CAPA Optimization in Near-Field Sensing

• Continuous-Aperture Arrays (CAPAs) are electromagnetic sensing systems that use continuous source currents to exploit full spatial degrees of freedom for accurate target localization.
• The CRB optimization framework minimizes position estimation errors by formulating a power-constrained, integral-based design problem and applying subspace manifold gradient descent.
• Simulation results demonstrate that CAPAs achieve up to a tenfold reduction in localization error over discrete arrays by optimally shaping the radiated field.

A Cramér-Rao bound (CRB) optimization framework for near-field sensing using continuous-aperture arrays (CAPAs) provides a systematic methodology for optimizing target localization accuracy in electromagnetic (EM) sensing scenarios where both the transmit and receive apertures are modeled as continuous surfaces. CAPAs emit EM probing signals through continuous source currents, thereby fully leveraging the spatial degrees of freedom (DoFs) inherent in continuous apertures. This framework addresses the non-convex, integral-based functional optimization problem of source current design, and demonstrates substantial improvements in location estimation performance over conventional spatially discrete arrays (SPDAs) (Jiang et al., 2024).

1. Cramér–Rao Bound Optimization Framework

The received electric field at a receiver point $\mathbf{q}$ can be expressed as

$$E(\mathbf{q}) = \int_{\mathcal{S}^{(\mathrm{tx})}} h(\mathbf{q}, \mathbf{p}) J(\mathbf{p})\;\mathrm{d}\mathbf{p}$$

where $h(\mathbf{q},\mathbf{p})$ encodes the continuous round-trip EM channel response from transmit point $\mathbf{p}$ to receive point $\mathbf{q}$ (including spherical wavefronts and reflection coefficients), and $J(\mathbf{p})$ is the source current distribution over the transmit aperture.

Given $N$ target positions $\mathbf{r} = [\mathbf{r}_1, \ldots, \mathbf{r}_N]$ and unknown target reflection parameters $\boldsymbol{\alpha}$, the Fisher information matrix (FIM) with respect to the parameter vector $$\boldsymbol{\xi} = [\mathbf{r}^\top, \Re\boldsymbol{\alpha}^\top, \Im\boldsymbol{\alpha}^\top]^\top$$ is partitioned as

$$J_\xi = \begin{bmatrix} J_{\mathbf{r}\mathbf{r}} & J_{\mathbf{r}\tilde{\alpha}} \ J_{\mathbf{r}\tilde{\alpha}}^\top & J_{\tilde{\alpha}\tilde{\alpha}} \end{bmatrix}$$

where, for example, $$[J_{\mathbf{r}\mathbf{r}}]_{m,n} = \frac{2}{\sigma^2}\int_{\mathcal{S}^{(\mathrm{rx})}}\Re\left\{\frac{\partial E(\mathbf{q})}{\partial [\mathbf{r}]_m}\frac{\partial E^*(\mathbf{q})}{\partial [\mathbf{r}]_n}\right\} \mathrm{d}\mathbf{q}$$. The CRB for position estimation is then

$$\mathrm{CRB}(J_{\mathbf{r}\mathbf{r}}) = [J_{\mathbf{r}\mathbf{r}} - J_{\mathbf{r}\tilde{\alpha}} J_{\tilde{\alpha}\tilde{\alpha}}^{-1} J_{\mathbf{r}\tilde{\alpha}}^\top]^{-1}$$

The CRB minimization problem for the transmit source current distribution is formulated as

$$\begin{aligned} &\min_{J(\mathbf{p})} \operatorname{Tr}\{\mathrm{CRB}(J_{\mathbf{r}\mathbf{r}})\} \ &\;\;\;\; \text{subject to:}\quad \int_{\mathcal{S}^{(\mathrm{tx})}} |J(\mathbf{p})|^2 \mathrm{d}\mathbf{p} \leq P \end{aligned}$$

This optimization targets minimizing the aggregate uncertainty in position estimates under a transmit power constraint.

2. Continuous-Function Source Currents and Spatial DoFs

CAPAs realize the transmit probing field via a continuous source current $J(\mathbf{p})$, as opposed to SPDAs which employ finite-dimensional current vectors. The continuous specification

• enables "full aperture utilization", i.e., an infinite number of element-equivalents,
• allows extremely fine shaping of the radiated EM field, and
• better models near-field wavefronts and their spatial diversity, crucial for precise near-field sensing.

This high spatial DoF leads to the observed order-of-magnitude improvement in localization accuracy (tenfold CRB reduction) over SPDAs, as demonstrated by the derived Fisher information and in simulation (Jiang et al., 2024).

3. Maximum Likelihood Estimation for Near-Field Sensing

Given noisy continuous-aperture measurements $y(\mathbf{q}) = E(\mathbf{q}) + n(\mathbf{q})$, the log-likelihood function for the targets’ locations and their scattering parameters is

$$\mathcal{L} = \int_{\mathcal{S}^{(\mathrm{rx})}} \left| y(\mathbf{q}) - \sum_{n=1}^{N} \alpha_n \tilde{E}_n(\mathbf{q}) \right|^2 \mathrm{d}\mathbf{q}$$

where $\tilde{E}_n(\mathbf{q})$ is the modeled echo from the $n$‑th target, incorporating continuous Green's function response.

The optimal $\boldsymbol{\alpha}$ are found by solving the linear equation $P\boldsymbol{\alpha} = m$, where the entries of $P$ and $m$ are surface integrals over $\mathcal{S}^{(\mathrm{rx})}$ involving the model echoes and measurements. In the single-target case, the MLE spectrum simplifies to

$$\mathcal{L}’ = \frac{ \left| \int_{\mathcal{S}^{(\mathrm{rx})}} \tilde{E}^*(\mathbf{q}) y(\mathbf{q}) \mathrm{d}\mathbf{q} \right|^2 }{ \int_{\mathcal{S}^{(\mathrm{rx})}} |\tilde{E}(\mathbf{q})|^2 \mathrm{d}\mathbf{q} }$$

Target positions are estimated by the locations of spikes in the spatial MLE spectrum.

4. Optimal Source Current Subspace Structure

The CRB trace minimization over continuous $J(\mathbf{p})$ is infinite-dimensional and non-convex. The key analytical insight is that the optimal $J(\mathbf{p})$ always lies in the subspace spanned by the transmit responses for the actual target positions: $$J(\mathbf{p}) = \sum_{n=1}^{N} w_n e^{j k_0 \|\mathbf{r}_n - \mathbf{p}\|_2}$$ That is, the optimal current is a linear combination of plane-wave "basis" functions from each target’s geometry, with complex weights $w_n$ to be optimized.

This subspace property (proved by contradiction) means that transmit power is always optimally directed along the functional basis defined by target-centric steering vectors, and all energy orthogonal to this subspace is "wasted" for localization accuracy.

5. Subspace Manifold Gradient Descent (SMGD) Method

Parameterizing $J(\mathbf{p})$ as above, the optimization reduces to finding the vector of weights $w$ on the constraint manifold $w^H B_0 w = P$, where $B_0$ is the Gram matrix of the basis functions: $$[B_0]_{m,n} = \int_{\mathcal{S}^{(\mathrm{tx})}} e^{j k_0 ( \| \mathbf{r}_m - \mathbf{p} \|_2 - \| \mathbf{r}_n - \mathbf{p} \|_2 ) } \mathrm{d}\mathbf{p}$$ The optimization of $w$ forms a complex ellipsoidal manifold.

To solve this efficiently, a subspace manifold gradient descent (SMGD) algorithm is used:

• The Euclidean gradient of the objective (the CRB trace, itself a matrix function of $w$) is projected onto the tangent space of the ellipsoidal manifold using the $B_0$ inner product.
• A conjugate gradient rule (e.g., Fletcher–Reeves) with Armijo backtracking is applied for step size selection.
• Upon each iteration, a retraction operator ensures $w$ stays feasible. Computational complexity is $\mathcal{O}(N^2)$ per iteration, compared to $\mathcal{O}(N^{4.5})$ for semi-definite relaxation.

6. Simulation Results and Comparative Performance

Extensive simulations demonstrate:

• SMGD's convergence to near-global optimality, nearly matching the performance of the computationally expensive SDR method.
• In multi-target positioning, the MLE spectrum shows clear discrimination between locations, and optimized $J(\mathbf{p})$ produces highly localized beam patterns focused at true target sites.
• The CRB achieved by optimized CAPA designs decreases monotonically with increasing transmit power and frequency.
• Critically, CAPA’s localization performance achieves dramatic improvement: a tenfold reduction in CRB (i.e., order-of-magnitude better accuracy) over SPDAs with half-wavelength antenna spacing, due to full exploitation of spatial DoFs.
• Robustness to initialization: the method remains effective even with moderate errors in initial target positions.

Summary

The CRB-optimization framework for near-field sensing with CAPAs utilizes continuous source currents to fully harvest aperture spatial DoFs, drastically improving position estimation accuracy relative to SPDAs. The optimal source pattern is shown to reside in the subspace of transmit responses to the targets, reducing the problem to a finite-dimensional complex quadratic program. The subspace manifold gradient descent method efficiently solves the CRB minimization, as confirmed by rapid convergence and order-of-magnitude performance gains in all simulation tests. These results establish the full-wave, continuous-aperture paradigm as an analytically rigorous and practically superior approach for high-accuracy near-field electromagnetic sensing (Jiang et al., 2024).