution vs SDR for optimizing the angle CRB. Here, we set $N_\text{T} = N_\text{R} = 20$ and $K = 2$.

Theorems & Definitions (4)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3