Near-Field DOA-Range and Polarization Estimation Based on Exact Propagation Model with COLD Arrays

Abstract

Many existing near-field source localization algorithms assume simplified models, for example, the Fresnel approximation model. Unlike these works, a new algorithm is herein proposed to localize multiple near-field electromagnetic sources under the exact source-array propagation model. Using the data measured by a linear (not necessarily uniform) cocentered orthogonal loop and dipole (COLD) array, three cumulant matrices are firstly defined to construct two matrix pencils. The magnitudes of the two matrix pencils’ generalized eigenvalues are then combined with their phases to extract the direction-of-arrival (DOA) and range estimates of the sources. The key idea of the new algorithm is to use a set of coarse estimates obtained from the magnitudes to resolve the set of ambiguous estimates obtained from the phases. In addition, the proposed algorithm estimates the polarizations without needing the prior estimation of the DOA-range parameters. This proposed algorithm is analytic, requires no iterative computations, and does not need to confine the inter-element spacing to be within a quarter wavelength.

Appendix A: Derivations of Eqs. (15)–(17)

Let ${\varvec{C}}_1(j, \ell )$ be the $(j, \ell )$th entry of ${\varvec{C}}_1$. Then, ${\varvec{C}}_1(j, \ell )$ can be expressed as

$$\begin{aligned} {\varvec{C}}_1(j, \ell )= & {} \sum _{i = 1}^2 \left\{ \mathrm{cum} \left[ \sum _{k = 1}^K a_{1,k} c_{i,k} s_k(t), \sum _{k = 1}^K a_{1,k}^*c_{i,k}^*s_k^*(t) \sum _{k = 1}^K b_{j,k} s_k(t) \sum _{k = 1}^K b_{\ell ,k}^*s_k^*(t) \right] \right\} \nonumber \\= & {} \sum _{i = 1}^2 \left\{ \left[ \sum _{k = 1}^K (a_{1,k} c_{i,k} a_{1,k}^*c_{i,k}^*b_{j,k} b_{\ell ,k}^*) \times \mathrm{cum}[s_k(t), s_k^*(t), s_k(t), s_k^*(t)]\right] \right\} \nonumber \\= & {} \sum _{k = 1}^K b_{j,k} b_{\ell ,k}^*\rho _k \end{aligned}$$

(41)

where $a_{m,k} = a_{m}(\theta _k, r_k)$ and $b_{j, k}$ is the (j, k)th entry of ${\varvec{B}}$. In establishing (41), the fact that $c_{1,k} c_{1,k}^*+ c_{2,k} c_{2,k}^*= 1$, $\forall k$, is used. Expressing ${\varvec{C}}_1(j, \ell )$ for all $j, \ell = 1, \ldots , 2M$ in matrix form, Eq. (15) is established.

Similarly, the $(j, \ell )$th entries of ${\varvec{C}}_2(j, \ell )$ and ${\varvec{C}}_3(j, \ell )$ can be expressed, respectively, as

$$\begin{aligned} {\varvec{C}}_2(j, \ell ) = \sum _{k = 1}^K b_{j,k} b_{\ell ,k}^*a_{p,k} \rho _k \end{aligned}$$

(42)

and

$$\begin{aligned} {\varvec{C}}_3(j, \ell ) = \sum _{k = 1}^K b_{j,k} b_{\ell ,k}^*a_{q,k} \rho _k \end{aligned}$$

(43)

For all $j, \ell = 1, \ldots , 2M$, $p \in [1, M]$, and $q \in [1, M]$, Eq. (42) yields (16) and eq. (43) yields (17).