CAPA: Continuous-Aperture Arrays for Revolutionizing 6G Wireless Communications

In this paper, a novel continuous-aperture array (CAPA)-based wireless communication architecture is proposed, which relies on an electrically large aperture with a continuous current distribution. First, an existing prototype of CAPA is reviewed, followed by the potential benefits and key motivations for employing CAPAs in wireless communications. Then, three practical hardware implementation approaches for CAPAs are introduced based on electronic, optical, and acoustic materials. Furthermore, several beamforming approaches are proposed to optimize the continuous current distributions of CAPAs, which are fundamentally different from those used for conventional spatially discrete arrays (SPDAs). Numerical results are provided to demonstrate their key features in low complexity and near-optimality. Based on these proposed approaches, the performance gains of CAPAs over SPDAs are revealed in terms of channel capacity as well as diversity-multiplexing gains. Finally, several open research problems in CAPA are highlighted.

Optimal Beamforming for Multi-User Continuous Aperture Array (CAPA) Systems

The optimal beamforming design for multi-user continuous aperture array (CAPA) systems is proposed. In contrast to conventional spatially discrete array (SPDA), the beamformer for CAPA is a continuous function rather than a discrete vector or matrix, rendering beamforming optimization a non-convex integral-based functional programming. To address this challenging issue, we first derive the closed-form optimal structure of the CAPA beamformer for maximizing generic system utility functions, by using the Lagrangian duality and the calculus of variations. The derived optimal structure is a linear combination of the continuous channel responses for CAPA, with the linear weights determined by the channel correlations. As a further advance, a monotonic optimization method is proposed for obtaining globally optimal CAPA beamforming based on the derived optimal structure. More particularly, a closed-form fixed-point iteration is proposed to obtain the globally optimal solution to the power minimization problem for CAPA beamforming. Furthermore, based on the optimal structure, the low-complexity maximum ratio transmission (MRT), zero-forcing (ZF), and minimum mean-squared error (MMSE) designs for CAPA beamforming are derived. It is theoretically proved that: 1) the MRT and ZF designs are asymptotically optimal in low and high signal-to-noise ratio (SNR) regimes, respectively, and 2) the MMSE design is optimal for signal-to-leakage-plus-noise ratio (SLNR) maximization. Our numerical results validate the effectiveness of the proposed designs and reveal that: i) CAPA achieves significant communication performance gain over SPDA, and ii) the MMSE design achieves nearly optimal performance in most cases, while the MRT and ZF designs achieve nearly optimal performance in specific cases.

Performance of Linear Receive Beamforming for Continuous Aperture Arrays (CAPAs)

The performance of linear receive beamforming in continuous aperture array (CAPA)-based uplink communications is analyzed. Continuous linear beamforming techniques are proposed for CAPA receivers under the criteria of maximum-ratio combining (MRC), zero-forcing (ZF), and minimum mean-square error (MMSE). i) For MRC beamforming, a closed-form expression for the beamformer is derived to maximize per-user signal power, and the achieved uplink rate and mean-square error (MSE) in detecting received data symbols are analyzed. ii) For ZF beamforming, a closed-form beamformer is derived using channel correlation to eliminate interference, with a function space interpretation demonstrating its optimality in maximizing signal power while ensuring zero inter-user interference. iii) For MMSE beamforming, it is proven to be the optimal linear receive approach for CAPAs in terms of maximizing per-user rate and minimizing MSE. A closed-form expression for the MMSE beamformer is then derived, along with the achievable sum-rate and sum-MSE. The proposed linear beamforming techniques are then compared with those for conventional spatially discrete arrays (SPDAs). Analytical and numerical results indicate that: i) for both CAPAs and SPDAs, the considered linear beamformers can be represented as weighted sums of each user's spatial response, with weights determined by channel correlation; ii) CAPAs achieve higher sum-rates and lower sum-MSEs than SPDAs under ZF and MMSE beamforming; and iii) SPDAs may outperform CAPAs with MRC beamforming in interference-dominated scenarios.

Beamforming Optimization for Continuous Aperture Array (CAPA)-based Communications

The beamforming optimization in continuous aperture array (CAPA)-based multi-user communications is studied. In contrast to conventional spatially discrete antenna arrays, CAPAs can exploit the full spatial degrees of freedoms (DoFs) by emitting information-bearing electromagnetic (EM) wave through continuous source current distributed across the aperture. Nevertheless, such operation renders the beamforming optimization problem as a non-convex integral-based functional programming problem, which is challenging for conventional discrete optimization methods. A couple of low-complexity approaches are proposed to solve the functional programming problem. 1) Calculus of variations (CoV)-based approach: Closed-form structure of the optimal continuous source patterns are derived based on CoV, inspiring a low-complexity integral-free iterative algorithm for solving the functional programming problem. 2) Correlation-based zero-forcing (Corr-ZF) approach: Closed-form ZF source current patterns that completely eliminate the interuser interference are derived based on the channel correlations. By using these patterns, the original functional programming problem is transformed to a simple power allocation problem, which can be solved using the classical water-filling approach with reduced complexity. Our numerical results validate the effectiveness of the proposed designs and reveal that: i) compared to the state-of-the-art Fourier-based discretization approach, the proposed CoV-based approach not only improves communication performance but also reduces computational complexity by up to hundreds of times for large CAPA apertures and high frequencies, and ii) the proposed Corr-ZF approach achieves asymptotically optimal performance compared to the CoV-based approach.

Diversity and Multiplexing for Continuous Aperture Array (CAPA)-Based Communications

The performance of multiplexing and diversity achieved by continuous aperture arrays (CAPAs) over fading channels is analyzed. Angular-domain fading models are derived for CAPA-based multiple-input single-output (MISO), single-input multiple-output (SIMO), and multiple-input multiple-output (MIMO) channels using the Fourier relationship between the spatial response and its angular-domain counterpart. Building on these models, angular-domain transmission frameworks are proposed to facilitate CAPA-based communications, under which the performance of multiplexing and diversity is analyzed. 1) For SIMO and MISO channels, closed-form expressions are derived for the average data rate (ADR) and outage probability (OP). Additionally, asymptotic analyses are performed in the high signal-to-noise ratio (SNR) regime to unveil the maximal multiplexing gain and maximal diversity gain. The diversity-multiplexing trade-off (DMT) is also characterized, along with the array gain within the DMT framework. 2) For MIMO channels, high-SNR approximations are derived for the ADR and OP, based on which the DMT and associated array gain are revealed. The performance of CAPAs is further compared with that of conventional spatially discrete arrays (SPDAs) to highlight the superiority of CAPAs. The analytical and numerical results demonstrate that: i) compared to SPDAs, CAPAs achieve a lower OP and higher ADR, resulting in better spectral efficiency; ii) CAPAs achieve the same DMT as SPDAs with half-wavelength antenna spacing while attaining a larger array gain; and iii) CAPAs achieve a better DMT than SPDAs with antenna spacing greater than half a wavelength.

Performance Analysis of Physical Layer Security: From Far-Field to Near-Field

The secrecy performance in both near-field and far-field communications is analyzed using two fundamental metrics: the secrecy capacity under a power constraint and the minimum power requirement to achieve a specified secrecy rate target. 1) For the secrecy capacity, a closed-form expression is derived under a discrete-time memoryless setup. This expression is further analyzed under several far-field and near-field channel models, and the capacity scaling law is revealed by assuming an infinitely large transmit array and an infinitely high power. A novel concept of "depth of insecurity" is proposed to evaluate the secrecy performance achieved by near-field beamfocusing. It is demonstrated that increasing the number of transmit antennas reduces this depth and thus improves the secrecy performance. 2) Regarding the minimum required power, a closed-form expression is derived and analyzed within far-field and near-field scenarios. Asymptotic analyses are performed by setting the number of transmit antennas to infinity to unveil the power scaling law. Numerical results are provided to demonstrate that: i) compared to far-field communications, near-field communications expand the areas where secure transmission is feasible, specifically when the eavesdropper is located in the same direction as the intended receiver; ii) as the number of transmit antennas increases, neither the secrecy capacity nor the minimum required power scales or vanishes unboundedly, adhering to the principle of energy conservation.

A Primer on Near-Field Communications for Next-Generation Multiple Access

Multiple-antenna technologies are advancing toward the development of extremely large aperture arrays and the utilization of extremely high frequencies, driving the progress of next-generation multiple access (NGMA). This evolution is accompanied by the emergence of near-field communications (NFC), characterized by spherical-wave propagation, which introduces additional range dimensions to the channel and enhances system throughput. In this context, a tutorial-based primer on NFC is presented, emphasizing its applications in multiuser communications and multiple access (MA). The following areas are investigated: \romannumeral1) the commonly used near-field channel models are reviewed along with their simplifications under various near-field conditions. \romannumeral2) Building upon these models, the information-theoretic capacity limits of NFC-MA are analyzed, including the derivation of sum-rate capacity and capacity region, and their upper limits for both downlink and uplink scenarios. \romannumeral3) A detailed investigation of near-field multiuser beamforming design is presented, offering low-complexity and effective NFC-MA design methodologies in both the spatial and wavenumber (angular) domains. Throughout these investigations, near-field MA is compared with its far-field counterpart to highlight its superiority and flexibility in terms of interference management, thereby laying the groundwork for achieving NGMA.

Continuous Aperture Array (CAPA)-Based Wireless Communications: Capacity Characterization

The capacity limits of continuous-aperture array (CAPA)-based wireless communications are characterized. To this end, an analytically tractable transmission framework is established for both uplink and downlink CAPA systems. Based on this framework, closed-form expressions for the single-user channel capacity are derived. The results are further extended to a multiuser case by characterizing the capacity limits of a two-user channel and proposing the associated capacity-achieving decoding and encoding schemes. 1) For the uplink case, the sum-rate capacity and capacity region, as well as the capacity-achieving detectors, are derived. 2) For the downlink case, the uplink-downlink duality is established by deriving the uplink-to-downlink and downlink-to-uplink transformations under the same power constraint, based on which the optimal power allocation policy and the achieved sum-rate capacity and capacity region are characterized. To gain further insights, several case studies are presented by specializing the derived results into various array structures, including the planar CAPA, linear CAPA, and planar spatially discrete array (SPDA). Numerical results are provided to reveal that: i) the channel capacity achieved by CAPAs converges towards a finite upper bound as the aperture size increases; and ii) CAPAs offer significant capacity gains over the conventional SPDAs.

Aperture Selection for CAP Arrays (CAPAs)

The concept of aperture selection is proposed for continuous aperture array (CAPA)-based communications. The achieved performance is analyzed in an uplink scenario by considering both line-of-sight (LoS) and non-line-of-sight (NLoS) scenarios. In the LoS scenario, the optimal selection strategy is demonstrated to follow the nearest neighbor criterion, and the resulting signal-to-noise ratio (SNR) is analyzed. In the NLoS scenario, the achieved outage probability along with the diversity order is revealed. Numerical results are provided to demonstrate that aperture selection effectively maintains satisfactory performance by leveraging selection diversity while simultaneously reducing the implementation complexity of CAPAs.

On the Performance of Continuous Aperture Array (CAPA)-Based Wireless Communications

The performance of continuous aperture array (CAPA)-based wireless communications is analyzed in an uplink scenario. An analytical framework is proposed to characterize uplink CAPA-based transmission using electromagnetic field theories. On this basis, new expressions are derived for the channel capacity in a single-user scenario and the sum-rate capacity in a multiuser scenario, along with the capacity-achieving decoding schemes. These findings are proved to differ greatly from those established for conventional spatially discrete (SPD) arrays. Numerical results are provided to demonstrate that CAPA offers significant capacity gains compared to the SPD array.