[Paper Review] Communicating with Extremely Large-Scale Array/Surface: Unified Modelling and Performance Analysis

Haiquan Lu, Yong Zeng|arXiv (Cornell University)|Apr 27, 2021

Antenna Design and AnalysisEngineering39 references17 citations

TL;DR

This paper proposes a unified physical model for extremely large-scale arrays (XL-arrays) and surfaces that explicitly accounts for element aperture, phase, and amplitude variations, moving beyond the conventional uniform plane wave (UPW) assumption. It derives a closed-form SNR expression showing SNR grows sub-linearly with array size M due to collective array properties like occupation ratio and dimensions, with a new 'uniform-power distance' criterion replacing Rayleigh distance for near-field modeling.

ABSTRACT

Wireless communications with extremely large-scale array (XL-array) correspond to systems whose antenna sizes are so large that conventional modelling assumptions, such as uniform plane wave (UPW) impingement, are longer valid. This paper studies the mathematical modelling and performance analysis of XL-array communications. By deviating from the conventional modelling approach that treats the array elements as sizeless points, we explicitly model their physical area/aperture, which enables a unified modelling for the classical discrete antenna arrays and the emerging continuous surfaces. As such, a generic array/surface model that accurately takes into account the variations of signal phase, amplitude and projected aperture across array elements is proposed. Based on the proposed model, a closed-form expression of the resulting SNR with the optimal single-user MRC/MRT beamforming is derived. The expression reveals that instead of scaling linearly with the antenna number M as in conventional UPW modelling, the SNR with the more generic model increases with M with diminishing return, which is governed by the collective properties of the array, such as the array occupation ratio and the physical sizes of the array along each dimension, while irrespective of the properties of the individual array element. Additionally, we have derived an alternative insightful expression for the optimal SNR in terms of the vertical and horizontal angular spans. Furthermore, we also show that our derived results include the far-field UPW modelling as a special case. One important finding during the study of far-field approximation is the necessity to introduce a new distance criterion to complement the classical Rayleigh distance, termed uniform-power distance (UPD), which concerns the signal amplitude/power variations across array elements, instead of phase variations as for Rayleigh distance.

Motivation & Objective

• Address the limitations of conventional uniform plane wave (UPW) modeling in extremely large-scale array (XL-array) systems where array size invalidates the UPW assumption.
• Develop a unified mathematical framework that models both discrete arrays and continuous surfaces by explicitly incorporating the physical aperture and spatial response of array elements.
• Analyze the performance of optimal single-user maximum ratio combining/transmission (MRC/MRT) beamforming in XL-array systems under realistic near-field conditions.
• Establish a new distance criterion—uniform-power distance (UPD)—to complement Rayleigh distance, focusing on amplitude/power variations rather than phase errors.
• Demonstrate that SNR scales sub-linearly with M, governed by collective array properties rather than individual element characteristics.

Proposed method

• Propose a generic array/surface model that explicitly models the physical aperture and spatial response of each array element, replacing the point-antenna assumption.
• Derive a closed-form expression for the signal-to-noise ratio (SNR) under optimal MRC/MRT beamforming using the new model, incorporating phase, amplitude, and projected aperture variations.
• Introduce the concept of 'uniform-power distance' (UPD), a new criterion based on signal power variation across elements, to define the near-field region, contrasting with the classical Rayleigh distance based on phase error.
• Express the optimal SNR in terms of geometric angular spans (vertical and horizontal) formed by the array and user location, providing an insightful link between array geometry and performance.
• Use asymptotic approximations and Taylor expansions to simplify complex integrals in the SNR derivation, enabling analytical tractability.
• Validate the model through extensive numerical comparisons with benchmark models, demonstrating the necessity of the proposed approach in XL-array regimes.

Experimental results

Research questions

• RQ1 How does the performance of XL-array systems change when the conventional uniform plane wave (UPW) assumption is no longer valid due to large array size?
• RQ2 What is the correct mathematical model for XL-arrays and continuous surfaces that accounts for physical aperture, phase, and amplitude variations across elements?
• RQ3 How does the SNR scale with the number of antennas M in XL-arrays under optimal beamforming, and what factors govern this scaling?
• RQ4 What is the appropriate distance criterion to define the near-field region in XL-arrays, and how does it differ from the classical Rayleigh distance?
• RQ5 To what extent do collective array properties (e.g., occupation ratio, dimensions) influence SNR, and how do they compare to individual element properties?

Key findings

• The proposed model generalizes conventional UPW-based modeling and reduces to it as a special case when the user is in the far-field.
• The optimal SNR scales sub-linearly with the number of antennas M, governed by collective array properties such as array occupation ratio and physical dimensions, not individual element characteristics.
• The SNR expression reveals that the gain from increasing M diminishes due to aperture and phase variations across the array, with the rate of increase determined by the array's geometric configuration.
• The uniform-power distance (UPD) is introduced as a more appropriate near-field criterion than Rayleigh distance, as it focuses on amplitude/power variations rather than phase errors.
• The SNR can be expressed in terms of the vertical and horizontal angular spans formed by the array and user location, offering a geometrically intuitive performance metric.
• Numerical results confirm that ignoring aperture and amplitude variations leads to significant performance overestimation, validating the necessity of the proposed model.