Continuous Aperture Array (CAPA) Fundamentals

• Continuous Aperture Array (CAPA) is a wireless architecture that models beamforming as a continuous current distribution over an electrically large aperture.
• It employs integral operators, Green’s functions, and calculus of variations to optimize beamforming functions in a function-space framework.
• By approximating ultradense discrete arrays with a continuous model, CAPA enhances energy collection, reliability, and secure communication with diverse hardware implementations.

Continuous Aperture Array (CAPA) denotes a wireless communication architecture in which transmission or reception is modeled over an electrically large aperture with a continuous current distribution, rather than over a finite set of discrete antenna ports. In CAPA, the beamformer is a continuous function over the aperture, the channel is represented by continuous spatial responses or integral operators, and core performance objectives become functionals on function spaces. CAPAs are used both as a physically grounded model for holographic or surface-based arrays and as a theoretical upper bound for densely packed discrete arrays, since they idealize the limiting case of aperture densification in which the full surface is used continuously rather than sampled by a finite number of elements (Liu et al., 2024, Inwood et al., 12 Mar 2026).

1. Conceptual foundations and relation to discrete arrays

A CAPA is fundamentally different from a spatially discrete array because the optimization variable is not a finite-dimensional beamforming vector or matrix, but a continuous source current distribution over a physical surface. In the survey formulation, CAPA is “a single electrically large aperture with a continuous current distribution,” whereas in an SPDA the array is modeled through a finite-dimensional vector of antenna excitations. This change is both electromagnetic and mathematical: physically, one shapes the radiating current over the aperture surface; algorithmically, one replaces matrix-based channel models by continuous functions and integral operators (Liu et al., 2024).

The literature repeatedly emphasizes that CAPA should not be reduced to the slogan “many antennas.” A CAPA is instead a different modeling and design paradigm in which the aperture current distribution itself is the basic object. This is why CAPA is used as a benchmark for future ultra-dense architectures, including holographic MIMO and related continuous electromagnetic apertures, and why it is described as a useful theoretical upper bound for densely packed discrete arrays (Inwood et al., 12 Mar 2026).

The “continuous” description is, however, an idealization. The survey reviews a Kymeta Corporation prototype consisting of approximately 70,000 radiating elements, each equipped with a tunable capacitor, arranged over a large circular aperture of diameter 82 cm, with software-adjustable amplitude and phase for each element. The point of such prototypes is not that hardware becomes literally continuous, but that sufficiently dense radiating structures approximate a continuous current sheet closely enough for continuous-aperture analysis to be meaningful (Liu et al., 2024).

This idealization is also what makes CAPA extensible. The rigid planar surface used in most early models is generalized by the flexible continuous aperture array (FCAPA), in which the surface is parameterized as

$$\mathbf{s} : \mathcal{U}\to\mathbb{R}^3,\qquad (u,v)\mapsto [u,g(u,v),v],$$

with conventional CAPA recovered when $g(u,v)=0$. In that sense, FCAPA treats CAPA as the special rigid case of a broader continuous-surface communication model (Ranasinghe et al., 12 Nov 2025).

2. Electromagnetic and signal models

The core signal model of CAPA is a continuous electromagnetic field model driven by Green’s functions. In the downlink multi-user setting, the transmitted signal density at each point $\mathbf{s}\in\mathcal S$ can be written as

$x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$

where $w_k(\mathbf s)$ is the beamforming function for user $k$. The received signal at user $k$ is then

$$y_k = \int_{\mathcal S} h_k^*(\mathbf s)\, x(\mathbf s)\, d\mathbf s + n_k = \sum_{i=1}^K \int_{\mathcal S} h_k^*(\mathbf s)\, w_i(\mathbf s)\, c_i\, d\mathbf s + n_k,$$

and the corresponding SINR is

$$\gamma_k = \frac{\left|\int_{\mathcal S} h_k^*(\mathbf s) w_k(\mathbf s)\, d\mathbf s\right|^2} {\sum_{i\neq k}\left|\int_{\mathcal S} h_k^*(\mathbf s) w_i(\mathbf s)\, d\mathbf s\right|^2+\sigma^2}.$$

This is the CAPA analogue of finite-dimensional MU-MIMO beamforming, but every inner product is an aperture integral rather than a vector product (Wang et al., 2024).

A closely related formulation appears in CAPA-aided ISAC, where the transmitter is a planar continuous radiation surface $\mathcal{S}_{\mathrm T}$ and the source current density is a function $g(u,v)=0$0. With dyadic Green’s function $g(u,v)=0$1, the electric field at user $g(u,v)=0$2 is

$g(u,v)=0$3

and the equivalent scalar channel becomes

$g(u,v)=0$4

In the far field, the resulting directional beam gain is

$g(u,v)=0$5

which is the continuous-aperture counterpart of a steering-vector inner product (Ye et al., 25 Nov 2025).

At the receive side, CAPA can also be modeled as a continuous matched-filtering device. For a one-dimensional aperture of length $g(u,v)=0$6, the received field is

$g(u,v)=0$7

and continuous matched filtering yields

$g(u,v)=0$8

The random variable

$g(u,v)=0$9

therefore governs outage probability and other reliability metrics. In this formulation, the difference from a discrete array is exact: matched filtering is an integral over a random process $\mathbf{s}\in\mathcal S$0, not a finite sum over a channel vector (Inwood et al., 12 Mar 2026).

3. Beamforming as functional optimization

Because CAPA beamformers are functions, the canonical beamforming problem is a non-convex integral-based functional programming problem. A general downlink utility-maximization model is

$\mathbf{s}\in\mathcal S$1

with $\mathbf{s}\in\mathcal S$2 strictly increasing in each SINR coordinate. The corresponding SINR-constrained power-minimization problem has the same optimal beamforming structure and is the main route used to derive it (Wang et al., 2024).

A central structural result is that the optimal CAPA beamformer lies in the span of the continuous channel responses. With

$\mathbf{s}\in\mathcal S$3

the channel correlation matrix $\mathbf{s}\in\mathcal S$4 plays the role of $\mathbf{s}\in\mathcal S$5 in SPDA. The optimal beamforming structure is

$\mathbf{s}\in\mathcal S$6

so each optimal beamforming function has the form

$\mathbf{s}\in\mathcal S$7

This is the continuous-aperture generalization of the classical channel-span result in finite-dimensional MU-MIMO (Wang et al., 2024).

The same literature derives continuous-domain analogues of familiar linear transceivers. In the low-SNR regime, MRT is asymptotically optimal: $\mathbf{s}\in\mathcal S$8 In the high-SNR regime, ZF is asymptotically optimal: $\mathbf{s}\in\mathcal S$9 Under equal power allocation, MMSE takes the regularized form

$x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$0

and is proved optimal for SLNR maximization (Wang et al., 2024).

Complementary low-complexity designs are obtained directly from calculus of variations and channel correlations. One line of work derives the closed-form structure of the optimal continuous source patterns through CoV, which in turn yields an integral-free iterative algorithm. Another derives correlation-based zero-forcing source current patterns that completely eliminate inter-user interference; after that reduction, the original functional programming problem becomes a power-allocation problem solvable by classical water-filling. Numerical results reported for these designs show that, compared to Fourier-based discretization, the CoV-based method both improves communication performance and reduces computational complexity by up to hundreds of times for large CAPA apertures and high frequencies, while Corr-ZF is asymptotically optimal relative to the CoV-based design (Wang et al., 2024).

4. Distributional analysis, reliability, and quantized reception

One strand of CAPA theory studies not only mean performance but the full distribution of the relevant random variables. For one-dimensional CAPAs in correlated Rayleigh fading, the matched-filter output admits the Karhunen–Loève expansion

$x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$1

which yields

$x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$2

The exact law is therefore an infinite weighted sum of independent exponentials. To make this tractable, the dominant $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$3 modes are retained as a hypoexponential distribution and the residual tail is approximated by a moment-matched Gamma variable. The resulting truncated hypoexponential model with a Gamma correction gives accurate PDF and CDF approximations under both sinc and Jakes correlation models, including the low-outage tail. The reported numerical study uses $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$4, $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$5 rays for Jakes, and $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$6 Monte Carlo replicates; the approximation remains highly accurate even at low outage probabilities of order $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$7, while a standard Gamma approximation substantially overestimates outage probability, especially for smaller aperture sizes (Inwood et al., 12 Mar 2026).

The same analysis quantifies the CAPA advantage over discrete arrays of the same physical length. In the reported comparison, two discrete-array baselines with 8 equally spaced elements are configured to capture either 50% or 80% of the energy of the continuous aperture, and the CAPA consistently shows better CDF performance. The paper attributes this to complete aperture utilization and the resulting energy collection, rather than to decorrelation alone. It also reports that CAPA median SNR grows nearly linearly with aperture length $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$8, with mean/median pairs $x(\mathbf{s}) = \sum_{k=1}^K w_k(\mathbf{s}) c_k,$9 for $w_k(\mathbf s)$0 and $w_k(\mathbf s)$1 for $w_k(\mathbf s)$2 (Inwood et al., 12 Mar 2026).

A separate line of work studies CAPA under low-resolution front ends. In a 1-bit receiver, the continuous field is first projected onto orthonormal spatial modes and only then quantized. Under Rayleigh fading, this leads to a moment-matching approximation in which the effective performance depends on the modal eigenvalue distribution $w_k(\mathbf s)$3. The key conclusion is that CAPA incurs a diversity-order penalty governed by Jensen’s inequality on the mode eigenvalues: nonuniform modal strengths reduce the effective diversity relative to an ideal i.i.d. discrete baseline. In perfectly aligned LoS, however, the result is qualitatively different. Under perfect spatial and phase alignment, all energy collapses into one dominant mode with $w_k(\mathbf s)$4, and the 1-bit CAPA achieves exactly the unquantized AWGN SEP: $w_k(\mathbf s)$5 The paper states this as complete elimination of the 1-bit penalty that forces conventional discrete systems to double their antenna count (Ranasinghe et al., 2 Apr 2026).

A common misconception is therefore that CAPA always improves all receiver architectures uniformly. The quantized-reception literature shows a more specific picture: in rich scattering, the quantization penalty is still present and is controlled by the spatial eigenvalue profile; in perfectly aligned LoS, the continuous-domain analog combining stage can make 1-bit quantization effectively lossless for QPSK hard detection (Ranasinghe et al., 2 Apr 2026).

5. Secure communications, ISAC, and modal transmission

Physical-layer security has been one of the most heavily developed CAPA application domains. For a single legitimate receiver and a single eavesdropper, continuous current distributions $w_k(\mathbf s)$6 are designed to maximize secrecy rate under a power constraint and to minimize the required power for a target secrecy rate. The resulting maximum secrecy rate (MSR) and minimum required power (MRP) admit closed-form expressions in terms of three channel descriptors $w_k(\mathbf s)$7, $w_k(\mathbf s)$8, and $w_k(\mathbf s)$9, and the corresponding optimal current distributions are continuous combinations of Bob’s and Eve’s channel responses. The asymptotics are particularly clear: for the MSR problem, the optimal current simplifies to MRT in the low-SNR regime and to ZF in the high-SNR regime; for the MRP problem, it simplifies to ZF in the high-target-rate regime. The same work specializes these results to planar CAPAs and planar SPDAs and shows superior secrecy performance for CAPAs (Zhao et al., 2024).

Secure CAPA beamforming has also been extended to artificial-noise design and to multiuser wiretap settings. In the single-user confidential-transmission model with artificial noise, the optimal information-bearing and AN current patterns are shown to lie in the span of the legitimate and eavesdropper channel responses,

$k$0

which turns the original infinite-dimensional functional program into a finite-dimensional coefficient optimization. A penalty-based SCA algorithm is then used for the exact design, while a two-stage ZF-MRT construction designs the AN pattern by ZF and the information pattern by MRT, followed by a one-dimensional power search; the paper states that this low-complexity design has negligible performance loss for the large transmit-power regime (Zhao et al., 15 Apr 2025). In the multiuser multi-eavesdropper case, CAPA secure current design is formulated through a weighted secrecy sum-rate, solved either by FP-based BCD with a continuous-function inversion theory or by a ZF-based zero-leakage design plus water-filling. Reported results include a 77% WSSR gain over the optimization-based MIMO method and a 117% gain of CAPA-ZF over MIMO-ZF at $k$1 (Sun et al., 9 Jan 2025).

CAPA has also been developed for integrated sensing and communication. In CAPA-aided ISAC, the aperture current $k$2 is optimized directly over the continuous surface. A reference sensing waveform is first designed in the wavenumber domain to maximize the minimum target beam gain, and the final joint problem minimizes a weighted combination of sensing-waveform mismatch and communication MUI under a total current-energy constraint. By Lagrangian transformation and calculus of variations, the optimal waveform satisfies a Fredholm integral equation of the second kind and can be written as a superposition of an interference-canceling channel-correlation term, a symbol-matched communication term, and a reference sensing term. The reported numerical results show nearly $k$3 dB BER gain for QPSK, about $k$4 dB gain for 16QAM at $k$5 dB, and up to threefold higher beam gains at target directions relative to SPDA baselines (Ye et al., 25 Nov 2025).

For circular apertures, CAPA has also been used to formulate OAM communication directly in electromagnetic-information-theoretic terms. The transmit current is expanded in Fourier/OAM basis functions

$k$6

and the continuous mode-to-mode channel is

$k$7

The paper states that, in the idealized continuous model, CAPA can synthesize an arbitrary number of mutually orthogonal OAM modes, but it also emphasizes that the practically useful number of streams is still bounded by the effective degrees of freedom (EDoF) of the continuous electromagnetic operator. This clarifies a common misunderstanding: mathematical orthogonality of OAM basis functions does not imply unbounded communication rate in a fixed geometry (Jin et al., 12 Feb 2025).

6. Implementations, reduced-complexity operation, and emerging extensions

The implementation literature distinguishes three practical CAPA hardware routes: electronic, optical, and acoustic. The survey identifies the metasurface-based leaky-wave antenna (MLWA), the optically driven tightly coupled array (OTCA), and the interdigital-transducer-based grating antenna (ITGA). MLWA is an RF-domain architecture with low hardware cost and commercialization already reported; OTCA uses optical-domain processing, photodiodes, and tightly coupled dipole arrays, offering weak frequency selectivity at high hardware cost; ITGA uses acoustic-domain processing through IDTs, with compact form factor and strong frequency selectivity. These proposals differ mainly in the intermediate analog wave domain used to generate the near-continuous current distribution (Liu et al., 2024).

A second implementation theme is reduced-complexity operation on a physically large continuous surface. Aperture selection activates only a sub-region $k$8 rather than the entire receive aperture. In LoS, for a rectangular $k$9 selected aperture, the optimal center follows the exact nearest-neighbor criterion,

$k$0

so the active sub-aperture is centered at the feasible point closest to the user projection on the array plane. In NLoS, segmented aperture selection yields diversity order equal to the rank of the segment correlation matrix,

$k$1

The numerical study reports that, in the near field, about 60% of full-aperture SNR is achieved by less than 40% active aperture, which motivates aperture selection as a low-complexity CAPA operating mode (Ouyang et al., 2024).

A third theme is that CAPA optimization can itself be learned as a continuous-function problem. In the multi-user downlink framework L-CAPA, the optimal current distribution is shown to lie in the span of the user channel functions,

$k$2

and the learning problem is decomposed into a GNN-based PolicyNet, ProjNet, and ValueNet. Reported inference time is 0.02 s for $k$3 and $k$4, compared with 9.6 s for the discretized WMMSE baseline (Guo et al., 2024). A different line of work treats CAPA beamforming itself as an implicit neural representation. BeaINR learns the beamforming function $k$5 directly, while CoefINR learns the coefficient function $k$6 in

$k$7

The reported inference times are 0.053 s for BeaINR and 0.039 s for CoefINR, both faster than WMMSE, Fourier, and SPDA baselines, with CoefINR offering lower sample and space complexity (Chen et al., 4 Jul 2025).

Finally, CAPA has already begun to expand beyond rigid planar surfaces. In FCAPA, the continuous current distribution and the surface geometry are optimized jointly. The weighted-sum-rate problem is posed over both $k$8 and the deformation field $k$9, with CAPA recovered when $$y_k = \int_{\mathcal S} h_k^*(\mathbf s)\, x(\mathbf s)\, d\mathbf s + n_k = \sum_{i=1}^K \int_{\mathcal S} h_k^*(\mathbf s)\, w_i(\mathbf s)\, c_i\, d\mathbf s + n_k,$$0. The reported result is that FCAPA outperforms typical CAPA systems by a wide margin, with performance increasing with increasing morphability (Ranasinghe et al., 12 Nov 2025).

Taken together, these directions suggest a consistent trajectory. CAPA began as a continuous-aperture benchmark for ultra-dense arrays, but the literature already treats it as a full electromagnetic communication architecture with its own hardware routes, function-space optimization theory, stochastic performance analysis, secure transmission mechanisms, sensing formulations, learning-based parameterizations, and geometry-aware extensions. A plausible implication is that future CAPA research will continue to move simultaneously in three directions already stated in the literature: more realistic hardware constraints, richer multi-user and multi-function operating modes, and broader continuous-domain mathematical tools for channels, current distributions, and aperture geometry (Liu et al., 2024).