⚡ electrical engineering

Analyzing URA Geometry for Enhanced Near-Field Beamfocusing and Spatial Degrees of Freedom

This paper analyzes how uniform rectangular array (URA) geometry influences near-field beamdepth and spatial degrees of freedom, introducing an effective beamfocusing Rayleigh distance (EBRD) to define near-field boundaries and demonstrating that elongated configurations like uniform linear arrays (ULAs) outperform square URAs in extending the effective near-field region and maximizing spatial multiplexing, which is leveraged to design a polar codebook achieving significant channel estimation improvements.

Original authors: Ahmed Hussain, Asmaa Abdallah, Abdulkadir Celik, Emil Björnson, Ahmed M. Eltawil

Published 2026-02-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ahmed Hussain, Asmaa Abdallah, Abdulkadir Celik, Emil Björnson, Ahmed M. Eltawil

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

Imagine you are trying to talk to a friend in a crowded, noisy room. In the old days (the "Far-Field" era), you had to shout in a specific direction, hoping your voice would reach them. If you shouted at the right angle, they heard you. But if two friends stood one behind the other in that same direction, your voice would hit both of them equally, and you couldn't tell them apart. You were limited to one conversation per direction.

Now, imagine you have a super-powered, magical megaphone (a massive antenna array) that can do something new: Near-Field Beamfocusing. Instead of just shouting in a direction, this megaphone can focus its sound into a specific bubble of space at a specific distance. You can whisper to the friend standing 5 meters away, while your other friend standing 10 meters away in the exact same direction hears nothing. You can have two conversations happening in the same direction, just at different distances!

This paper is about figuring out the rules of this magical megaphone, specifically when it's shaped like a rectangle (like a TV screen) rather than a long stick or a perfect square.

Here is the breakdown of their discoveries using simple analogies:

1. The "Bubble" of Focus (Beamdepth)

When you focus a beam in the near-field, it doesn't go on forever like a laser pointer. It has a start and an end. The authors call this the Beamdepth.

The Analogy: Think of a flashlight beam. In the far-field, the light spreads out and goes on forever. In the near-field, the light is like a focused spotlight that illuminates a specific patch of wall. If you stand too close or too far from that patch, the light is too dim to see. The "Beamdepth" is the thickness of that illuminated patch.
The Discovery: The shape of your antenna array changes how thick this patch is.
- Square Arrays: Like a square tile. They create a "fat" patch. It's good for covering a wide area, but you can't fit many distinct "bubbles" in a row.
- Elongated Arrays (like a long stick): These create a very "thin" patch. This is great because you can stack many thin bubbles one after another along the same line, allowing you to talk to many more people in the same direction.

2. The "Magic Boundary" (EBRD)

There is a limit to how far this focusing trick works. Beyond a certain distance, the beam spreads out and loses its ability to focus on specific depths. The authors call this limit the Effective Beamfocusing Rayleigh Distance (EBRD).

The Analogy: Imagine you are throwing a frisbee. You can aim it perfectly at a specific spot on the ground for the first 20 meters. But after 20 meters, the wind takes over, and it starts wobbling and drifting. You can't control exactly where it lands anymore. The EBRD is that 20-meter mark.
The Discovery:
- If you have a fixed number of antenna elements (like a fixed budget for parts), making the array long and skinny pushes this "20-meter mark" much further away. You get a longer "magic zone."
- If you have a fixed physical size (like a fixed wall space), the shape matters less, but making it bigger (physically larger) is the best way to extend the magic zone.

3. The "Resolution" Problem (Side Lobes)

When you focus a beam, you don't just get a perfect circle of light. You get a bright center (the main beam) and some faint, annoying "ghost" lights around the edges (called Side Lobes).

The Analogy: Think of a spotlight on a stage. You want the actor to be bright, but you don't want the light spilling onto the audience in the front row.
The Twist: In the near-field, these "ghost lights" appear in two directions:
1. Sideways (Angular): The usual problem.
2. Forward/Backward (Axial): New ghost lights appear in front of and behind your target bubble.
The Solution: The authors found a mathematical trick (a "modified window") to dim these ghost lights. However, there's a catch: if you dim the forward ghost lights, the sideways ones get a bit brighter, and vice versa. It's a trade-off, like turning down the bass on your stereo to hear the vocals better, but then the vocals get a bit quieter. You have to find the perfect balance.

4. The "Secret Codebook" (Channel Estimation)

To use this technology, the phone and the tower need to agree on where to focus the beam. They need a "dictionary" or a "codebook" of all possible beam shapes.

The Problem: Existing dictionaries were built for the old "Far-Field" world. They are too big and inefficient for this new "Near-Field" world.
The Solution: The authors designed a Polar Codebook.
- The Analogy: Imagine you are trying to find a lost hiker in a forest.
  - Old Method: You check every single tree in the forest (inefficient, takes forever).
  - New Method: You know the hiker is only in the "Near-Field" zone (the EBRD). You only check the trees in that specific zone, and you space your checks out perfectly so you don't miss anyone but don't waste time checking the same spot twice.
- The Result: This new codebook is smaller (saves memory), faster (saves battery), and finds the user more accurately (better signal quality). In tests, it improved signal quality by 2 dB, which is a huge deal in wireless tech.

5. The "Capacity" Boost (Spatial Degrees of Freedom)

Finally, the paper asks: "How many people can we talk to at once?"

The Discovery: In the old world, you could only talk to one person per direction. In this new world, because you can focus on different distances, you can talk to multiple people in the same direction.
The Winner: The Long and Skinny (Elongated) antenna arrays are the champions here. They allow for the most "layers" of communication, meaning the network can handle way more data traffic than a square array could.

Summary

This paper is the rulebook for the next generation of wireless networks. It tells engineers:

Shape matters: Long, skinny antenna arrays are better for focusing beams at specific distances.
Distance matters: There is a specific "magic zone" where this focusing works, and we can calculate exactly how big that zone is.
Balance matters: You have to balance the "ghost lights" (interference) to get the clearest signal.
Efficiency matters: By using these new rules, we can build a "dictionary" for the network that is smaller, faster, and much smarter, allowing our future 6G networks to handle massive amounts of data with crystal-clear precision.

1. Problem Statement

With the advent of Ultra-Massive MIMO (UM-MIMO) and higher frequency bands (mmWave/THz), future wireless systems will increasingly operate in the radiative near-field region. Unlike the far-field, where beams are planar and steered only by angle, near-field beams are spherical and can be focused on specific spatial regions with finite depth.

The paper addresses three critical gaps in current research:

Lack of Generalized Analysis: Existing beamdepth analyses are often limited to square apertures or continuous apertures, lacking closed-form expressions for general Uniform Rectangular Arrays (URAs) with arbitrary aspect ratios.
Geometry Impact: The influence of array geometry (e.g., square vs. elongated) on near-field beamdepth and the effective near-field boundary under practical constraints (fixed element count vs. fixed aperture size) is not well understood.
Channel Estimation: Efficient channel estimation in the near-field requires codebooks tailored to the polar domain (angle + range). Existing codebooks designed for Uniform Linear Arrays (ULAs) or square arrays do not directly apply to general URAs due to complex coupling between azimuth, elevation, and range.

2. Methodology

The authors employ a rigorous analytical approach combining electromagnetic theory, array signal processing, and compressed sensing:

System Model: A narrow-band system with a Base Station (BS) equipped with a generalized URA ( $N_1 \times N_2$ elements) and a single-antenna User Equipment (UE). The channel is modeled using a Uniform Spherical Wave (USW) model, utilizing a second-order Taylor expansion for distance approximation.
Beamforming Analysis: The authors derive the normalized array gain using Fresnel functions. They define the Beamdepth ( $r_{BD}$ ) as the distance interval where the array gain drops by no more than 3 dB from its peak.
Metric Definition: They introduce the Effective Beamfocusing Rayleigh Distance (EBRD), a new metric defining the maximum distance at which effective beamfocusing and spatial multiplexing are possible for a specific angle and array geometry.
Spatial Degrees of Freedom (DoF): A new expression for Effective DoF (EDoF) is derived based on a Fresnel-based gain function and DFT-based spatial frequency sampling to quantify the number of independent sub-channels in the near-field.
Codebook Design: Leveraging the derived beamdepth and EBRD, a Polar Codebook is constructed. The design uses non-uniform sampling in the range dimension based on the intersection of 3 dB beamdepth points to minimize column coherence while ensuring full coverage.

3. Key Contributions

A. Theoretical Derivations

Closed-Form Beamdepth & EBRD: Derived exact expressions for beamdepth and EBRD for a generalized URA with aspect ratio $\eta = N_1/N_2$ $η = N_{1} / N_{2}$ .
- Result: Beamdepth is maximized for square arrays ( $\eta=1$ ) under a fixed element count, while elongated arrays (approaching ULA) yield narrower beamdepths and extended near-field regions.
Angle-Dependent Limits: Established that the EBRD is angle-dependent, reaching its maximum at boresight and decreasing toward endfire directions.

B. Geometric Insights

Fixed Element Count: Under a fixed number of antennas, elongated configurations (e.g., ULA-like) significantly reduce beamdepth and extend the EBRD compared to square arrays, enhancing spatial multiplexing.
Fixed Aperture Length: Under a fixed physical aperture, the impact of geometry on beamdepth diminishes, but aperture size ( $D$ ) remains the dominant factor. Increasing $D$ quadratically expands the near-field region (via $r_{RD} \propto D^2$ ), whereas increasing frequency only has a linear effect.

C. Beam Pattern & Sidelobe Analysis

Axial vs. Lateral Resolution: Demonstrated that lateral (angular) resolution is inherently finer than axial (range) resolution in the near-field.
Sidelobe Suppression: Proposed a transformation to map standard far-field window functions (e.g., Hamming) to the near-field range domain. This "modified Hamming" window effectively suppresses axial forelobes (which are higher than lateral sidelobes) without degrading lateral resolution.

D. Spatial Degrees of Freedom (DoF)

Derived an analytical expression for EDoF showing that elongated arrays (ULA configurations) achieve higher EDoF than square URAs in Line-of-Sight (LoS) near-field channels.
Identified that the EDoF reduces to 1 (far-field behavior) at the boundary defined by $r_{RD1} = D_t D_r / \lambda$ , providing a precise limit for near-field gains.

E. Polar Codebook Design

Designed a Polar Codebook specifically for URAs where codewords are spaced by the beamdepth in the distance dimension.
This design guarantees complete near-field coverage up to the EBRD and achieves low column coherence ( $\mu \approx 0.26$ ), outperforming existing methods (P-SOMP, EB-SOMP).

4. Results

Beamdepth Accuracy: The analytical beamdepth expressions closely match numerical simulations, with deviations only occurring at very short focal distances ( $< 2D$ ).
EBRD vs. Geometry: Simulations confirm that for a fixed element count ( $N_{BS}=4096$ ), transitioning from a square array ( $\eta=1$ ) to an elongated array ( $\eta=16$ ) significantly increases the EBRD, allowing more users to be served in the near-field.
Spectral Efficiency (SE): In multiuser scenarios, elongated arrays combined with the proposed polar codebook achieve significantly higher SE (up to 5.8 bps/Hz) compared to DFT codebooks (2 bps/Hz) within the EBRD region.
Channel Estimation Performance:
- NMSE Improvement: The proposed Beam-Focused SOMP (BF-SOMP) algorithm using the new codebook achieves a 2 dB improvement in Normalized Mean Squared Error (NMSE) over state-of-the-art methods at high SNR.
- Pilot Efficiency: The proposed method continues to improve with increasing pilot length, whereas existing methods saturate.
- Complexity: The proposed codebook requires approximately 50% less storage (codebook size) than P-SOMP for large arrays (e.g., $N_{BS}=2048$ ) due to optimized range sampling within the EBRD.

5. Significance

This paper provides a foundational framework for designing next-generation near-field communication systems. Its significance lies in:

Guiding Hardware Deployment: It clarifies that for near-field gains, elongated array geometries are superior to square ones for maximizing spatial multiplexing and extending the effective near-field region.
Defining Operational Limits: The introduction of EBRD offers a more accurate boundary for near-field operation than the classical Rayleigh distance, preventing the overestimation of near-field capabilities.
Enabling Practical Estimation: The proposed Polar Codebook solves the critical bottleneck of channel estimation in UM-MIMO near-field systems, offering a solution that is computationally efficient, storage-light, and highly accurate.
Sidelobe Management: The novel windowing technique provides a practical method to mitigate interference from axial forelobes, a previously overlooked issue in near-field beamforming.

In summary, the work bridges the gap between theoretical near-field electromagnetic properties and practical system design, offering specific guidelines for array geometry, codebook construction, and channel estimation to unlock the full potential of near-field spatial multiplexing.