Fast Fractional Programming for Multi-Cell Integrated Sensing and Communications

TL;DR

A nonhomogeneous bound is developed and used in conjunction with the FP technique to solve the ISAC beamforming problem without the need to invert any large matrices.

Abstract

This paper concerns the coordinate multi-cell beamforming design for integrated sensing and communications (ISAC). In particular, we assume that each base station (BS) has massive antennas. The optimization objective is to maximize a weighted sum of the data rates (for communications) and the Fisher information (for sensing). We first show that the conventional beamforming method for the multiple-input multiple-output (MIMO) transmission, i.e., the weighted minimum mean square error (WMMSE) algorithm, works for the ISAC problem case from a fractional programming (FP) perspective. However, the WMMSE algorithm frequently requires computing the $N\times N$ matrix inverse, where $N$ is the number of transmit or receive antennas, so the algorithm becomes quite costly when antennas are massively deployed. To address this issue, we develop a nonhomogeneous bound and use it in conjunction with the FP technique to solve the ISAC beamforming problem without the need to invert any large matrices. It is further shown that the resulting new FP algorithm has an intimate connection with gradient projection, based on which we can accelerate the convergence via Nesterov's gradient extrapolation.

Figures (12)

• Figure 1: A multi-cell ISAC system with $L=3$ and $K=3$. The circle is the point target to sense. The arrows are the transmit signals and the echo signals. Each BS $i$ aims to acquire $\theta_i$ independently.
• Figure 2: In the $\tau$th iteration, $f_q$ and $f_o$ have the same gradient with respect to each $\bm W_{\ell k}$ after the updates of the auxiliary variables $(\underline\bm\Gamma,\underline\bm Y,\underline{\widetilde{\bm Y}})$.
• Figure 3: Nesterov's gradient ascent with extrapolation Nesterov_book.
• Figure 4: The convergence behaviors of the different ISAC beamforming algorithms when $N_r=N_t=128$.
• Figure 5: The convergence behaviors of the different ISAC beamforming algorithms when $N_r=N_t=4$.

Theorems & Definitions (13)

• Proposition 1: Lagrangian Dual Transform shen2019optimization
• Proposition 2: Quadratic Transform shen2019optimization
• Remark 1
• Remark 2
• Remark 3
• Lemma 1: Nonhomogeneous Bound
• Remark 4
• Example 1
• Proposition 3: Convergence Analysis
• Proposition 4