⚡ electrical engineering

Near-field Boundary Distance in mmWave and THz Communications with Misaligned Antenna Arrays

This paper presents a generalized mathematical framework and derives closed-form expressions to characterize the radiative near-field boundary distance in mmWave and THz communications, specifically accounting for realistic antenna array misalignments that significantly alter traditional near-field assumptions.

Original authors: Peng Zhang, Vitaly Petrov, Emil Björnson

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Peng Zhang, Vitaly Petrov, Emil Björnson

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 have a conversation with a friend across a large, noisy field. To make sure you hear each other clearly, you both use high-tech, directional megaphones (antenna arrays) that focus your voices into tight beams.

In the world of 5G and future 6G networks, these "megaphones" operate at incredibly high frequencies (millimeter waves and Terahertz). Because the sound waves are so tiny, the area where the sound is "focused" but not yet perfectly straight is surprisingly large. We call this the Near-Field.

Here is the problem: Most old textbooks assume that if you stand still and point your megaphones perfectly at each other, the rules are simple. But in the real world, people move. Your friend might tilt their head, turn their body, or walk slightly to the side. This paper asks: "What happens to the 'safe zone' for communication when our megaphones aren't pointing perfectly straight at each other?"

The Core Concept: The "Perfect Alignment" Myth

Think of the Near-Field as a "fuzzy zone" right in front of your megaphone. Inside this zone, the sound waves are curved, like ripples in a pond. Outside this zone (the Far-Field), the waves flatten out and travel in straight lines, like a laser beam.

Engineers need to know exactly where the "fuzzy zone" ends and the "straight line" begins. This distance is called the Near-Field Boundary.

The Old Way: Previous research assumed that both you and your friend were standing perfectly still and pointing your megaphones directly at each other. They calculated the boundary based on this perfect scenario.
The Real World: In reality, your friend is walking, maybe looking at a different angle, or their phone is tilted in their hand. This is Misalignment.

The Paper's Big Discovery

The authors of this paper realized that misalignment changes the size of the "fuzzy zone."

Imagine you are holding a flashlight. If you shine it straight at a wall, the light spreads out in a predictable circle. But if you tilt the flashlight slightly, the shape of the light on the wall changes. It might stretch out or shrink.

Similarly, when the transmitting and receiving antennas are tilted relative to each other:

The "Fuzzy Zone" Shifts: The distance where the waves are curved can become significantly longer or shorter than the textbooks predict.
The Impact is Huge: In some cases, ignoring this tilt can lead to errors of tens of meters. In a high-speed network, being off by 20 meters could mean the difference between using a simple, fast connection (Far-Field) and needing a complex, slow, energy-hungry connection (Near-Field).

The "Recipe" They Created

The authors didn't just point out the problem; they wrote a new "recipe" (a mathematical framework) to calculate the correct boundary distance for any situation.

The Ingredients: They considered different shapes of antennas (long strips called ULAs and flat squares called UPAs) and how they might be rotated in 3D space (tilted up/down, left/right).
The Method: They used a concept called Phase Error. Imagine the sound waves as a marching band. In the Far-Field, everyone marches in perfect step. In the Near-Field, the people at the edges of the band are slightly out of step with the people in the middle. The paper calculates exactly how far away you need to be so that, even with the antennas tilted, the band is still marching in step enough to be considered "Far-Field."

Why This Matters for Your Future Phone

You might wonder, "Why should I care about a few meters?"

Battery Life: If your phone thinks it's in the "Far-Field" (straight lines) but it's actually in the "Near-Field" (curved waves), it will try to use a simple beamforming technique. This might fail, causing the signal to drop or the phone to use more power trying to fix it.
Speed: Knowing the exact boundary helps engineers design networks that switch between "simple" and "complex" modes at the right moment. If we get the math wrong because we ignored the tilt, the network might switch too late, slowing down your video call.
Realistic Design: As we move to 6G, devices will be everywhere—on drones, in cars, on moving trains. They will rarely be perfectly aligned. This paper provides the tools to build networks that work reliably even when everything is moving and tilting.

The Takeaway

Think of this paper as a new GPS for wireless signals.

Old maps said, "If you are 50 meters away, the signal is straight."
This new map says, "If you are 50 meters away, but your friend is tilting their phone 30 degrees, the signal is actually still curved, and you are still in the 'fuzzy zone.' You need to be 70 meters away to be safe."

By accounting for the fact that antennas aren't perfect and people aren't robots, this research ensures that the ultra-fast wireless networks of tomorrow won't stumble over simple geometry errors.

1. Problem Statement

Millimeter-wave (mmWave) and Terahertz (THz) communications utilize large frequency bands to achieve ultra-high data rates. However, the extremely short wavelengths of these signals, combined with the deployment of large-scale antenna arrays (to maintain link budget), significantly extend the radiative near-field region. In this region, the conventional far-field assumption (plane-wave propagation) fails, and spherical wave models are required.

The Core Gap:
Existing literature characterizes the near-field boundary distance (the transition point where far-field approximations become valid) primarily under the assumption of perfect alignment between transmitter (Tx) and receiver (Rx) antenna arrays. In realistic deployments, particularly with mobile nodes, antenna arrays inevitably suffer from misalignment (rotations and off-boresight angles).

Current models do not account for these rotations.
Misalignment significantly alters the phase distribution across the array, potentially shifting the near-field boundary by tens of meters.
Ignoring this can lead to incorrect system design, such as applying far-field beamsteering in a near-field regime (causing gain loss) or unnecessarily using complex near-field focusing.

2. Methodology

The authors propose a generalized mathematical framework to characterize the near-field boundary under arbitrary array misalignments.

Generalized Phase-Error Criterion:
Instead of relying solely on the canonical Fraunhofer criterion (max phase deviation $\pi/8$ ), the paper introduces a tunable threshold $\phi \in (0, \pi]$ . The near-field boundary $r_F$ is defined as the minimum distance where the maximum residual phase difference between any two Tx-Rx element pairs (after applying beamsteering/phase compensation) does not exceed $\phi$ .
Geometry-Driven Derivation:
The authors model the system using 3D geometry, accounting for:
- Tx/Rx Rotations: Vertical ( $\theta$ ) and horizontal ( $\phi$ ) rotations of the Tx array.
- Off-boresight Angles ( $\alpha$ ): The angle between the array centers and the boresight direction.
- Effective Distance: They derive the "effective distance" $r'$ by combining the physical propagation distance with the phase pre-compensation applied by Tx and Rx beamsteering.
- Taylor Approximation: A third-order Taylor expansion is used to approximate the spherical wave phase delay, isolating the dominant terms dependent on array aperture and rotation angles.
Configurations Analyzed:
The framework is applied to derive closed-form expressions for:
1. ULA-to-ULA (L2L): Uniform Linear Arrays.
2. UPA-to-UPA (P2P): Uniform Planar Arrays (considering both single and dual-angle rotations).
3. Mixed/Point Scenarios: ULA-to-Point and UPA-to-Point setups.

3. Key Contributions

Unified Framework for Misalignment: The paper establishes a unified approach to calculate near-field boundaries that explicitly incorporates Tx/Rx rotation angles and off-boresight displacements, moving beyond the "perfect alignment" assumption.
Closed-Form Expressions:
- For L2L (On-boresight): $r_F \propto (D_1 \cos \theta + D_2)^2$ .
- For L2L (Off-boresight): A dominant expression is derived showing dependence on the relative rotation mismatch $(\theta - \alpha)$ .
- For P2P (Planar): The authors derive complex expressions showing that UPA boundaries are the sum of two Fraunhofer-type terms (one for each dimension), revealing a non-separable coupling between azimuth and elevation rotations.
- Mixed Arrays: The framework is shown to be extensible to mixed UPA-to-ULA links.
Threshold Analysis: The paper proves that for practical mmWave/THz parameters, a single dominant expression usually suffices for L2L off-boresight cases, whereas P2P cases require a "max" of two branches due to 3D orientation coupling.
Generalized Criterion: The introduction of the tunable parameter $\phi$ allows the model to adapt to different system requirements (e.g., stricter phase accuracy for high-order modulation).

4. Key Results

Extensive numerical simulations (validated against an in-house simulator) reveal the following:

Significant Deviation: Antenna misalignment causes non-negligible deviations in the near-field boundary.
- In practical setups, ignoring rotation can lead to errors of 7% to 28% for typical device sizes (smartphones/tablets).
- In extreme 3D rotation scenarios, deviations can reach up to 70%, translating to shifts of several tens of meters in the boundary distance.
UPA vs. ULA:
- UPAs generally exhibit larger near-field distances than ULAs due to their 2D aperture structure.
- UPAs show a unique sensitivity pattern where one dimension can remain dominant even if the other is heavily misaligned, unlike ULAs where rotation directly reduces the projected aperture.
Impact of Rotation:
- As the rotation angle $|\theta|$ increases, the projected aperture decreases, generally reducing the near-field boundary distance.
- However, the relationship is not always monotonic or symmetric in 3D (UPA) cases due to the coupling of azimuth and elevation angles.
Statistical Variability: Under random misalignment (modeled via truncated von Mises distribution), the near-field distance distribution shifts significantly away from the no-rotation baseline, particularly at THz frequencies (1 THz).

5. Significance and Impact

This work is critical for the design and optimization of 6G and beyond communication systems:

System Design: Accurate determination of the near-field boundary is essential for deciding whether to use far-field beamsteering (distance-independent) or near-field beamfocusing (distance-dependent). Misclassifying the regime leads to performance degradation.
Robustness: The derived expressions allow engineers to design robust alignment strategies and codebooks that account for realistic node mobility and orientation errors.
Standardization: The generalized framework provides a more accurate basis for defining near-field/far-field transition criteria in future wireless standards, moving beyond the simplistic Fraunhofer distance ( $2D^2/\lambda$ ) which assumes perfect alignment.

In summary, the paper demonstrates that ignoring antenna misalignment in mmWave/THz near-field analysis is a critical error that can lead to substantial system inefficiencies. The proposed closed-form expressions provide the necessary tools to accurately model these effects for next-generation network planning.