Ziv-Zakai Bound for Near-Field Localization and Sensing

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.

💡 Research Summary

The paper investigates the fundamental performance limits of near‑field localization and sensing when a massive antenna array (an Extremely Large Aperture Array, ELAA) is employed. While the Cramér‑Rao Bound (CRB) is the standard benchmark for high‑SNR regimes, it fails to capture the error behavior in low‑SNR or threshold regions where the likelihood function exhibits multiple lobes and global ambiguities. To address this gap, the authors derive the Ziv‑Zakai Bound (ZZB), a Bayesian lower bound that incorporates prior knowledge of the unknown parameters, for the joint estimation of distance d and angle of arrival θ in a 2‑D scenario.

System model. A single‑antenna user equipment (UE) transmits a narrowband carrier (e.g., an OFDM sub‑carrier) at frequency f_c. The signal is received by a uniform linear array (ULA) with K elements, spacing δ, and total aperture D_a = (K‑1)δ. The received vector is r = α s(d,θ)+n, where s(d,θ)∈ℂ^K is the near‑field steering vector that depends on the true Euclidean distance d and AoA θ through the exact spherical‑wave distance d_k = ‖p − p_k‖. Using the Fresnel approximation, the authors recover the closed‑form CRB expressions for distance and AoA that have appeared in prior work.

ZZB formulation. Assuming a uniform prior over the admissible ranges d∈