Low-Complexity DOA Estimation via Synthetic Coprime Polarization Sensitive Array with Reduced Mutual Coupling in Nonuniform Noise

Abstract

To improve the performance of underdetermined direction of arrival (DOA) estimation in the scenario of strong mutual coupling and unknown nonuniform noise, this paper presents a novel coprime polarization sensitive array (PSA) with limited antenna number and proposes a low-complexity DOA estimation method via iterative covariance matrix reconstruction. The proposed synthetic array is composed of two single-polarized subarrays with dipole or loop antennas. Owing to the polarization domain, it can efficiently reduce the mutual coupling between subarrays, which dominates the effect of mutual coupling in the existing coprime arrays. For effectively eliminating the nonuniform noise, the cross-covariance matrix between subarrays is utilized. Based on it, the full aperture and all degrees of freedoms of the difference coarray with holes are obtained, and then, initial DOAs are estimated by using orthogonal matching pursuit algorithm. The oblique projection operators depending on initial DOAs are constructed to fill holes in difference coarray, in order to generate a larger covariance matrix for refined angle estimation. Then, a fast iterative procedure associated with covariance matrix reconstruction is given to achieve final DOA estimation, for further enhancing estimation performance. Theoretical analysis and simulation results verify the effectiveness of the proposed array and method.

Appendix A Proof of Theorem 1

Proof

For the kth signal source, suppose difference $m^k$ being the position of (cross-) difference coarray ${\varvec{v}}_k$ in (17). Thus, $m^k$ is expressed as

$$\begin{aligned} m^k_{ij} = N[x]_i - M[y]_j, \end{aligned}$$

(22)

where $[x]_i \in [0 \ 1 \ \ldots \ M - 1]$ denotes the location of the ith antenna of D-subarray, and $[y]_j \in [0 \ 1 \ \ldots \ N - 1]$ denotes the location of the jth antenna of L-subarray. Recalling to (19), the difference $m^k_{{\text {hole}}}$, i.e., position of hole ${{u}}_k(m)$ in difference coarray, is available as

$$\begin{aligned} m^k_{{\text {hole}}} = m^k_{ab} - m^k_{cd} = (N[x]_a - M[y]_b) - (N[x]_c - M[y]_d). \end{aligned}$$

(23)

It is worth noting that difference $m^k_{ab} - m^k_{cd}$ corresponds to one pair of combination, i.e., ${\varvec{v}}_k(m_{2h - 1})/{\varvec{v}}_k(m_{2h})$.

For the proposed array, the maximum continuous holes in the difference coarray can be obtained in the case of $N = M + 1$. In this condition, the maximum continuous holes are between items at $[M(M - 1)$ and $N(M - 1)$ (i.e., $[M(M - 1) + 1, \ N(M - 1) - 1]$) in the difference coarray.

The difference corresponding to the item at $N(M - 1)$ can be given as $N[x]_{M} - M[y]_1 = N(M - 1) - 0$. Note that the difference $m^k_o$ between the qth antennas of both D- and L-subarrays is q. Thus, we have $m^k_o \in [0 \ 1 \ 2 \ \ldots \ M - 1]$. Then, the difference $m^k_{{\text {hole}}}$ corresponding to the hole in the range of $[M(M - 1) + 1, \ N(M - 1) - 1]$ can be expressed as

$$\begin{aligned} m^k_{{\text {hole}}} = m^k_{ab} - m^k_{cd} = (N(M - 1) - 0) - [m^k_o]_i. \end{aligned}$$

(24)

It is obvious that all holes in the range of $[M(M - 1) + 1, \ N(M - 1) - 1]$ can be filled by (24). By this means, only one pair of combination, i.e., $H = 1$, is enough to fill all holes in the cross-difference coarray of the synthetic coprime PSA. $\square$