Cramer-Rao Bound for Angle of Arrival Estimates in True-Time-Delay Systems

TL;DR

This work derives the Cramér-Rao bound for angle-of-arrival estimation in true-time-delay (TTD) systems within a joint communication and sensing context, using a uniform linear array to model the received signal across frequency samples. The Fisher information is analyzed to produce a bound that scales with the signal energy, number of frequency samples, and array-specific terms via $\kappa_0$, with a narrow-band simplification giving $\mathrm{Var}(\hat{\phi}) \ge \frac{\sigma_v^2}{2 E_s N \kappa_0 (2\pi f_c d/c \cos\phi)^2}$. Numerical evaluation compares maximum-likelihood (ML) AoA estimation against a peak-based estimator, showing ML can approach the CRB while the peak method is constrained by frequency resolution; increasing the number of antennas $M$ improves the bound through $\kappa_0$, and up-sampling (e.g., sinc interpolation) can reduce the resolution gap to achieve RMSE below $1^\circ$ under the tested parameters. The results inform fast beam training and interference mitigation in JC&S systems, guiding design choices for bandwidth, antenna count, and processing complexity in practical 6G-like scenarios.

Abstract

In the context of joint communication and sensing JC&S, the challenge of obtaining accurate parameter estimates is of interest. Parameter estimates, such as the AoA can be utilized for solving the initial access problem, interference mitigation, localization of users or monitoring of the environment and synchronization of MIMO systems. Recently, TTD systems have gained attention for fast beam training during initial access and mitigation of beam squinting. This work derives the CRB for angle estimates in typical TTD systems. Properties of the CRB and the Fisher information are investigated and numerically evaluated. Finally, methods for angle estimation such as ML and established estimators are utilized to solve the angle estimation problem using a uniform linear array.

Figures (7)

• Figure 1: SIMO system model Coll202509_2 with an omni-directional transmitting antenna and a ULA receiver consisting of $M$ elements spaced by distance $d$. The vector $\vec{k}$ denotes the direction from the transmitter (TX) to the receiver (RX), while the vector $\vec{n}$ represents the array normal.
• Figure 2: $10\log_\text{10}(\kappa(f,\phi))$ for $M=8,f_\text{c}= 3.75GHz, B=20MHz, \tau_\text{d}=1/B$
• Figure 3: Numerical evaluation of $\kappa_0$ for different $\tau_\text{d}$, $\kappa_0$ is even in regards to $\phi$
• Figure 4: Numerical evaluation of $\kappa_1$ for different $\tau_\text{d}$, $\kappa_1$ is uneven in regards to $\phi$
• Figure 5: Numerical evaluation of $\kappa_2$ for different $\tau_\text{d}$, $\kappa_2$ is approx. even in regards to $\phi$