A Fast and High-Resolution Multi-Target Localization Approach in MIMO Radar

Abstract

This paper presents a fast and high-resolution estimation approach using polarization information combined with angle information for multi-target localization in bistatic multiple-input multiple-output (MIMO) radar. The propagator method (PM) is extended to jointly estimate the direction of departure (DOD), the direction of arrival (DOA) and the polarization parameters. The PM avoids the singular value decomposition (SVD) of the covariance matrix of the received signals so that the computational complexity is reduced. In addition, the closely spaced targets can be well distinguished by polarization diversity. The Cramer-Rao bounds (CRBs) of the estimated parameters are derived. The position of a target is calculated based on the estimated angles. The simulation results demonstrate that the proposed approach can achieve better performance compared with conventional methods of target localization.

1. Introduction

Multiple-input multiple-output (MIMO) radar [1] and its applications in localization and direction–finding [2]-[3] have recently been the focus of intensive research. Specifically, many target localization methods for bistatic MIMO radar have been investigated to estimate both direction of departure (DOD) and direction of arrival (DOA) [4]–[13]. However, when multiple targets are closely spaced and cannot be well distinguished from the spatial domain, the resolution of these algorithms is greatly degraded. In recent years, polarimetric radar has reflected the tremendous advantages in target estimation, detection and tracking [14]. Considering that target echoes at different locations have different states of polarization, an ESPRIT-based DOD-DOA-polarization estimation method was proposed by us in [15] for bistatic MIMO radar, in which polarization information can be made full use of to improve the target resolution.

In array signal processing, the propagator method (PM) [16] is a fast DOA estimation approach, which can avoid singular value decomposition (SVD) of the covariance matrix. However, spectral peak searching through all the two-dimensional (2D) space is the main computational burden if the parameters of both DOD and DOA need to be obtained. In [11], a fast multi-target localization algorithm was proposed using PM-based DOD and DOA estimation. In this paper, a novel fast and high-resolution estimation approach is presented using polarization information combined with angle information for multi-target localization. A traditional PM algorithm is extended herein to jointly estimate the DOD, the DOA and the polarization parameters, which can avoid the calculation of the SVD of the covariance matrix, as well as peak searching. Therefore, the computational complexity is further reduced and fast multi-target localization can be implemented. Also, by using polarization diversity, the closely spaced targets can be well distinguished with a high resolution. To evaluate the performance of the proposed approach, the Cramer-Rao bounds (CRBs) of the estimated parameters are derived. The position of a target is calculated based on the estimated angles.

The paper is organized as follows. In the next section, the signal model of bistatic MIMO radar based on polarization sensitive receiver array is described. In Section 3, a fast and high-resolution multi-target localization approach is presented. In Section 4, the CRBs of the estimated parameters are derived. The simulation results are given in Section 5. Finally, a conclusion is drawn in Section 6.

2. Polarization Sensitive Array-based Bistatic MIMO Radar Signal Model

As described in Fig. 1, we consider a bistatic MIMO radar system having a uniform linear transmitter array with M elements and a polarization-sensitive receiver array with N pairs of crossed dipoles to measure the polarization information of targets. d _t and d _r are the inter-element spacing at the transmitter and receiver, respectively, which are no more than half a wavelength. The target's range is assumed to be much larger than the aperture of the transmitter and receiver arrays. M elements of the transmitter simultaneously transmit waveforms, which are orthogonal, to each other. These signals are reflected by P targets with different Doppler frequencies and complex amplitudes. For the pth target, p = 1, 2, ·, P, its angles, say, DOD and DOA, are denoted by θ _p and ϕ _p and its polarization information is parameterized by two polarization phase factors γ _p and η _p , respectively.

Figure 1.

Diagram of a bistatic MIMO radar having a uniform linear transmitter array with M elements and a polarization-sensitive receiver array with N pairs of crossed dipoles.

In the lth snapshot, upon the reflections by P targets, the signal matrix X ^(l) ∊ ℂ^2N×K received by the 2N-element polarization-sensitive array is determined by [15]

$$X^{(l)}={\displaystyle \sum _{p=1}^P{a}_r({\varphi }_p,{\gamma }_p,{\eta }_p){\beta }_p{a}_t^T({\theta }_p)\left[\begin{array}{c}{s}_1\\ ⋮\\ s_M\end{array}\right]}{\mathrm{e}}^{\mathrm{j}2\text{π}{f}_{dp}{t}_l}+W^{(l)}$$

(1)

l = 1,2, ·, L, where s _m ∊ ℂ^1×K for m = 1, 2, · M denotes a normalized transmitted baseband coded signal vector for the mth element, having a length of K and $s_m{s}_m^{\mathrm{H}}=1$ , and t _l is slow time with the index l. Here f _dp and β _p denote the Doppler frequency and amplitude of the pth target, respectively. β _p is proportional to the radar cross sections (RCSs) of the pth target. W _(l) denotes the noise matrix, which is complex, white Gaussian distributed with a variance of σ² _w and a _t (θ _p ) ∊ ℂ ^M×1 is the pth steering vector of the transmitter array, given by

$$a_t\left({\theta }_p\right)={\left[1,e^{-\mathrm{j}2\text{π}\left(d_t/\lambda \right)\sin {\theta }_p},\dots \text{ ,}{e}^{-\mathrm{j}2\text{π}\left(M-1\right)\left(d_t/\lambda \right)\sin {\theta }_p}\right]}^{\mathrm{T}}$$

(2)

a _r (ϕ _p , γ _p , η _p ) ∊ ℂ^2N×1 denotes the pth manifold vectors of the receiver array, which can be written as

$$a_r({\varphi }_p,{\gamma }_p,{\eta }_p)=q_p\otimes v_p$$

(3)

where ⊗ denotes the Kronecker product. q _p ∊ ℂ^N×1 is the pth steering vector of the arrival signal

$$q_p={\left[1,e^{-\mathrm{j}2\text{π}\left(d_y/\lambda \right)\sin {\varphi }_p},\dots ,e^{-\mathrm{j}2\text{π}\left(N-1\right)\left(d_y/\lambda \right)\sin {\varphi }_p}\right]}^{\mathrm{T}}$$

(4)

and ν _p ∊ ℂ^2×1 is the pth polarization vector of the arrival signal

$$v_p=\left[\begin{array}{l}-\cos {\gamma }_p\\ \sin {\gamma }_p\cos {\varphi }_p{\mathrm{e}}^{\mathrm{j}{\eta }_p}\end{array}\right]$$

(5)

for 0 ≤ γ _p ≤ π/2, −π ≤ η _p < π. We assume that there are M matched filters at the receiver. After being matched with transmitted waveforms, the output of the mth matched filter is $y_m^{(l)}=X^{(l)}{s}_m^{\mathrm{H}}$ . Thus, the output of M matched filters can be composed as a vector, i.e.,

$$y^{\left(l\right)}=\left[\begin{array}{c}{y}_1^{(l)}\\ ⋮\\ y_M^{(l)}\end{array}\right]=\left[\begin{array}{c}{A}_r(\varphi ,\gamma ,\eta ){\mathrm{\Phi }}_{t1}(\theta )b^{(l)}\\ ⋮\\ A_r(\varphi ,\gamma ,\eta ){\mathrm{\Phi }}_{tM}(\theta )b^{(l)}\end{array}\right]+\left[\begin{array}{c}{v}_1^{(l)}\\ ⋮\\ v_M^{(l)}\end{array}\right]$$

(6)

where $A_r(\varphi ,\gamma ,\eta )=[a_r({\varphi }_1,{\gamma }_1,{\eta }_1),\dots ,a_r({\varphi }_P,{\gamma }_P,{\eta }_P)]$ is the manifold matrix of the receiver array. ${\mathrm{\Phi }}_{tm}(\theta )=\text{diag}\left[e^{-\mathrm{j}2\text{π}\left(m-1\right)\left(d_t/\lambda \right)\sin {\theta }_1},\dots ,e^{-\mathrm{j}2\text{π}\left(m-1\right)\left(d_t/\lambda \right)\sin {\theta }_p}\right]$ is a diagonal matrix composed of the phase shifts of P targets at the mth element of transmitter array. $b^{(l)}={\left[{\beta }_1{\mathrm{e}}^{\mathrm{j}2\text{π}{f}_{d1}{t}_l},\dots ,{\beta }_P^{}{\mathrm{e}}^{\mathrm{j}2\text{π}{f}_{dP}{t}_l}\right]}^{\mathrm{T}}$ denotes the target signal vector. $v_m^{(l)}=(1/\sqrt{K})W^{(l)}{s}_m^{\mathrm{H}}$ denotes the noise matrix after matched filters with the m-th transmitted waveforms. (6) can be further denoted as

$$y^{\left(l\right)}=A(\theta ,\varphi ,\gamma ,\eta )b^{(l)}+v^{\left(l\right)}$$

(7)

where $A(\theta ,\varphi ,\gamma ,\eta )∊{ℂ}^{2MN\times P}$ denotes the total manifold matrix with respect to the transmitter and receiver arrays, $A(\theta ,\varphi ,\gamma ,\eta )=\left[a_1({\theta }_1,{\varphi }_1,{\gamma }_1,{\eta }_1),\cdot \cdot \cdot ,a_P({\theta }_P,{\varphi }_P,{\gamma }_P,{\eta }_P)\right]$ where $a_p({\theta }_p,{\varphi }_p,{\gamma }_p,{\eta }_p)=a_t({\theta }_p)\otimes a_r({\varphi }_p,{\gamma }_p,{\eta }_p)$ . We form a matrix $Y∊{ℂ}^{2MN\times L}$ by composing (7) in L snapshots

$$Y=A(\theta ,\varphi ,\gamma ,\eta )B+V$$

(8)

where $Y={\left[y^{(1)},\dots ,y^{(L)}\right]}^{\mathrm{T}}$ , and $B={\left[b^{(1)},\dots ,b^{(L)}\right]}^{\mathrm{T}}$ and $V={\left[v^{(1)},\dots ,v^{(L)}\right]}^{\mathrm{T}}$ . Thus, based on the signal model in (8), our goal is to estimate multiple parameters of targets, including DOD parameter θ _p , DOA parameter ϕ _p and polarization parameters γ _p , σ _p .

3. Fast and High-Resolution Multi-Target Localization Approach

In this section, a fast and high-resolution multiple target localization approach is described in detail. The total structure of the proposed scheme is shown in Fig. 2.

Figure 2.

The total structure of the proposed scheme.

3.1 DOA/DOA/Polarization Estimation

The approach employs the propagator method. However, we extend it by exploiting the characteristics of signal subspace of different matrix blocks and the polarization array processing technology. The DOD, DOA and two polarization parameters are estimated step-by-step based on matrix Y .

First, in order to estimate the DOD parameters, let Y _t1 and Y _t2 be the 2N(M − 1)xL submatrices of Y , consisting of the first and last 2N(M − 1) rows of Y , i.e., $Y_{t1}=A_{t1}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V_{t1}$ and $Y_{t2}=A_{t2}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V_{t2}$ , where $A_{t1}\left(\theta ,\varphi ,\gamma ,\eta \right)$ and $A_{t2}\left(\theta ,\varphi ,\gamma ,\eta \right)$ are the first and last 2N(M − 1) rows of $A(\theta ,\varphi ,\gamma ,\eta )$ , respectively. Then we have $A_{t2}\left(\theta ,\varphi ,\gamma ,\eta \right)=A_{t1}\left(\theta ,\varphi ,\gamma ,\eta \right){\mathrm{\Phi }}_t$ , where ${\mathrm{\Phi }}_t=\text{diag}\left\{e^{-\mathrm{j}2\text{π}\left(m-1\right)\left(d_t/\lambda \right)\sin {\varphi }_1},\dots ,e^{-\mathrm{j}2\text{π}\left(m-1\right)\left(d_t/\lambda \right)\sin {\varphi }_P}\right\}$ . Therefore, the estimation of DOD can be derived from the diagonal matrix Φ _t . We compose the data for Y_t1 and Y_t2 as follows.

$$Z_t=\left[\begin{array}{l}{Y}_{t1}\\ Y_{t2}\end{array}\right]=\left[\begin{array}{l}{A}_{t1}\\ A_{t1}{\mathrm{\Phi }}_t\end{array}\right]B+\left[\begin{array}{l}{V}_{t1}\\ V_{t2}\end{array}\right]=\left[\begin{array}{l}{A}_{t11}\\ A_{t12}\\ A_{t11}{\mathrm{\Phi }}_t\\ A_{t12}{\mathrm{\Phi }}_t\end{array}\right]B+\left[\begin{array}{l}{V}_{t1}\\ V_{t2}\end{array}\right]=K_{t}B+V_t$$

(9)

Where A _t11 and A _t12 are the first 2P rows and the last 2N(M − 1) − 2P rows of A _t1. Suppose K _t1 and K _t2 are the first 2P rows and the last 4N(M − 1) − 2P rows of K _t , then there exists a matrix V ^H _t , which satisfies

$$V_t^H{K}_{t1}=K_{t2}$$

(10)

Here we call V ^H _t as a propagator operator [14]. Let Z _t1 and Z _t2 be the first 2P rows and the last 4M(N − 1) − 2P rows of Z _t . If the noises are ignored, then

$$Z_{t1}=K_{t1}B,Z_{t2}=K_{t2}B$$

(11)

By pre-multiplying Z _t1 by V ^H _t , we obtain

$$V_t^H{Z}_{t1}=V_t^H{K}_{t1}B=K_{t2}B=Z_{t2}$$

(12)

If there exist noises, we can solve V ^H _t by $V_t^H=Z_{t2}{Z}_{t1}{}^H{\left(Z_{t1}{Z}_{t1}{}^H\right)}^{-1}$ . Let ${\overline{V}}_t=\left[\begin{array}{l}{I}_{2P}\\ V_t^H\end{array}\right]$ , where I _2P is a 2P × 2P identity matrix, then

$${\overline{V}}_t{K}_{t1}=\left[\begin{array}{l}{I}_{2P}\\ V_t^H\end{array}\right]K_{t1}=\left[\begin{array}{l}{K}_{t1}\\ K_{t2}\end{array}\right]=K_t=\left[\begin{array}{l}{A}_{t1}\\ A_{t1}{\mathrm{\Phi }}_t\end{array}\right]$$

(13)

From (13) we can see that ${\overline{V}}_t$ can span the signal subspace. Equally dividing ${\overline{V}}_t$ into two matrices according to row, we name them as ${\overline{V}}_{t1}$ and ${\overline{V}}_{t2}$ , then ${\overline{V}}_{t1}{K}_{t1}=A_{t1}$ , ${\overline{V}}_{t2}{K}_{t1}=A_{t1}{\mathrm{\Phi }}_t$ . Therefore,

$${\overline{V}}_{t2}{\overline{V}}_{t1}{}^{+}{A}_{t1}={\Psi }_t{A}_{t1}=A_{t1}{\mathrm{\Phi }}_t$$

(14)

where ${\Psi }_t={\overline{V}}_{t2}{\overline{V}}_{t1}{}^{+}={\left({\overline{V}}_{t1}{}^H{\overline{V}}_{t1}\right)}^{-1}{\overline{V}}_{t1}{}^H$ . From (14), we perform an Eigenvalue decomposition of matrix Ψ _t to yield P major Eigenvalues λ_t1, λ_t2, · λ ^tP , which are the diagonal elements of Φ _t . Thus, the DOD estimation of P targets can be obtained from the Eigenvalue of matrix Ψ _t .

Second, in order to estimate the DOA parameter, at first we form a new data matrix Y' from the output data matrix Y , i.e.,

$$Y\text{'}\left(\theta ,\varphi ,\gamma ,\eta \right)=TY\left(\theta ,\varphi ,\gamma ,\eta \right)=A\text{'}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V\text{'}$$

(15)

where T is a Kronnecker product permutation matrix, $A\text{'}(\theta ,\varphi ,\gamma ,\eta )=TA(\theta ,\varphi ,\gamma ,\eta )=\left[a_1\text{'}({\theta }_1,{\varphi }_1,{\gamma }_1,{\eta }_1),\cdot \cdot \cdot ,a_P\text{'}({\theta }_P,{\varphi }_P,{\gamma }_P,{\eta }_P)\right]$ , with $a_p\text{'}({\theta }_p,{\varphi }_p,{\gamma }_p,{\eta }_p)=a_r({\varphi }_p,{\gamma }_p,{\eta }_p)\otimes a_t({\theta }_p)$ and V ' = TV . Similarly, let Y _r1 and Y _r2 be the 2M(N − 1) × L submatrices of Y' consisting of the first and last 2M(N − 1) rows of Y' , i.e., $Y_{r1}=A_{r1}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V_{r1}$ and $Y_{r2}=A_{r2}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V_{r2}$ , where $A_{r1}\left(\theta ,\varphi ,\gamma ,\eta \right)$ and $A_r{}_2\left(\theta ,\varphi ,\gamma ,\eta \right)$ are the first and last 2M (N − 1) rows of $A\text{'}(\theta ,\varphi ,\gamma ,\eta )$ , respectively. Then, $A_{r2}\left(\theta ,\varphi ,\gamma ,\eta \right)=A_{r1}\left(\theta ,\varphi ,\gamma ,\eta \right){\mathrm{\Phi }}_r$ , where ${\mathrm{\Phi }}_r=\text{diag}\left\{e^{-\mathrm{j}2\text{π}\left(m-1\right)\left(d_r/\lambda \right)\sin {\varphi }_1},\dots ,e^{-\mathrm{j}2\text{π}\left(m-1\right)\left(d_r/\lambda \right)\sin {\varphi }_P}\right\}$ . The estimation of DOA can be derived from the diagonal matrix Φ _r . We proceed the data of Y _r1 and Y _r2 in the same way as Y _t1 and Y _t2, thus, the propagator operator is $V_r^H=Z_{r2}{Z}_{r1}{}^H{\left(Z_{r1}{Z}_{r1}{}^H\right)}^{-1}$ , where Z _r1 and Z _r2 are the first 2P rows and the last 4M(N − 1) − 2P rows of $Z_r=\left[\begin{array}{l}{Y}_{r1}\\ Y_{r2}\end{array}\right]$ . Lastly, we have ${\Psi }_r{A}_{r1}=A_{r1}{\mathrm{\Phi }}_r$ , where ${\Psi }_r={\overline{V}}_{r2}{\overline{V}}_{r1}{}^{+}$ and ${\overline{V}}_{r2}$ are two equally divided submatrices of ${\overline{V}}_r=\left[\begin{array}{l}{I}_{2P}\\ V_r^H\end{array}\right]$ according to row. Thus, the DOA estimation of P targets can be obtained from the P major Eigenvalues λ_r1, λ_r2, ·, λ _rP of matrix Ψ _r . Therefore, θ _p and ϕ _p for the p-th target can be written as

$${\theta }_p=\arcsin \left(-\lambda \cdot angle\left({\lambda }_{tp}\right)/2\pi d_t\right)$$

(16)

$${\varphi }_p=\arcsin \left(-\lambda \cdot angle\left({\lambda }_{rp}\right)/2\pi d_r\right)$$

(17)

Third, in order to estimate polarization information, let Y _v1 and Y _v2 be the N(M − 1)xL submatrices of Y , consisting of the even and odd rows of Y , i.e., $Y_{v1}=A_{v1}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V_{v1}$ and $Y_{v2}=A_{v2}\left(\theta ,\varphi ,\gamma ,\eta \right)B+V_{v2}$ , where $A_{v1}\left(\theta ,\varphi ,\gamma ,\eta \right)$ and $A_{v2}\left(\theta ,\varphi ,\gamma ,\eta \right)$ are the even and odd rows of $A(\theta ,\varphi ,\gamma ,\eta )$ , respectively. Then we have $A_{v2}\left(\theta ,\varphi ,\gamma ,\eta \right)=A_{v1}\left(\theta ,\varphi ,\gamma ,\eta \right){\mathrm{\Phi }}_v$ , where Φ _v = diag {r₁, r₂,…,r _P } and r _p is the ratio of the first element and the second element of matrix v _p in (5), denoted as [17]

$$r_p=\frac{-\cos {\gamma }_p}{\sin {\gamma }_p\cos {\varphi }_p{\mathrm{e}}^{\mathrm{j}{\eta }_p}}$$

(18)

Therefore, the estimation of polarizations can be derived from the diagonal matrix Φ _v . Proceed the data of Y _v1 and Y _v2 in the same way as Y _t1 and Y _t2, so the propagator operator is $V_v^H=Z_{v2}{Z}_{v1}{}^H{\left(Z_{v1}{Z}_{v1}{}^H\right)}^{-1}$ , where Z _v1 and Z _v2 are the first 2P rows and the last 2MN − 2P rows of $Z_v=\left[\begin{array}{l}{Y}_{v1}\\ Y_v{}_2\end{array}\right]$ . Finally, we have ${\Psi }_v{A}_{v1}=A_{v1}{\mathrm{\Phi }}_v$ , where ${\Psi }_v={\overline{V}}_{v2}{\overline{V}}_{v1}{}^{+}$ and ${\overline{V}}_{v2}$ are two equally divided submatrices of ${\overline{V}}_v=\left[\begin{array}{l}{I}_{2P}\\ V_v^H\end{array}\right]$ according to row. Then perform Eigenvalue decomposition of Ψ_v to yield Φ _v ; which is a diagonal matrix composed of the P major Eigenvalues λ_v1, λ_v2, ·, λ _vP . The polarization parameters λ _p and η _p of the pth target are calculated by

$${\gamma }_p={\tan }^{-1}\left(\left|{\omega }_p\right|\right)$$

(19)

$${\eta }_p=\text{angle}\left({\omega }_p\right)$$

(20)

where ${\omega }_p=-\frac{1}{r_p\cos {\varphi }_p}$ .

3.2 Target Localization

A target can be located if its DOD, DOA and the distance D between transmitter and receiver arrays are known in a bistatic MIMO radar system. For the pth target, θ _p and ϕ _p are used to determine its position via trilateration and γ _p and η _p are used to identify multiple targets and improve the resolution. The coordinate diagram for locating a target in a bistatic MIMO radar system is shown in Fig. 3.

Figure 3.

The coordinate diagram for locating a target in a bistatic MIMO radar system.

From Fig. 3, the coordinate of the pth target (x _p , y _p ) can be determined by

$$\{\begin{array}{l}{x}_p=D\frac{\sin {\theta }_p\cos {\varphi }_p}{\sin ({\theta }_p+{\varphi }_p)}\\ y_p=D\frac{\cos {\theta }_p\cos {\varphi }_p}{\sin ({\theta }_p+{\varphi }_p)}\end{array}$$

(21)

4. Cramer-Rao bound

From the received data in (8), the Fisher information matrix with respect to θ _p , ϕ _p , γ _p and η _p can be denoted as [18]

$$F=\left[\begin{array}{l}{F}_{11}{F}_{12}{F}_{13}{F}_{14}\\ F_{21}{F}_{22}{F}_{23}{F}_{24}\\ F_{31}{F}_{32}{F}_{33}{F}_{34}\\ F_{41}{F}_{42}{F}_{43}{F}_{44}\end{array}\right]$$

(22)

where the (i, j) th item of F ₁₁ is given by

$$\begin{array}{l}{F}_{11}\left({\theta }_i,{\theta }_j\right)=2\mathrm{Re}tr\left[{\left(\frac{\partial A\left(\theta ,\varphi ,\gamma ,\eta \right)B}{\partial {\theta }_i}\right)}^H{R}_V^{}{}^{-1}\frac{\partial A\left(\theta ,\varphi ,\gamma ,\eta \right)B}{\partial {\theta }_j}\right]\\ \begin{array}{ccc}& & \end{array}=2\mathrm{Re}tr\left[{\left({\dot{A}}_{\theta }{e}_i{e}_i{}^{T}B\right)}^H{R}_V^{}{}^{-1}\left({\dot{A}}_{\theta }{e}_j{e}_j{}^{T}B\right)\right]\\ \begin{array}{ccc}& & \end{array}=2\mathrm{Re}tr\left[\left(e_i{}^T{\dot{A}}_{\theta }{R}_V^{}{}^{-1}{\dot{A}}_{\theta }{e}_j\right)\left(e_j{}^{T}B{B}^H{e}_i\right)\right]\\ \begin{array}{ccc}& & \end{array}=2L\mathrm{Re}\left[{\left({\dot{A}}_{\theta }{}^H{R}_V^{}{}^{-1}{\dot{A}}_{\theta }\right)}_{ij}{\left(R_B^T\right)}_{ij}\right]\end{array}$$

(23)

where tr (•) denotes the trace of a matrix, Re(•) denotes the real part, e_i denotes the i-th column of the identity matrix and

$${\dot{A}}_{\theta }=\left[\frac{\partial a_t\left({\theta }_1\right)}{\partial {\theta }_1}\otimes a_r\left({\varphi }_1,{\gamma }_1,{\eta }_1\right),\dots ,\frac{\partial a_t\left({\theta }_P\right)}{\partial {\theta }_p}\otimes a_r\left({\varphi }_P,{\gamma }_p,{\eta }_p\right)\right]$$

(24)

$${\dot{A}}_{\varphi }=\left[a_t\left({\theta }_1\right)\otimes \frac{\partial a_r\left({\varphi }_1,{\gamma }_1,{\eta }_1\right)}{\partial {\varphi }_1},\dots ,a_t\left({\theta }_P\right)\otimes \frac{\partial a_r\left({\varphi }_P,{\gamma }_P,{\eta }_p\right)}{\partial {\varphi }_P}\right]$$

(25)

$${\dot{A}}_{\gamma }=\left[a_t\left({\theta }_1\right)\otimes \frac{\partial a_r\left({\varphi }_1,{\gamma }_1,{\eta }_1\right)}{\partial {\gamma }_1},\dots ,a_t\left({\theta }_P\right)\otimes \frac{\partial a_r\left({\varphi }_P,{\gamma }_P,{\eta }_p\right)}{\partial {\gamma }_P}\right]$$

(26)

$${\dot{A}}_{\eta }=\left[a_t\left({\theta }_1\right)\otimes \frac{\partial a_r\left({\varphi }_1,{\gamma }_1,{\eta }_1\right)}{\partial {\eta }_1},\dots ,a_t\left({\theta }_P\right)\otimes \frac{\partial a_r\left({\varphi }_P,{\gamma }_P,{\eta }_p\right)}{\partial {\eta }_P}\right]$$

(27)

$$R_B\triangleq \frac{1}{L}B{B}^H$$

(28)

where $R_V^{}=E\left[V{V}^H\right]$ is the covariance matrix of the noise matrix V . Then,

$$\begin{array}{l}{F}_{11}=2L\mathrm{Re}\left({\dot{A}}_{\theta }^H{R}_V{}^{-1}{\dot{A}}_{\theta }\odot R_B^T\right)\\ F_{12}=2L\mathrm{Re}\left({\dot{A}}_{\theta }^H{R}_V{}^{-1}{\dot{A}}_{\gamma }\odot R_B^T\right)\\ F_{13}=2L\mathrm{Re}\left({\dot{A}}_{\theta }^H{R}_V{}^{-1}{\dot{A}}_{\gamma }\odot R_B^T\right)\\ F_{14}=2L\mathrm{Re}\left({\dot{A}}_{\theta }^H{R}_V{}^{-1}{\dot{A}}_{\eta }\odot R_B^T\right)\\ F_{21}=2L\mathrm{Re}\left({\dot{A}}_{\varphi }^H{R}_V{}^{-1}{\dot{A}}_{\theta }\odot R_B^T\right)\\ F_{22}=2L\mathrm{Re}\left({\dot{A}}_{\varphi }^H{R}_V{}^{-1}{\dot{A}}_{\varphi }\odot R_B^T\right)\\ F_{23}=2L\mathrm{Re}\left({\dot{A}}_{\varphi }^H{R}_V{}^{-1}{\dot{A}}_{\gamma }\odot R_B^T\right)\\ F_{24}=2L\mathrm{Re}\left({\dot{A}}_{\varphi }^H{R}_V{}^{-1}{\dot{A}}_{\eta }\odot R_B^T\right)\\ F_{31}=2L\mathrm{Re}\left({\dot{A}}_{\gamma }^H{R}_V{}^{-1}{\dot{A}}_{\theta }\odot R_B^T\right)\\ F_{32}=2L\mathrm{Re}\left({\dot{A}}_{\gamma }^H{R}_V{}^{-1}{\dot{A}}_{\varphi }\odot R_B^T\right)\\ F_{33}=2L\mathrm{Re}\left({\dot{A}}_{\gamma }^H{R}_V{}^{-1}{\dot{A}}_{\gamma }\odot R_B^T\right)\\ F_{34}=2L\mathrm{Re}\left({\dot{A}}_{\gamma }^H{R}_V{}^{-1}{\dot{A}}_{\eta }\odot R_B^T\right)\\ F_{41}=2L\mathrm{Re}\left({\dot{A}}_{\eta }^H{R}_V{}^{-1}{\dot{A}}_{\theta }\odot R_B^T\right)\\ F_{42}=2L\mathrm{Re}\left({\dot{A}}_{\eta }^H{R}_V{}^{-1}{\dot{A}}_{\varphi }\odot R_B^T\right)\\ F_{43}=2L\mathrm{Re}\left({\dot{A}}_{\eta }^H{R}_V{}^{-1}{\dot{A}}_{\gamma }\odot R_B^T\right)\\ F_{44}=2L\mathrm{Re}\left({\dot{A}}_{\eta }^H{R}_V{}^{-1}{\dot{A}}_{\eta }\odot R_B^T\right)\end{array}$$

(29)

where ⊙ denotes the Hadamard Product. Therefore, the CRB matrix is calculated by

$$CRB=F^{-1}$$

(30)

5. Simulation Results and Analysis

Assuming that the transmitter array consists of M = 6 elements and the receiver array has N = 6 pairs of elements of crossed dipoles, d _t and d _r are all half a wavelength. Walsh sequences with a length of 1024 are selected as the orthogonally transmitted waveform. All the P targets have RCS amplitudes of one and different Doppler frequencies. The target echoes are elliptically polarized with different polarization angles. The duration of a snapshot is 5µs. The number of snapshots is 500 and the estimation performance in all the simulations is obtained by 100 Monte Carlo trials.

5.1 Simulation 1

Assuming that there are P = 3 targets, their parameters of DOD, DOA and polarization are θ = (10°, 40°, 60°, ϕ = (20°, 50°, 70°), γ = (20°, 40°, 60°), η = (0°, 20°, 70°). Their Doppler frequencies are 1000Hz, 2550Hz and 5000Hz, respectively. Fig. 4 (a) and (b) respectively present the estimation results of DOD-DOA and two polarization parameters while SNR is 15dB. It is shown in Fig. 4 that the estimate approximates to the real value. The proposed approach can effectively estimate DOD, DOA and polarizations for accurate identification and location of the multiple targets in the bistatic MIMO radar.

Figure 4.

DOD, DOA and polarizations estimation results when SNR=15 dB.

The parameter estimation performance of the algorithm is evaluated through root-mean-square error (RMSE) and root CRB. The RMSE and root CRB versus SNR for DOD, DOA and polarizations estimation are shown in Fig. 5 (a) - (d).

The results in Fig. 5 indicate that the RMSEs of DOD, DOA and polarizations decrease with the increase of SNR. The RMSE is close to the root of CRB, which is a lower bound of the unbiased estimator of variance.

Figure 5.

The RMSE and Root CRB versus SNR for DOD, DOA and polarizations estimation.

5.2 Simulation 2

At first, the simulation investigates the situation that there are two targets being closely spaced. Assume they have the same DOA and 1° DOD difference, θ = (10°, 11°) and ϕ = (30°, 30°). Their polarization states are different, γ = (30°, 60°), η = (0°, 45°). SNR=15dB. The resolution of DOD-DOA estimation is compared between the proposed approach and Chen's Method in [11], as shown in Table 1.

Table 1.

DOD-DOA Estimation result when two targets are closely spaced (º)

Parameters	Proposed method		Method in [11]
DOD/DOA	target 1	target 2	target 1	target 2
θ	10	11	10	11
$\widehat{\theta }$	10.0068	10.9969	10.5002	5.3686
|Δθ|	0.0068	0.0031	0.5002	5.6314
ϕ	30	30	30	30
$\widehat{\varphi }$	29.9468	30.0572	30.0018	30.6315
|Δϕ|	0.0532	0.0572	0.0018	0.6315

Furthermore, a situation is investigated where there are three closely spaced targets. Suppose that they have θ = (10°, 12°, 15°) and ϕ = (30°, 31°, 32°). Their polarization states are γ = (30°, 60°, 90°) and η = (0°, 45°, 120°). SNR=15dB. The resolution of DOD and DOA estimation is compared between the proposed approach and Chen's Method in [11], as shown in Table 2.

Table 2.

DOD-DOA Estimation result when three targets are closely spaced (º)

Parameters	Proposed method			Method in [11]
DOD/DOA	target 1	target 2	target 3	target 1	target 2	target 3
θ	10	12	15	10	12	15
$\widehat{\theta }$	10.3898	12.2706	14.2042	12.4334	12.5053	4.7493
|Δθ|	0.3898	0.2706	0.7958	2.4334	0.5053	10.2507
ϕ	30	31	32	30	31	32
$\widehat{\varphi }$	29.7795	30.7990	32.7444	31.045 5	31.0726	−1.5003
|Δϕ|	0.2205	0.2010	0.7444	1.0455	0.0726	33.5003

From the data for the DOA-DOD estimation in Table 1 and Table 2, we observe that when the radar targets are too closely spaced to be distinguished, the proposed method can effectively identify different targets with high-resolution by taking advantage of the different states of polarization of the echoes. However, the method in [11] cannot identify multiple closely spaced targets. For example, in Table 1, the estimate of DOD for target 2 is 5.3686º, which deviates from its real value of 11º. Also, in Table 2, the estimates of DOD and DOA for target 3 are 4.7493º and −1.5003º, which greatly deviate from the real values of 15º and 32º.

5.3 Localization accuracy

Assume that the distance between transmitter array and receiver array is D=100m. There are three closely spaced targets whose coordinates in x-y plane are (63, 35), (66, 40), (70, 45) in metres and SNR=15dB. The localization results of the three targets are shown in Fig. 6.

Figure. 6.

The localization results of three closely spaced targets.

It can be seen from Fig. 6 that the estimates of the coordinates of the three targets are close to their real values. The proposed approach can effectively locate closely spaced targets.

The localization errors versus SNR for the three targets are shown in Fig. 7. The result indicates that the proposed approach has high localization accuracy in positioning multiple closely spaced targets.

Figure. 7.

Localization error versus SNR for three closely spaced targets.

6. Conclusion

In this paper, we propose a fast and high-resolution multi-target localization approach in a bistatic MIMO radar system by extending the traditional propagator method to estimate DOD, DOA and polarization parameters. The proposed method can effectively combine polarization and angle information to accurately identify and locate multiple targets. The closely spaced targets can be well distinguished by polarization diversity. Also, the propagator method avoids SVD in the covariance matrix and spectrum peak searching; therefore the computational complexity is reduced. Based on the angle and polarization information, the two-dimensional (2D) position of a target can be calculated via trilateration. In future research, the three-dimensional (3D) position of a target will be further determined by estimating the elevation-azimuth angles of DOD and DOA and the Doppler shift can be estimated to locate and track moving targets.