⚡ electrical engineering

Joint Range-Angle Estimation in Near-Field ISAC System using Uniform Circular Array

This paper investigates joint range-angle estimation and communication in near-field ISAC systems using a Uniform Circular Array, deriving a continuous-time channel model and CRLB to reveal a fundamental aperture-versus-SNR trade-off where a 0.5 m radius array achieves optimal performance by balancing estimation accuracy with received signal strength.

Original authors: Lorenzo Zaniboni, Mark F. Flanagan

Published 2026-04-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lorenzo Zaniboni, Mark F. Flanagan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Super-Sharp Eye and a Loud Voice

Imagine a future 6G cell tower (the "Base Station") that does two things at once: it talks to your phone (Communication) and acts like a radar to figure out exactly where your phone is (Sensing).

Usually, these towers use a straight line of antennas (like a row of soldiers). This paper suggests a better shape: a circle of antennas (like a round table). The researchers wanted to see if this circular shape could help the tower "see" objects much closer to it with incredible precision, while still talking to them clearly.

The Problem: The "Flat World" vs. The "Curved World"

The Old Way (Far-Field): When you are far away from a speaker, sound waves hit your ears as flat sheets. In this "Far-Field," antennas can only tell you which direction something is, but not how far away it is.
The New Way (Near-Field): When you are close to a speaker, the sound waves are curved (like ripples in a pond). In this "Near-Field," the antennas can tell you both the direction and the distance. This is crucial for 6G, where devices might be very close to the tower.

The Innovation: The Round Table (Uniform Circular Array)

The researchers compared a straight line of antennas (ULA) to a circle of antennas (UCA).

The Straight Line (ULA): Imagine a flashlight beam. It's very sharp if you look straight at it, but if you look from the side, the beam gets blurry and weak. Similarly, a straight antenna line works great for objects directly in front of it, but its ability to "see" distance fades quickly as the object moves to the side.
The Circle (UCA): Imagine a lighthouse. No matter which way you turn, the light is equally strong. Because the antennas are in a circle, the tower has the same "vision" in every direction. It creates a giant, invisible bubble of high-precision sensing that covers the whole area, not just the front.

The Discovery: The "Big Lens" Paradox

This is the most interesting part of the paper. The researchers found a tricky trade-off, like trying to use a giant telescope.

The Theory (The Big Lens): If you make the circle of antennas huge (a large radius), the math says you should be able to pinpoint a location with microscopic accuracy. It's like having a massive telescope lens; in theory, it should see everything perfectly.
The Reality (The Signal Loss): However, when you spread the antennas out too far, the signal gets weaker at each individual antenna. It's like trying to hear a whisper with a giant, spread-out net; the net catches the sound, but the sound is so faint that the noise of the wind drowns it out.

The Result:

Huge Circle (5 meters): The math says "Perfect accuracy!" But in practice, the signal is so weak that the computer gets confused by the noise. It guesses wildly, and the accuracy is actually bad.
Medium Circle (0.5 meters): The math says "Good accuracy." But because the antennas are closer together, the signal is strong and clear. The computer can hear the whisper clearly over the wind. This turned out to be the winner.

The Solution: How They Fixed It

To make the system work, the researchers had to build a very smart "brain" for the tower:

The Beamformer (The Spotlight): They designed a special way to focus the signal (like a spotlight) so it hits the phone exactly where it is, maximizing the strength of the connection.
The Estimator (The Detective): They created a mathematical detective (Maximum Likelihood Estimator) that looks at the returning signals to guess the phone's location.
- The Challenge: The "clues" (signals) can be confusing, leading the detective to the wrong answer (a "local minimum").
- The Fix: They taught the detective to do a quick "sweep" of the whole area first (a grid search) to find the most promising spots, and then zoom in for a detailed look. This prevents it from getting stuck guessing the wrong location.

The Conclusion: Quality Over Size

The paper concludes that bigger isn't always better in the near-field world.

While a giant antenna circle looks impressive on paper, it actually performs worse because the signal gets too weak to be useful. The "Goldilocks" zone—a medium-sized circle (0.5 meters)—provided the best balance. It was big enough to see distance and direction clearly, but small enough to keep the signal strong and loud.

In short: To build the best 6G radar-communication system, don't just make the antenna array huge. Make it the right size so the signal stays strong enough to be heard above the noise.

1. Problem Statement

The paper addresses the challenges of Integrated Sensing and Communication (ISAC) in the Near-Field (NF) regime for 6G wireless networks.

Context: Future 6G systems utilize Extremely Large Antenna Arrays (ELAA) and mmWave frequencies, pushing operation into the NF region where spherical wavefronts (rather than planar) dominate. This allows for joint range and angle estimation, unlike Far-Field (FF) systems which only estimate angle.
Limitations of Existing Work:
- Most existing NF-ISAC studies use Uniform Linear Arrays (ULA). ULAs suffer from an angle-dependent effective aperture; as the target moves away from the broadside, the effective aperture shrinks, reducing the NF region and estimation accuracy.
- Prior models often rely on narrowband assumptions or discrete-time channels, neglecting critical wideband effects like per-element delay variations and non-uniform Doppler shifts inherent in spherical wave propagation.
Objective: To develop a unified framework for a Uniform Circular Array (UCA) that leverages its rotational symmetry to provide an angle-invariant NF region, while accounting for full wideband propagation physics (delay, Doppler, spherical geometry) to optimize joint estimation and communication performance.

2. Methodology

A. System Model

Geometry: A Base Station (BS) equipped with a UCA of radius $R$ and $N_a$ elements serves a single User Equipment (UE) acting as a point target (monostatic sensing).
Channel Modeling: The authors propose a continuous-time wideband channel model incorporating:
- Spherical Wavefronts: Per-element distance $r_k$ varies across the array, leading to non-uniform path loss ( $g_k \propto 1/r_k$ ) and phase shifts.
- Wideband Effects: Explicit modeling of time-delay ( $\tau_0 = 2d/c$ ) and Doppler shifts ( $\nu_0$ ) for both communication (downlink) and sensing (round-trip) channels.
- Signaling: Uses OFDM waveforms with pilots and data symbols.

B. Theoretical Derivations

Cramér-Rao Lower Bound (CRLB):
- Derived a closed-form CRLB for the joint estimation of distance ( $d$ ) and azimuth angle ( $\theta$ ).
- Calculated the Fisher Information Matrix (FIM) by differentiating the signal model with respect to $d$ and $\theta$ , accounting for the complex interplay between array geometry and spherical wave propagation.
Optimal Beamforming:
- Formulated an optimization problem to minimize the trace of the CRLB matrix (minimizing total estimation variance).
- Solved using Riemannian Gradient Descent on the unit sphere to find the optimal transmit beamforming vector $\mathbf{f}$ , subject to a power constraint.

C. Estimation Algorithm

Maximum Likelihood (ML) Estimator:
- Formulated an ML estimator for $d$ and $\theta$ based on the received signal model.
- Optimization Strategy: Due to the non-convex nature of the likelihood function (multiple local minima), a two-stage approach was used:
  1. Coarse Search: A 2D grid search to identify promising basins.
  2. Refinement: The Levenberg–Marquardt algorithm initialized with the grid search results to converge to the global optimum.

3. Key Contributions

Wideband UCA Channel Model: Extended prior narrowband UCA models to a continuous-time wideband framework, explicitly capturing per-element delays, Doppler shifts, and spherical wavefront geometry.
Closed-Form CRLB: Derived analytical expressions for the joint range-angle CRLB, revealing the fundamental trade-off between array aperture size and received Signal-to-Noise Ratio (SNR) in the NF regime.
Optimization Framework: Designed an optimal transmit beamformer via Riemannian gradient descent and a robust ML estimator initialized by grid search.
Discovery of the Aperture-SNR Trade-off: Identified that while larger array radii theoretically tighten the CRLB (better resolution), they drastically reduce the received SNR at a fixed distance due to element spreading. This reduction pushes practical estimators into the "threshold regime," degrading performance.

4. Numerical Results

Simulations were conducted with $N_a=64$ elements at 60 GHz, testing radii $R \in \{0.5, 1, 2, 5\}$ meters over distances $d \in [10, 400]$ m.

CRLB vs. RMSE:
- Theoretical: Larger radii (e.g., $R=5$ m) yield significantly tighter CRLBs (e.g., range error bound drops from 16.5 mm to 0.16 mm at 10 m).
- Practical: All configurations operate in the threshold regime (where SNR is too low for the estimator to converge to the CRLB). Consequently, the Root Mean Square Error (RMSE) for larger radii is orders of magnitude worse than the CRLB (up to 20,000 $\times$ degradation for $R=5$ m).
Performance Winner:
- $R = 0.5$ m achieved the best practical performance (lowest RMSE and highest communication rate).
- Reasoning: The smaller radius maintains higher received SNR across the distance range, keeping the system above the critical threshold where the ML estimator fails. Larger radii suffer from low SNR, causing the estimator to converge to local minima (outliers).
Communication Rate:
- The achievable rate ( $C_{est}$ ) for larger radii was suppressed near zero due to beam misalignment caused by estimation failures.
- $R=0.5$ m sustained the highest communication rates because the estimation remained robust.

5. Significance and Conclusion

Paradigm Shift: The paper challenges the intuition that "larger aperture is always better" in NF-ISAC. It demonstrates that in the near-field, increasing aperture size without increasing transmit power can be detrimental due to the SNR aperture trade-off.
Geometry Advantage: The UCA is shown to be superior to the ULA for monostatic sensing because it provides an angle-invariant near-field region, ensuring consistent performance regardless of the target's azimuth.
Practical Implication: For current ISAC deployments, optimizing for robustness to the threshold effect (via moderate aperture sizes that maintain high SNR) is more critical than maximizing theoretical resolution bounds. The study suggests that a radius of 0.5 m is optimal for the evaluated 60 GHz, 64-element configuration.

In summary, this work provides a rigorous wideband framework for UCA-based ISAC, proving that practical system design must balance theoretical resolution gains against the SNR penalties introduced by large apertures in the near-field.