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Low-Complexity Cramér-Rao Lower Bound and Sum Rate Optimization in ISAC Systems

Tianyu Fang, Nhan Thanh Nguyen, Markku Juntti

TL;DR

This work tackles joint beamforming in ISAC by optimizing a weighted objective $\delta\sum_{k=1}^K R_k - \mathrm{tr}(\mathbf F^{-1})$ under a transmit-power constraint to balance communications and sensing accuracy. It develops a low-complexity algorithm, SCA-SGPI, by first constructing convex surrogates of the non-convex objective via successive convex approximation and then efficiently solving the resulting quadratic subproblems with a shifted generalized power iteration. Empirical results show that SCA-SGPI achieves a favorable SR–CRLB tradeoff with substantially reduced runtime compared to SDR-based methods, and scales more favorably with the number of users. The method is practically appealing for ISAC deployments and can be extended to multi-target scenarios in future work.

Abstract

While Cramér-Rao lower bound is an important metric in sensing functions in integrated sensing and communications (ISAC) designs, its optimization usually involves a computationally expensive solution such as semidefinite relaxation. In this paper, we aim to develop a low-complexity yet efficient algorithm for CRLB optimization. We focus on a beamforming design that maximizes the weighted sum between the communications sum rate and the sensing CRLB, subject to a transmit power constraint. Given the non-convexity of this problem, we propose a novel method that combines successive convex approximation (SCA) with a shifted generalized power iteration (SGPI) approach, termed SCA-SGPI. The SCA technique is utilized to approximate the non-convex objective function with convex surrogates, while the SGPI efficiently solves the resulting quadratic subproblems. Simulation results demonstrate that the proposed SCA-SGPI algorithm not only achieves superior tradeoff performance compared to existing method but also significantly reduces computational time, making it a promising solution for practical ISAC applications.

Low-Complexity Cramér-Rao Lower Bound and Sum Rate Optimization in ISAC Systems

TL;DR

This work tackles joint beamforming in ISAC by optimizing a weighted objective under a transmit-power constraint to balance communications and sensing accuracy. It develops a low-complexity algorithm, SCA-SGPI, by first constructing convex surrogates of the non-convex objective via successive convex approximation and then efficiently solving the resulting quadratic subproblems with a shifted generalized power iteration. Empirical results show that SCA-SGPI achieves a favorable SR–CRLB tradeoff with substantially reduced runtime compared to SDR-based methods, and scales more favorably with the number of users. The method is practically appealing for ISAC deployments and can be extended to multi-target scenarios in future work.

Abstract

While Cramér-Rao lower bound is an important metric in sensing functions in integrated sensing and communications (ISAC) designs, its optimization usually involves a computationally expensive solution such as semidefinite relaxation. In this paper, we aim to develop a low-complexity yet efficient algorithm for CRLB optimization. We focus on a beamforming design that maximizes the weighted sum between the communications sum rate and the sensing CRLB, subject to a transmit power constraint. Given the non-convexity of this problem, we propose a novel method that combines successive convex approximation (SCA) with a shifted generalized power iteration (SGPI) approach, termed SCA-SGPI. The SCA technique is utilized to approximate the non-convex objective function with convex surrogates, while the SGPI efficiently solves the resulting quadratic subproblems. Simulation results demonstrate that the proposed SCA-SGPI algorithm not only achieves superior tradeoff performance compared to existing method but also significantly reduces computational time, making it a promising solution for practical ISAC applications.

Paper Structure

This paper contains 12 sections, 3 theorems, 20 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Signal Model
  4. Communications Signal Model
  5. Sensing Signal Model
  6. Problem formulation
  7. Beamforming Optimization via Successive Convex Approximation
  8. The SCA Method
  9. The SGPI Method
  10. Complexity Analysis
  11. Numerical Results
  12. Conclusion

Key Result

Lemma 1

Function $\log\left( 1+\frac{|z|^2}{d}\right)$ with $z\in\mathbb C$ and $d\in\mathbb R_{++}$ can be lower bounded by its first-order Taylor expansion as follows fang2023optimal: The equality achieved at $(z,d)=(z_0,d_0)$, where $(z_0,d_0)$ is a given feasible point.

Figures (3)

  • Figure 1: Convergence of the inner and outer loops of Algorithm \ref{['al1']} with $N_{\rm t} =16$, $N_{\rm r} =20$, $K=4$, and $L=30$.
  • Figure 2: Tradeoff region between communications and sensing with $N_{\rm t} =16$, $N_{\rm r} =20$, $K=4$, and $L=30$.
  • Figure 3: Performance versus number of communications users.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3