Continuous Aperture Array-Assisted Integrated Communication and Navigation in LEO Satellite Constellations

Abstract: This paper proposes a novel continuous aperture array (CAPA)-assisted integrated communication and navigation (ICAN) framework for low Earth orbit (LEO) satellite constellations. Within this framework, an electromagnetic-based collaborative transmission model is developed, in which multiple satellites equipped with CAPAs simultaneously radiate downlink data streams and navigation reference signals over shared spectrum. Building upon this, the achievable communication rate and the navigation Cramer-Rao bound (CRB) are derived, which explicitly characterize the intrinsic coupling between the dual-function beamformers and system performance. To improve the positioning accuracy with communication quality of service guarantee, a joint beamforming optimization problem is formulated to minimize the average CRB subject to transmit power budgets and minimum rate constraints. To tackle the inherent infinite-dimensionality of the CAPA beamformer design, an ICAN channel subspace is introduced to equivalently transform the formulation into a tractable finite-dimensional problem, which is then efficiently solved via an iterative convex optimization algorithm. Finally, numerical results demonstrate that the proposed CAPA-assisted beamforming design algorithm significantly outperforms conventional discrete phased array architectures and other benchmark schemes, yielding notable improvements in ICAN performance.

• The paper develops a lossless channel-subspace reduction that converts infinite-dimensional CAPA beamforming into a finite-dimensional design with M+L coefficients per satellite, then solves it using SDR, BCD, and SCA.
• The proposed method lowers navigation CRB versus discrete phased arrays and zero-forcing designs, closely approaches the navigation-only bound at medium-to-high power, and converges in roughly 8–10 iterations.
• The results show that larger satellite groups and lower altitudes improve positioning, while communication-user load and oversized apertures impose diminishing returns and highlight practical payload and interference trade-offs.

System architecture and electromagnetic model

This paper develops a continuous aperture array (CAPA)-assisted integrated communication and navigation (ICAN) framework for LEO satellite constellations, in which a cooperative service group of $K$ satellites jointly serves $M$ communication user equipments (CUEs) and $L$ navigation user equipments (NUEs) over shared spectrum. The constellation follows a Walker Delta configuration analogous to Starlink, and each satellite carries a nadir-pointing CAPA modeled as a continuous radiating surface supporting a spatially continuous sinusoidal current distribution. The transmit signal at each satellite is a linear superposition of $M$ downlink data streams and one deterministic pseudo-random navigation reference, mapped to the aperture through continuous beamformers $w_{k,m}(\mathbf{s}'_k)$ and $v_k(\mathbf{s}'_k)$.

Propagation is characterized via far-field dyadic Green's functions with log-normal rain attenuation, Doppler assumed compensated by receiver synchronization. Effective scalar channel responses are obtained by projecting the Green's function onto uni-polarized transmit and receive polarization vectors. Two performance metrics follow directly: the CUE SINR and achievable rate, which include both inter-CUE interference and an intrinsic navigation-to-communication interference term; and the navigation CRB under a direct position estimation (DPE) framework, where NUEs apply satellite-specific matched filtering and stack the resulting observations into a $K$-dimensional vector. A notable modeling choice is that the effective interference-plus-noise variance $\sigma_{\mathrm{eff},l}^2$ depends on both the communication beamformers and the position itself; the authors treat it as constant within each estimation iteration, an approximation they acknowledge as standard but which couples the CRB to the beamforming design through both the Jacobian of the mean vector and the noise floor.

Problem formulation and subspace-based solution

The joint design problem minimizes the average CRB trace across NUEs subject to per-satellite power budgets on each CAPA surface and minimum achievable rate constraints for all CUEs. Because the optimization variables are functions defined over continuous apertures, the problem is infinite-dimensional. The central methodological contribution is Theorem 1, which establishes that an optimal solution exists entirely within the ICAN channel subspace $\mathcal{C}_k$, spanned by the conjugates of the communication and navigation channel responses associated with each satellite's aperture. The proof proceeds by orthogonal decomposition: components outside $\mathcal{C}_k$ contribute nothing to any rate, mean-vector, or CRB term while consuming transmit power, so discarding them preserves feasibility and optimality. This yields a finite-dimensional parameterization by coefficient vectors $M$0 and $M$1 of dimension $M$2 per satellite — a strictly lossless alternative to Fourier-series discretization, whose required basis size grows rapidly with aperture size and carrier frequency.

The resulting problem remains non-convex due to quadratic coupling between variables. The authors apply semidefinite relaxation (SDR), Schur-complement reformulation of the matrix-inversion objective into linear matrix inequalities (LMIs), block coordinate descent (BCD) with a damped update of $M$3, and a successive convex approximation (SCA) penalty method based on the trace-minus-maximum-eigenvalue characterization of rank-one constraints. Convergence is guaranteed by monotone boundedness of the objective sequence, with limit points satisfying KKT conditions of the penalized SDR problem. The worst-case per-iteration complexity is polynomial in $M$4, $M$5, and the subspace dimension, typical of SDP-based designs.

Numerical results

Simulations use a 72-plane, 18-satellites-per-plane Walker Delta constellation at 550 km altitude with 35 GHz carrier, $M$6 serving satellites, $M$7 CUEs, $M$8 NUEs, and $M$9 apertures. Key findings:

Comparison	Behavior
Proposed vs. discrete phased array (DPA)	Consistently lower average CRB; DPA limited by reduced spatial degrees of freedom
Proposed vs. Fourier-based scheme	Near-equal accuracy but far lower computational complexity
Proposed vs. ZF-oriented scheme	ZF worst in low-power regime due to stringent nulling constraints
Navigation-centric bound	Proposed algorithm closely tracks this lower bound at medium-to-high power

The algorithm converges within roughly 8–10 iterations, and increasing the service group size from $L$0 to $L$1 monotonically reduces the converged average CRB despite a less favorable starting point caused by stronger mutual interference. Additional sensitivity analyses show that the CRB grows steeply with the number of CUEs (rate constraints divert power and degrees of freedom away from navigation), that larger apertures yield diminishing returns beyond roughly $L$2 with the paper explicitly cautioning against maximizing aperture size given payload cost, and that lower orbital altitudes consistently improve positioning accuracy with constellation density exhibiting diminishing marginal returns.

Limitations and open questions

Several assumptions constrain the generality of these results. The framework presumes perfect CSI and real-time ephemeris exchange over inter-satellite links, ideal LoS propagation with only log-normal rain attenuation, and Doppler fully compensated — none of which hold exactly under orbital dynamics and atmospheric variability. The effective-noise-variance freezing approximation, while standard, introduces a gap between the analyzed CRB and the exact likelihood. SDR with Gaussian randomization does not guarantee rank-one solutions, so recovered beamformers are suboptimal in general. Hardware feasibility of continuously controllable current distributions at 35 GHz, calibration requirements, and robustness to position errors in the DPE initialization remain unexamined. Open questions include extending the design to imperfect CSI, multi-polarized receivers, and time-varying service-group selection as satellites move through the coverage area.

Conclusion

The paper establishes a multi-satellite CAPA-assisted ICAN framework grounded in electromagnetic information theory, proves a lossless subspace reduction that renders the infinite-dimensional dual-function beamforming problem tractable, and demonstrates through simulation that the resulting design outperforms discrete phased arrays, Fourier discretization, and zero-forcing benchmarks while closely approaching the navigation-only performance bound under communication QoS constraints.