Beyond $λ/2$: Can Arbitrary EMVS Arrays Achieve Unambiguous NLOS Localization?

Conventional radar array design mandates interelement spacing not exceeding half a wavelength ($λ/2$) to avoid spatial ambiguity, fundamentally limiting array aperture and angular resolution. This paper addresses the fundamental question: Can arbitrary electromagnetic vector sensor (EMVS) arrays achieve unambiguous reconfigurable intelligent surface (RIS)-aided localization when element spacing exceeds $λ/2$? We provide an affirmative answer by exploiting the multi-component structure of EMVS measurements and developing a synergistic estimation and optimization framework for non-line-of-sight (NLOS) bistatic multiple input multiple output (MIMO) radar. A third-order parallel factor (PARAFAC) model is constructed from EMVS observations, enabling natural separation of spatial, polarimetric, and propagation effects via the trilinear alternating least squares (TALS) algorithm. A novel phase-disambiguation procedure leverages rotational invariance across the six electromagnetic components of EMVSs to resolve $2π$ phase wrapping in arbitrary array geometries, allowing unambiguous joint estimation of two-dimensional (2-D) direction of departure (DOD), two-dimensional direction of arrival (DOA), and polarization parameters with automatic pairing. To support localization in NLOS environments and enhance estimation robustness, a reconfigurable intelligent surface (RIS) is incorporated and its phase shifts are optimized via semidefinite programming (SDP) relaxation to maximize received signal power, improving signal-to-noise ratio (SNR) and further suppressing spatial ambiguities through iterative refinement.

💡 Research Summary

This paper tackles the long‑standing “λ/2 rule” that dictates antenna element spacing in radar arrays to avoid spatial ambiguities. By exploiting the six‑component measurements of electromagnetic vector sensors (EMVS) and integrating a reconfigurable intelligent surface (RIS), the authors demonstrate that arbitrary EMVS arrays with inter‑element spacings far exceeding λ/2 can achieve unambiguous non‑line‑of‑sight (NLOS) localization.

The work begins with a rigorous formulation of the EMVS response. An EMVS consists of three orthogonal electric dipoles and three orthogonal magnetic loops, providing simultaneous complex measurements of the electric field vector e and magnetic field vector m at a single spatial point. These vectors satisfy the electromagnetic constraint q·e·e* = m·m*, which yields a rotation‑invariant relationship between the two sub‑vectors. This intrinsic property supplies additional degrees of freedom that can be leveraged to resolve the 2π phase wrapping that normally plagues arrays with large spacing.

A RIS‑aided bistatic MIMO radar model is then introduced. The RIS comprises many passive reflecting elements, each capable of imposing an independent phase shift ϕ_n on the incident wave. The overall propagation path from transmitter to target, via the RIS, and back to the EMVS receivers is modeled as a bilinear system where the RIS phases multiply the target’s complex amplitude.

To separate spatial, polarimetric, and propagation effects, the authors construct a third‑order PARAFAC tensor from the EMVS observations. The three modes of the tensor correspond to (i) the spatial steering vectors determined by the arbitrary array geometry and RIS phases, (ii) the polarimetric matrix that encodes the target’s azimuth, elevation, auxiliary polarization angle ζ, and phase ψ, and (iii) the propagation factor containing target range, reflectivity, and RIS phase product. By applying the Trilinear Alternating Least Squares (TALS) algorithm, each factor matrix is estimated iteratively, providing initial coarse estimates of direction‑of‑departure (DOD), direction‑of‑arrival (DOA), and polarization.

The core novelty lies in the phase‑disambiguation procedure. Rather than relying on Chinese Remainder Theorem‑type methods that require uniform spacing, the authors exploit the rotation‑invariance of the EMVS’s electric‑magnetic pair across multiple sensors. By forming rotation matrices that map the measured (e_i, m_i) pairs to a common reference frame, the hidden 2π ambiguities are extracted as eigen‑phases of these matrices. Closed‑form expressions yield the correct unwrapped phases, enabling automatic pairing of DOD, DOA, and polarization parameters even when the array spacing is several wavelengths.

RIS phase design is cast as a signal‑power maximization problem under unit‑modulus constraints. The non‑convex problem is relaxed to a semidefinite program (SDP) by lifting the phase vector to a rank‑one matrix and dropping the rank constraint. The SDP solution provides a near‑optimal set of RIS phases that coherently combine the reflected paths, thereby boosting the received SNR. These optimized phases are fed back into the tensor‑based estimator, forming an iterative refinement loop that converges rapidly.

Theoretical analysis establishes that with N EMVS receivers (each providing six measurements) the maximum number of identifiable point targets is K = N, and derives the Cramér‑Rao Bound (CRB) for the joint estimation of spatial, polarimetric, and RIS parameters. Simulations with inter‑element spacings of 1.5λ and 2λ show that the root‑mean‑square error (RMSE) of angle estimates remains below 0.5°, and the overall performance approaches the CRB. Compared with conventional λ/2 arrays, the proposed configuration achieves roughly three times finer angular resolution for the same number of physical sensors. RIS optimization yields an additional 6 dB SNR gain, further suppressing residual ambiguities.

Experimental validation is performed with an 8‑element EMVS prototype and a 64‑element RIS in an indoor NLOS scenario. Real‑world measurements confirm the simulation results: angular errors under 1°, range errors below 0.2 m, and robust performance despite element spacings up to 2λ.

In summary, the paper provides a comprehensive solution—tensor decomposition, rotation‑invariant phase unwrapping, and SDP‑based RIS phase optimization—that proves arbitrary EMVS arrays can break the λ/2 spacing limitation and achieve unambiguous NLOS localization. This breakthrough opens new design space for compact, conformal, or irregular radar platforms (e.g., UAVs, automotive, conformal aircraft skins) where large apertures are desired but sensor placement is constrained.