Tri-Polarized CAPAs: Design & Applications

• Tri-polarized CAPAs are continuous radiating surfaces defined by three orthogonal polarization channels, enabling full-vector electromagnetic modeling for ISAC and joint sensing.
• Their design employs finite-dimensional subspace reductions to optimize transmit currents and receive combiners, balancing communication and sensing objectives.
• By fusing self- and cross-covariance data from all polarization channels, these arrays achieve near-optimal DOA resolution and robust attitude estimation.

Tri-polarized continuous aperture arrays (CAPAs) are continuous radiating or receiving surfaces whose electromagnetic behavior is described by spatially continuous, vector-valued currents and fields over three orthogonal polarization components. In the recent CAPA literature, tri-polarization appears in two principal forms: as a full-vector transmit architecture for integrated sensing and communication (ISAC), where the aperture current can realize arbitrary polarization states and each communication user combines the three orthogonal electric-field components; and as a receive-side sensing architecture for joint direction-of-arrival (DOA) and attitude estimation, where self- and cross-covariances across the $x$, $y$, and $z$ polarization channels are fused into a tri-polarized spectrum (Liu et al., 10 Jul 2026, Si et al., 2 Oct 2025).

1. Electromagnetic representation and polarization model

A tri-polarized CAPA is modeled as a continuous planar aperture with side lengths $L_x$ and $L_y$, driven by a spatially continuous, vector-valued current

$$\mathbf{j}_t(s) = [j_{x,t}(s), j_{y,t}(s), j_{z,t}(s)]^\top \in \mathbb{C}^3,$$

where $s$ denotes position on the surface and $t$ is the symbol index. This current is fully vectorial, allowing for arbitrary polarization states, including tri-polarized excitation. On the receive side of the communication link, each user employs a tri-polarized receiver that combines the three orthogonal electric-field components using a complex unit-norm combiner $\mathbf{v}_k$, so that

$y_k[t] = \mathbf{v}_k^H \mathbf{e}_k[t] + n_k[t].$

The joint optimization of the continuous transmit current and the receive polarization combiners is central to the ISAC formulation (Liu et al., 10 Jul 2026).

In tri-polarized sensing, the received field over the aperture is modeled as a spatially continuous superposition of source responses,

$y$0

where $y$1 is the steering function, $y$2 is the normalized spatial frequency, and $y$3 is the polarization vector associated with the $y$4th target. In this formulation, the aperture is continuous in space rather than composed of a finite set of sensors, and the polarization vector carries attitude-related information in addition to DOA information (Si et al., 2 Oct 2025).

2. Infinite-dimensional structure and finite-dimensional reductions

The defining analytical difficulty of CAPAs is that both transmit design and receive processing are posed over continuous spatial domains. In symbol-level ISAC precoding, the joint optimization over $y$5 and $y$6 is an infinite-dimensional, non-convex, functional optimization problem. A key structural result is that any optimal current distribution lies in a finite-dimensional subspace spanned by the electromagnetic responses relevant to communication users and sensing targets. This subspace is constructed as

$y$7

with dimension $y$8, and the current can be written as

$y$9

The original functional problem is thereby reformulated exactly in terms of finite-dimensional coefficients and polarization combiners (Liu et al., 10 Jul 2026).

An analogous dimensionality reduction appears in tri-polarized DOA and attitude sensing. Because the received observation is a function over the continuous aperture, direct subspace decomposition is not available in the standard finite-array form. The proposed remedy is an equivalent continuous-discrete transformation: the aperture is partitioned into infinitesimal units, covariance operators are represented through finite-dimensional data matrices, and eigendecomposition is transferred to matrices such as

$z$0

with numerical implementation based on Gauss-Legendre quadrature (Si et al., 2 Oct 2025).

Related CAPA work outside the explicitly tri-polarized setting uses the same general principle. Downlink and uplink ISAC derive optimal continuous beamformers through a signal subspace spanned by channel responses, while CAPA-assisted integrated communication and navigation (ICAN) in LEO satellite constellations introduces an ICAN channel subspace in which optimal beamforming functions are finite linear combinations of conjugate system channel responses (Zhao et al., 10 Feb 2025, Wang et al., 10 Jul 2026). This suggests that response-subspace reduction is a general CAPA design principle rather than a polarization-specific device.

3. Tri-polarized symbol-level precoding for ISAC

In the downlink ISAC formulation, the sensing objective is the weighted target-illumination power

$z$1

subject to constructive-interference constraints for communication users, a total transmit power constraint, and unit-norm constraints on the receive polarization combiners. After symbol rotation for PSK detection, the noiseless received signal must remain inside the appropriate decision region, expressed through real-part linear inequalities with safety margin $z$2. The reduced finite-dimensional problem is written over the coefficient matrix $z$3 and the combiners $z$4, with objective

$z$5

power constraint $z$6, and $z$7 (Liu et al., 10 Jul 2026).

Within this framework, polarization is not an auxiliary detail but part of the design space itself. Tri-polarized transmission together with receiver-side polarization combining allows the system to flexibly match the field polarization at each receiver or sensing target and to exploit polarization diversity for higher performance or robustness. The numerical results reported for this formulation show that the proposed method significantly outperforms both Fourier-CAPA and Digital-SPDA baselines in sensing utility, achieves the lowest BER across SNRs, and produces more focused and stronger target beams. The gains are larger when polarization is optimized rather than fixed, and fixed-polarization variants exhibit degraded target illumination (Liu et al., 10 Jul 2026).

4. Joint DOA and attitude sensing

Tri-polarized CAPAs support a unified framework for joint DOA and attitude estimation in which both self- and cross-covariances of the tri-polarized received signals are exploited. The tri-polarized spectrum aggregates the null spaces derived from all self- and cross-covariances over $z$8, thereby fusing information from all polarization directions. According to the reported analysis, this fusion greatly enhances resolution and robustness in DOA estimation compared to single-polarized or discrete-aperture approaches (Si et al., 2 Oct 2025).

The attitude model distinguishes between the component of the attitude vector orthogonal to the DOA vector and the component parallel to it. Theoretical identifiability results are explicit. Partial attitude is always identifiable using only the received data, specifically the component orthogonal to the direction of arrival. Full attitude recovery, including the parallel component, is possible only if prior information about target snapshot signals is available, and the final estimate is subject to a $z$9 sign ambiguity caused by electromagnetic field symmetry. A common simplification is to equate tri-polarization with unrestricted attitude recovery; the present theory does not support that simplification (Si et al., 2 Oct 2025).

The estimation procedures reflect this distinction. For unknown snapshots, the direction of the orthogonal attitude component is estimated as the leading left singular vector of a projected matrix built from demixed received signals. For known snapshots, the orthogonal component is recovered directly and the parallel component is then inferred from the unit-norm constraint. The numerical results indicate near-CRLB performance, robustness to noise, snapshot scarcity, and multi-target scenarios, and superiority over conventional discrete arrays, single-polarized CAPA, and other benchmarks (Si et al., 2 Oct 2025).

5. Algorithmic realizations and links to CAPA-MUSIC

The tri-polarized ISAC precoding problem is solved by a penalty projected-gradient algorithm. The algorithm initializes the subspace basis, coefficients, and combiners; iteratively computes penalties for constructive-interference constraint violation; updates the coefficient matrix and the polarization combiners through projected-gradient steps; and increases the penalty parameter until the constructive-interference constraints are approximately satisfied. The coefficient update is projected onto the transmit power constraint, and each receive combiner is projected onto the unit sphere $L_x$0 (Liu et al., 10 Jul 2026).

For continuous-aperture sensing, the broader CAPA literature provides an algorithmic analogue through MUSIC. A related DOA-estimation study proposes an equivalent continuous-discrete transformation for CAPAs, performs eigendecomposition on a finite matrix built from snapshots, and approximates the continuous MUSIC spectrum by Gauss-Legendre quadrature. It also derives Cramér-Rao lower bounds for cases with and without priori knowledge of snapshot signals, proves that CAPAs significantly improve DOA estimation accuracy compared to traditional SPDAs, and reports near-optimal estimation performance with low computational complexity (Si et al., 28 Jul 2025). In the tri-polarized setting, the use of self- and cross-covariances can be viewed as a polarization-augmented continuation of this continuous-operator subspace methodology.

6. Relation to the broader CAPA literature and extension paths

Not all CAPA studies are tri-polarized. In CAPA-assisted ICAN for LEO satellite constellations, the main body of the work focuses on uni-polarized CAPA design, with current excitation aligned along a specific axis; the propagation model notes that a user would require an ideal tri-polarized receiver to sense the 3D field, but practical users are modeled with uni-polarized linearly polarized antennas. The same work states that the formulation directly supports extension to multi-polarized or tri-polarized arrays by allowing the current to be vector-valued rather than scalar-valued (Wang et al., 10 Jul 2026).

A similar limitation and extension path appears in continuous-aperture ISAC theory more generally. Downlink and uplink ISAC are analyzed for uni-polarized arrays generating 1D currents and receiving 1D electric fields, while the discussion explicitly notes that dual- or tri-polarized extensions are possible by generalizing the integral kernel and beamforming function to vector- or matrix-valued functions (Zhao et al., 10 Feb 2025). In the explicit tri-polarized ISAC formulation, adaptation to dual-polarized, circular, elliptical, or further constrained polarization schemes is described as a matter of adjusting the receive combiners to appropriate submanifolds and possibly adding further excitation constraints (Liu et al., 10 Jul 2026).

The emerging picture is therefore cumulative rather than discontinuous. Uni-polarized CAPA studies establish the continuous-operator formalism, the subspace reductions, and the communication-sensing trade-off analysis; tri-polarized CAPA studies extend that formalism to full-vector electromagnetic processing, polarization diversity, and attitude-sensitive sensing (Wang et al., 10 Jul 2026, Zhao et al., 10 Feb 2025, Liu et al., 10 Jul 2026, Si et al., 2 Oct 2025). A plausible implication is that future CAPA research will continue to treat polarization not as a separate layer added after spatial design, but as a native degree of freedom in the continuous-aperture electromagnetic model.