Continuous Wavefront Design via Virtual Point Sources: A Holographic Paradigm for Near-Field XL-MIMO

Abstract: Beamforming design for extremely large-scale multiple-input multiple-output (XL-MIMO) systems is challenging due to prohibitive computational complexity and complex near-field propagation effects. To address this, this paper introduces a holographic beamforming paradigm that reformulates the design from optimizing variables at spatially discrete antenna locations to shaping a continuous electromagnetic wave function over the array aperture, effectively mitigating the growth of algorithmic complexity as the array scale increases. We apply this paradigm to the challenging dual near-field (DNF) scenario, where strong transceiver coupling severely degrades conventional iterative algorithms. In this case, we propose a novel Virtual Point Source (VPS) method, which approximates the ideal wave function with a single and analytically tractable spherical-wave. A rigorous geometric-optical analysis is provided to show that the optimal VPS location can be determined in a fully non-iterative manner, thus decoupling the coupled DNF problem. The proposed method is demonstrated in an intelligent reflecting surfaces (IRS)-assisted system, where simulation results show that our non-iterative approach achieves performance comparable to converged alternating-optimization (AO) algorithms, while incurring significantly lower complexity and avoiding convergence uncertainty. This work offers a new theoretical framework for holographic beamforming design in XL-MIMO systems.

• The paper introduces a holographic beamforming approach that leverages virtual point sources to simplify XL-MIMO design via a continuous wavefront model.
• It develops a spectral-domain framework incorporating interference, diffraction, and spatial filtering, yielding closed-form solutions in near-field scenarios.
• The VPS method achieves robust, non-iterative optimization for IRS-assisted and LoS channels, significantly reducing computational complexity.

Continuous Wavefront Design via Virtual Point Sources: A Holographic Paradigm for Near-Field XL-MIMO

Introduction and Motivation

The proliferation of 6G and beyond wireless networks is driving a dramatic escalation in the use of extremely large-scale MIMO (XL-MIMO) systems, particularly in the mmWave and THz bands. In these high-frequency regimes, the resulting electrical aperture sizes fundamentally alter array propagation physics and introduce strong near-field effects, which invalidate traditional far-field planar-wave models. This paradigm shift mandates fundamentally new approaches to beamforming design, as conventional optimization frameworks predicated on discrete antenna-centric models fail to scale both computationally and physically. The paper "Continuous Wavefront Design via Virtual Point Sources: A Holographic Paradigm for Near-Field XL-MIMO" (2604.08908) undertakes a comprehensive re-formulation of the XL-MIMO beamforming problem based on electromagnetic field theory, introducing a continuous, wavefront-centric, or "holographic" methodology. This approach transcends element-wise optimization and provides a rigorous, computationally efficient strategy tailored for the unique challenges of near-field, coupled multi-array environments.

Analytical Framework: Continuous-Space XL-MIMO Modeling

The paper develops a spectral-domain framework for analyzing beamforming in finite- and infinite-aperture MIMO arrays. Key physical effects are decomposed into:

• Interference spectrum: Captures the underlying angular content of the idealized array excitation.
• Diffraction spectrum: Models the waveform distortion arising from finite physical apertures, primarily described via the sinc function in the Fourier domain.
• Spatial filtering (channel passband): Realized via array geometry, only certain directional components propagate efficiently between TX/RX.

A pivotal result in the analysis is the diffraction degradation principle: As the electrical aperture $D/\lambda \rightarrow \infty$, the main lobe of the diffraction pattern narrows to a delta function, enabling "perfect projection" of continuous wavefronts with arbitrary spatial precision. In this regime, the discrete antenna problem transitions to an effectively continuous one, decoupling computational complexity from aperture size, and fundamentally altering design methodology.

In near-field dual-coupled (DNF) channels, this leads to channel-induced strong coupling of the angular spectrum across TX/RX, rendering classic iterative AO algorithms highly non-convex and initialization-sensitive. In contrast, for far-field or single near-field (SNF) channels, the spatial filtering collapses to a point (delta function or localized "focal region"), enabling fully decoupled, closed-form solutions.

Holographic Array Definition and Path-Centric Design

A new, physically-motivated definition of a holographic array is introduced, based on:

• Infinite electrical aperture ($D/\lambda \rightarrow \infty$) with a fixed physical aperture, achieved via increasing frequency.
• Critical (Nyquist) spatial sampling ($d = \lambda/2$), avoiding both angular spectrum aliasing and unnecessary spatial domain coupling.

This definition contrasts with prior holographic MIMO literature, which often invoked arbitrarily dense sub-wavelength sampling.

The accompanying holographic methodology is structurally two-step:

1. Use Maxwellian wave physics to identify a small parameter set describing dominant propagation paths (e.g., wavefront curvature and direction).
2. Synthesize the required continuous phase profile across the aperture, which can then be discretized to configure the actual TX/RX or IRS elements.

The key insight is that for LoS/sparse channels, the bulk of field synthesis can be reduced to constructing a spatially continuous wave function—removing dependence on the sheer number of physical elements, and instead making complexity a function of path and geometric parameters.

Virtual Point Source (VPS) Paradigm for DNF Decoupling

The main technical innovation is the Virtual Point Source (VPS) algorithm for DNF MIMO and IRS-aided systems. The approach proceeds as follows:

• Replace the high-dimensional joint optimization of discrete element phase/amplitude with the lower-dimensional task of placing a single "virtual point source" (VPS) in space.
• The optimal VPS location is determined using non-iterative geometric-optical analysis, maximizing the intersection (Common Visible Region, CVR) of the solid angles subtended by TX and RX/IRS elements.
• Once the VPS is fixed, closed-form continuous wave functions are synthesized at both endpoints, generating phase profiles purely by evaluating geometric delay relative to the VPS.
• The method thus transforms the NP-hard coupled iterative optimization into a decoupled, direct procedure; complexity is invariant to aperture size and only depends on geometric scene parameters.

The VPS method is rigorously justified for LoS-dominated XL-MIMO, where amplitude non-uniformity is negligible compared to rapid phase variations, as shown via asymptotic analysis. For more complex multipath environments, the method generalizes to multiple VPSs.

Practical Application: IRS-Assisted DNF

The IRS case considers coupling between a BS, IRS, and a user in near-field. The paper details a non-iterative VPS-based algorithm:

1. Use the opposing-triangles construction to analytically determine the optimal VPS position between BS and IRS.
2. Configure the BS array to focus energy on the VPS.
3. Set IRS element phases to coherently combine incoming energy from the VPS and re-radiate towards the user, again determined by geometric distance to the VPS and user.

This non-iterative, geometry-driven framework robustly overcomes convergence and optimality limitations of alternating-optimization procedures.

Numerical Results

Extensive simulation results are provided, including:

• Convergence studies: VPS-initialized AO always outperforms far-field, zero, and random initializations. The non-iterative VPS design yields performance at least as high as the best-case AO after optimization, and is substantially more robust to initial conditions.
• Frequency scaling: The VPS method’s efficacy increases with electrical aperture (higher carrier frequency, fixed size), validating the theoretical foundations.
• Angular/siting robustness: The VPS approach demonstrates extremely stable performance against array misalignment.
• Distance scaling: Performance gains are invariant to global scene scale, confirming the method's generalizability.

Implications and Future Research

This work introduces a physically-grounded, computationally efficient paradigm for XL-MIMO and IRS-aided system design in 6G and beyond. The formalization of the diffraction degradation principle and the continuous/holographic array model provides a theoretical basis for rethinking large-scale array synthesis, moving away from element-centric optimization towards direct field control via geometric optics.

Possible future directions include:

• Extending the VPS framework to multi-user, non-LoS, or multipath-dense environments, via parametric VPS superposition.
• Theoretical analysis of quantization errors induced by practical element discretization in the continuous-to-discrete mapping.
• Joint optimization of analog/hybrid transceiver architectures leveraging the continuous wavefront model.
• Near-field localization and LDMA design using the holographic framework.

Conclusion

The VPS-based holographic methodology provides a formal, physically-consistent, and scalable approach to near-field XL-MIMO beamforming design. By reframing the optimization problem in terms of continuous wavefront design over the array aperture and using geometric-optical principles for non-iterative solution construction, this paradigm achieves performance competitive with, and often superior to, conventional AO-based methods while incurring dramatically reduced complexity. This theoretical advance lays the groundwork for practical, high-performance, and tractable deployment of XL-MIMO and IRS-assisted systems in future wireless networks (2604.08908).