Ziv-Zakai Bound for Near-Field Localization and Sensing

TL;DR

The paper tackles near-field localization with extremely large aperture arrays (ELAA) and shows that traditional CRB analyses fail to capture threshold behavior at low to moderate SNRs. It develops a Ziv-Zakai Bound (ZZB) framework for 2D estimation of distance $d$ and AoA $\theta$, using a uniform prior and a binary-detection perspective based on the cross-correlation of near-field steering vectors, with explicit expressions for $P_{min}$ and $\rho$. The analysis yields both low- and high-SNR asymptotics, revealing that $\mathsf{ZZB}^{(\hat d;\theta)} \to T_d^2/12$ at low SNR and that, for $\theta=0$, $\mathsf{ZZB}^{(\hat d;\theta=0)} \to \frac{18\lambda^2 (d_{max}^5-d_{min}^5)}{\pi^2 K \mathsf{SNR} D_a^4 T_d}$ at high SNR, matching the global CRB averaged over the prior. Numerical results corroborate that the ZZB tightly bounds the MLE performance across a broad SNR range, quantifies SNR thresholds, and demonstrates how aperture, number of elements, and AoA priors influence near-field sensing performance. These insights guide design and performance evaluation for next-generation near-field array processing in localization and sensing applications.

Abstract

The increasing carrier frequencies and growing physical dimensions of antenna arrays in modern wireless systems are driving renewed interest in localization and sensing under near-field conditions. In this paper, we analyze the Ziv-Zakai Bound (ZZB) for near-field localization and sensing operated with large antenna arrays, which offers a tighter characterization of estimation accuracy compared to traditional bounds such as the Cramér-Rao Bound (CRB), especially in low signal-to-noise ratio or threshold regions. Leveraging spherical wavefront and array geometry in the signal model, we evaluate the ZZB for distance and angle estimation, investigating the dependence of the accuracy on key signal and system parameters such as array geometry, wavelength, and target position. Our analysis highlights the transition behavior of the ZZB and underscores the fundamental limitations and opportunities for accurate near-field sensing.

Figures (3)

• Figure 1: Considered 2D scenario, where an ELAA is employed to localize a single antenna transmitting UE in ${\bf p}=[d,\theta]^{\top}$.
• Figure 2: ZZB for distance estimation (bold lines) as a function of the SNR for different number of antennas, with known AoA . Comparison with the CRB (-$\,$-) and with MSE of the MLE ($\cdot\cdot$).
• Figure 3: ZZB for distance estimation as a function of the SNR for $K=21$ with unknown AoA span $[\theta_{\mathrm{min}}, \theta_{\mathrm{max}}]$. Comparison with the CRB (- -).