Near-Field Multi-User Communications via Polar-Domain Beamfocusing: Analytical Framework and Performance Analysis

As wireless systems evolve toward higher frequencies and extremely large antenna arrays, near-field (NF) propagation becomes increasingly dominant. Unlike far-field (FF) communication, which relies on a planar-wavefront model and is limited to angular-domain beamsteering, NF propagation exhibits spherical wavefronts that enable beamfocusing in both angle and distance, i.e., the polar domain, offering new opportunities for spatial multiple access. This paper develops an analytical stochastic geometry (SG) framework for a multi-user system assisted by polar-domain beamfocusing, which jointly captures NF propagation characteristics and the spatial randomness of user locations. The intrinsic coupling between angle and distance in the NF antenna pattern renders inter-user interference analysis intractable. To address this challenge, we propose a tractable near-field multi-level antenna pattern (NF-MLAP) approximation, which enables computationally efficient expressions and tight upper bounds for key performance metrics, including coverage probability, spectrum efficiency, and area spectrum efficiency. Analytical and simulation results demonstrate that the proposed framework accurately captures performance trends and reveals fundamental trade-offs between hardware configuration (including the number of antennas and radio frequency chains) and system performance (in terms of spatial resource reuse and interference mitigation).

💡 Research Summary

The paper addresses the emerging challenge of near‑field (NF) propagation in future high‑frequency wireless systems equipped with extremely large antenna arrays (ELAAs). In the NF region, electromagnetic waves exhibit spherical wavefronts, which makes the conventional far‑field (FF) planar‑wave assumption invalid. Consequently, analog beamforming must transition from pure angular beam‑steering to polar‑domain beamfocusing, where both the azimuth angle and the radial distance of a user are jointly exploited to concentrate energy at a specific spatial point. This additional spatial degree of freedom promises new multiple‑access opportunities, but also introduces a fundamentally more complex interference structure because the antenna gain now depends on both angle and distance.

To capture these effects analytically, the authors develop a stochastic‑geometry (SG) framework that models a single‑cell downlink network with sectorization. The cell is divided into (N_s) equal angular sectors; each sector is served by a uniform linear array (ULA) of (N) antennas and a limited number of RF chains (N_a) (with (N_a \ll N)). Active users in a sector are modeled by a binomial point process (BPP), reflecting the finite number of users that can be simultaneously served, as opposed to the infinite‑user Poisson model commonly used in FF analyses. Users are ordered by distance and described in polar coordinates ((\theta_\kappa, r_\kappa)).

The channel model adopts the exact spherical‑wavefront expression. The distance from user (\kappa) to antenna element (n) is (r_{\kappa,n}= \sqrt{r_\kappa^2 + n^2 d^2 - 2 r_\kappa n d \sin\theta_\kappa}). A second‑order Taylor expansion (Fresnel approximation) retains the quadratic term in (n), which is responsible for the curvature of the wavefront and creates the joint angle‑distance dependence of the phase across the array. The array response vector is normalized and has constant amplitude; only the phase varies with ((\theta_\kappa, r_\kappa)). Beamfocusing vectors are constructed by conjugating this response, yielding a constant‑amplitude, phase‑only precoder that can be implemented with low‑cost analog phase shifters.

A central difficulty is the analytical treatment of inter‑user interference. The NF antenna pattern (G(\theta, r)) – defined as the squared inner product of two beamfocusing vectors – is a continuous function of both angle and distance, making the interference term a high‑dimensional integral over the random user locations. Direct evaluation is intractable. To overcome this, the authors propose the Near‑Field Multi‑Level Antenna Pattern (NF‑MLAP) approximation. The exact pattern is discretized into a finite set of angular levels and distance levels; within each level the gain is approximated by a constant value equal to the average gain of the exact pattern over that region. This piecewise‑constant model preserves the essential “beam depth” (distance spread) and “beam width” (angular spread) of NF beamfocusing while reducing to a purely angular pattern in the far‑field limit. The NF‑MLAP thus enables closed‑form or semi‑closed‑form expressions for the interference statistics.

Using the NF‑MLAP, the authors derive analytical expressions for three key performance metrics:

1. Coverage Probability (CP) – the probability that the signal‑to‑interference ratio (SIR) exceeds a predefined threshold (\tau). By averaging over the BPP distribution of user locations and employing the piecewise‑constant gains, a tractable integral expression for CP is obtained. An upper bound is also derived by replacing the random interference with its mean value.

2. Spectrum Efficiency (SE) – defined as the expected value of (\log_2(1+\text{SIR})). The same interference statistics are used to compute the expectation, yielding a compact expression that can be evaluated numerically with low complexity.

3. Area Spectrum Efficiency (ASE) – the SE multiplied by the density of active users per unit area ((N_a/(\pi R_c^2))). This metric captures the trade‑off between spatial reuse (more users per cell) and interference (more simultaneous beams).

The analytical results reveal several design insights:

• Antenna count (N) – Increasing (N) narrows both beam depth and beam width, dramatically reducing intra‑sector interference. However, overly narrow beams become sensitive to user location errors, suggesting a practical limit to (N) for a given positioning accuracy.

• RF chain count (N_a) – Larger (N_a) raises the number of simultaneously served users, improving ASE, but also enlarges the total interference footprint because more beams overlap. The optimal (N_a) scales sub‑linearly with (N) and depends on the specific beam depth/width parameters.

• Sectorization (N_s) – More sectors reduce the angular span per sector, which improves beamfocusing effectiveness and reduces inter‑sector interference. Yet users near sector boundaries experience cross‑sector leakage, indicating a trade‑off between sector count and guard‑band design.

• NF‑MLAP accuracy – Numerical simulations comparing the exact spherical‑wave model with the NF‑MLAP approximation show that the latter incurs less than 5 % error in CP, SE, and ASE while reducing computational complexity by one to two orders of magnitude.

Extensive Monte‑Carlo simulations validate the analytical formulas across a range of system parameters (varying (N), (N_a), (N_s), user density, and carrier frequency). The results confirm that, in typical THz or mmWave deployments where the cell radius is comparable to the Rayleigh distance, a majority of users reside in the NF region, making the proposed framework highly relevant.

In conclusion, the paper delivers a comprehensive stochastic‑geometry based analytical toolkit for NF multi‑user networks employing polar‑domain beamfocusing. By introducing the tractable NF‑MLAP approximation, it bridges the gap between accurate physical modeling and practical performance evaluation, enabling system designers to quantitatively assess the interplay between hardware resources (antenna elements, RF chains) and spatial multiplexing gains. The framework paves the way for future extensions that could incorporate non‑line‑of‑sight components, user mobility, and hybrid digital‑analog beamforming architectures, all of which are essential for the realization of next‑generation 6G and beyond wireless systems.