ESPRIT-Like Two-Dimensional DOA Estimation for Monostatic MIMO Radar with Electromagnetic Vector Received Sensors under the Condition of Gain and Phase Uncertainties and Mutual Coupling

Since its first appearance more than a decade ago, Multiple-Input Multiple-Output (MIMO) radar has become a research hotspot [1,2,3]. It makes use of multiple antennas to emit orthogonal waveforms and utilizes multiple antennas to receive the echoes which will bring unique advantages [4]. It also can emit correlated waveforms to achieve more flexible beam pattern designs [5]. There are many methods for beampattern design, such as the classic Capon beamforming method and its improved form [6,7,8]. For MIMO radar, we can realize the beampattern design through flexible waveform design [9]. For example, Ahmed et al. [10] propose a transmit beampattern design method based on one step unconstrained optimization, which avoids the design of waveform covariance matrix and reduces the complexity. The robust waveform covariance matrix design with minimum sidelobe level or minimum integrated sidelobe level is studied in [11] and it also gives a design method which has polynomial time complexity. There are two categories of MIMO radar. One is statistical MIMO radar whose array spacing is very large. It can achieve diversity gains and overcome the effect of target scintillation [12,13]. The other is colocated MIMO whose array elements are closely spaced [14,15,16]. This can improve the parameter estimation performance significantly. Angle estimation is very important in radar system [17], so in this paper, we pay attention to the 2D-DOA estimation of collocated monostatic MIMO radars.

There are many algorithms for direction of arrival (DOA) estimation, such as the Capon method [18], multiple signal classification (MUSIC) algorithm [19], and the estimation of signal parameter via rotational invariance technique (ESPRIT) algorithm [20]. In addition, the sparse signal reconstruction-based DOA estimation algorithm has become a hot research topic [21]. These algorithms can achieve high estimation accuracy and have been successfully applied to the DOA and direction of departure (DOD) estimation in MIMO radar [22,23,24]. Among them, the ESPRIT algorithm has received extensive attention because of its high computational efficiency.

Polarization sensitive array signal processing is a new bench of array signal processing [25] that has received a lot of attention recently [26,27]. A polarization array is an array consisting of multiple polarization- sensitive units which are spatially distributed in a certain structure and are not identical in polarization [28]. Unlike conventional scalar sensor arrays, polarization-sensitive arrays can acquire polarization direction information, that is, they have polarization diversity capability. This outstanding feature provides a new and useful means for information acquisition and transmission [29].

The combination of a MIMO array and a polarization-sensitive array will bring much more performance improvement in parameter estimation [30]. Based on this, Jiang et al. [31] proposed polarimetric MIMO radar. Its transmitting array consists of scalar antennas and its receiving array is composed of cross-dipole vector antennas. Then it utilizes ESPRIT algorithm to estimate DOD, DOA and polarization station angle (PSA). Gu et al. [30] used a six-component electromagnetic vector sensor (EMVS) to increase the information of a single vector antenna. As we all know, the spatial electromagnetic wave signal is a vector signal. The complete electric and magnetic information of a vector signal is a complex vector with six dimensions. Most of the existing radar systems use a single polarization scalar array to obtain only one-dimensional electromagnetic wave signal information. The electromagnetic vector sensor can get all or at least more than one dimension information of space electromagnetic wave signals. For example, Jiang et al. [31] adopted cross-dipole antennas. Therefore, the vector antennas only obtain two-component electric-field of source. Gu et al. [30] adopted a six-component electromagnetic vector-sensor which uses three orthogonal electric-dipoles and magnetic-loops to measure the three electric-field- and three magnetic-field parameters of the incident signals, respectively. Thus the EMVS can get complete electromagnetic vector information. The EMVS is a new type of single input multiple output antenna which receiving close attention from academia and industry. Because it uses more information, we can get more advantages in DOA estimation. For example, we can use the relationship between these electromagnetic components to estimate DOA. Besides, we can combine it with the ESPRIT results to improve the estimation accuracy of DOA. Then, Zheng et al. [26] extended EMVS to the MIMO radar. It uses the rotational invariance of the spatial-polarimetric domain (SPD) and proposes a corresponding ESPRIT-like method to estimate the 2D-DOA and PSA in monostatic MIMO radar. It is suitable for arbitrary array configurations.

However, all the above methods are based on the hypothesis that there are no gain and phase uncertainties (GPU) and the mutual coupling (MC) of EMVS components is also not considered. There are too many uncertainties in the actual production of the antenna, the amplitude and the phase gain of different array elements cannot be exactly the same, so amplitude and phase errors and mutual coupling are very common in practical situations. When the element spacing is relatively close, the mutual coupling between the elements is unavoidable, so in practice, the array manifold is inevitably affected by mutual coupling and array gain or phase uncertainties. This results in significant distortion of the amplitude and phase of the signals received from the array, so the direct use of these methods will lead to a serious degradation of estimation accuracy or even failure. How to eliminate this effect is a meaningful task. There are many people researching the problem of GPU and MC. Wang et al. [32] solved the MC problem of a uniform linear array (ULA) based on subspace theory. In [33], a robust direction-of-arrival (DOA) estimation algorithm for quasi-stationary signals based on the Khatri-Rao (KR) subspace approach was put forward. It can solve the GPU problem of DOA estimation. Si et al. [34] tackled the GPU and MC problem for ULAs or uniform circular arrays (UCAs) by applying a group of auxiliary sensors. However, for the problem of GPU and MC in MIMO-EMVS, as far as we know, there are no corresponding references, so in this paper, we studied the GPU and MC problem of MIMO-EMVS.

Firstly, we propose a spatial-polarimetric domain ESPRIT-like algorithm based on centralized EMVS (C-SPD ESPRIT-like algorithm) for the estimation of 2D-DOA and PSA in MIMO-EMVSs under the condition of GPU. The EMVS adopts a centralized configuration. This algorithm can be applied to an arbitrary array configuration. Then, considering the mutual coupling of components of EMVS, we revise the C-SPD ESPRIT-like algorithm and apply it to a class of separated EMVSs. The GPU and MC problem are solved simultaneously. The proposed methods have low complexity for avoiding the angle searching operation and the array configuration can be arbitrary. And the effectiveness of proposed algorithms is verified by the simulation results.

The remainder of the paper is organized as follows: Section 2 is devoted to the introduction of the signal model of polarimetric MIMO radar. In Section 3, the two proposed methods for estimation of DOA under non-ideal conditions are described, and some discussions are presented. Section 4 gives a comparison of these methods. Simulation results are presented in Section 5 to evaluate the advantages of the proposed methods. At last, conclusions are drawn in Section 6.

Assuming there is a monostatic MIMO radar which hass M transmitters and N − 1 receivers. Each receiver is composed of a six-component centralized EMVS which is shown in Figure 1. As we all know, the spatial electromagnetic wave signal is a vector signal. The complete electric and magnetic information of a vector signal is the complex vector of six dimensions. This six dimensional vector is related to the DOA and polarization station angle of the target. The polarization station angle is composed of a polarization phase angle and the polarization phase difference. The EMVS consists of the phase centered coincidence of three orthogonal electric dipoles and three orthogonal magnetic rings that can get complete electromagnetic vector information, so each EMVS has six component outputs, representing the electric field and magnetic field components of the incident electromagnetic wave. In our model, the position of the m-th array element in the transmitting array is $\left(x_{tm},y_{tm},z_{tm}\right),m=1,2,\dots ,M$ and the position of the n-th array element in the receiving array is $\left(x_{tn},y_{tn},z_{tn}\right),n=1,2,\dots ,N-1$. There is no limitation of the array configuration of both the the transmitting array and the receiving array. That is to say they are arbitrary array configurations and later we will show that our algorithm has no requirement for a specific array configuration.

Assuming the transmitting waveforms are normalized orthogonal signals, i.e., $\left(1/\Gamma \right){\displaystyle {\sum }_{\tau =1}^{\Gamma }s\left(\tau \right)s^H\left(\tau \right)}=I_M$, where $s\left(\tau \right)={\left[s_1\left(\tau \right),s_2\left(\tau \right),\dots ,s_M\left(\tau \right)\right]}^T\in {ℂ}^{M\times 1}$ denotes $M$ transmitted signals, $\tau =1,2,\dots ,\Gamma$ is the sampling index in fast time. There are $K$ far field uncorrelated point targets. The 2D-DOA of incident signals can be expressed as azimuth angle $\varphi \in [0,2\pi )$ and elevation angle $\theta \in [0,\pi ]$. They are measured from the positive $x$-axis and positive $z$-axis, respectively. The PSA of targets are auxiliary polarization angle $\gamma \in [0,\pi /2]$ and polarization phase difference $\eta \in (-\pi ,\pi ]$, respectively. The received data can be expressed as follows:where $t$ is the slow-time, i.e., snapshot. ${\Lambda }_t$ is a $M\times M$ complex diagonal matrix which represents the GPU of transmitting sensors. ${\Lambda }_r$ is a $6\left(N-1\right)\times 6\left(N-1\right)$ complex diagonal matrix which represents the GPU of receiving sensors. $A_r\left(\theta ,\varphi \right)=[a_r({\theta }_1,{\varphi }_1),\dots ,a_r({\theta }_K,{\varphi }_K)]\in {ℂ}^{\left(N-1\right)\times K}$ is the receiving spatially steering matrix. $A_t\left(\theta ,\varphi \right)=[a_t({\theta }_1,{\varphi }_1),\dots ,a_t({\theta }_K,{\varphi }_K)]\in {ℂ}^{M\times K}$ is the transmitting spatially steering matrix. $A_{pol}\left(\theta ,\varphi \right)=[a_{pol}({\theta }_1,{\varphi }_1,{\gamma }_1,{\eta }_1),\dots ,$ $a_{pol}({\theta }_K,{\varphi }_K,{\gamma }_K,{\eta }_K)]\in {ℂ}^{6\times K}$ is the spatial-polarimetric domain (SPD) steering matrix of a single centralized EMVS. $\oplus$ is the Khatri-Rao product. $a_t({\theta }_k,{\varphi }_k)$, $a_r({\theta }_k,{\varphi }_k)$ and $a_{pol}({\theta }_k,{\varphi }_k,{\gamma }_k,{\eta }_k)$ are expressed as follows: where $e$ (the first three elements of $a_{pol}({\theta }_k,{\varphi }_k,{\gamma }_k,{\eta }_k)$) and $h$ (the last three elements of $a_{pol}({\theta }_k,{\varphi }_k,{\gamma }_k,{\eta }_k)$) represent the electric-field component and magnetic-field component, respectively. They are related to the 2D-DOA and PSA of targets. $b\left(t\right)={\left[{\rho }_1\left(t\right),\dots ,{\rho }_K\left(t\right)\right]}^T\in {ℂ}^{K\times 1}$ represents reflection coefficient of all targets, where ${\rho }_k\left(t\right)={\alpha }_k\exp \left(j2\pi f_{k}t\right)$. The amplitude of the reflection coefficient ${\alpha }_k$ is related to the target radar cross section (RCS). The phase of reflection coefficient is related to the Doppler frequency of the target. Note that, here $t$ represents the slow-time. $w\left(t,\tau \right)\in {ℂ}^{6\left(N-1\right)\times 1}$ represents additive white Gaussian noise whose mean value is zero and covariance matrix is ${\sigma }_n^2{I}_{6\left(N-1\right)}$.

Through matched filtering, the received data can be represented as where $N\left(t\right)=\frac{1}{\Gamma }{\displaystyle \sum _{\tau =1}^{\Gamma }w\left(t,\tau \right)s^H\left(\tau \right)}$. Perform vectorization to (5), we can obtain:where $A\left(\theta ,\varphi ,\gamma ,\eta \right)=\left[A_t\left(\theta ,\varphi \right)\oplus A_r\left(\theta ,\varphi \right)\oplus A_{pol}\left(\theta ,\varphi ,\gamma ,\eta \right)\right]$ $\in {ℂ}^{6M\left(N-1\right)\times K}$. And its columns are the virtual array steering vector $a\left(\theta ,\varphi ,\gamma ,\eta \right)=\left[a_t\left(\theta ,\varphi \right)\otimes a_r\left(\theta ,\varphi \right)\otimes a_{pol}\left(\theta ,\varphi ,\gamma ,\eta \right)\right]$. $\Lambda ={\Lambda }_t\otimes {\Lambda }_r$. $\otimes$ is the Kronecker product. And $n\left(t\right)=\mathrm{vec}\left[N\left(t\right)\right]$. $\mathrm{vec}(·)$ represents the vectorization operation. The noise corresponding to m-th transmitting antenna and the n-th receiving antenna is $n_{n,m}=w_n{s}_m^H$. The elements of the noise covariance matrix are equal to:

We can see that $n\left(t\right)$ is still a complex white Gaussian noise with zero mean and covariance matrix ${\sigma }_n^2{I}_{6M\left(N-1\right)}$.

When there are no GPU and MC, we can use the method in [26] to estimate the 2D-DOA of targets. For virtual array after matched filtering, it extracts the component with the same orientation in EMVSs to construct a subarray. Then we can obtain six spatially identical subarrays. According to the SPD rotational invariance of the six subarrays, we can get $C_6^2=15$ rotation invariant factors (RIFs) by using the ESPRIT algorithm. According to the knowledge of [35], we can calculate the incident electromagnetic wave’s electric-field and magnetic-field by only selecting five RIFs. Then we can perform vector cross-product between electric-field and magnetic-field to obtain the pointing vector of source. Last, 2D-DOA can be obtained by the triangle operation with the pointing vector. This is the method which is used in [26] and it is different with the spatial ESPRIT algorithm. It uses the SPD invariance property to construct the electromagnetic component and uses the vector cross-product of these components to get the DOA of targets, while the spatial ESPRIT directly uses the spatial invariance property to estimate the DOA of targets.

When GPU and MC occur, it results in significant distortion of the amplitude and phase of the signals received from the array. This spoils the invariance property of the SPD, so the direct use of the SPD ESPRIT algorithm will lead to wrong estimation of RIFs which will cause DOA estimation errors, therefore, the ESPRIT algorithm in [26] which is computationally efficient cannot be used directly. Based on this, we research the 2D-DOA estimation of MIMO-EMVSs and put forward the ESPRIT-like 2D-DOA estimation algorithm.

The main content of Section 3 includes the following two aspects: firstly, considering the problem of GPU, we propose a novel C-SPD ESPRIT-like algorithm with centralized EMVSs in Section 3.1. It utilizes the ISM to estimate the 2D-DOA and PSA of targets. The C-SPD ESPRIT-like algorithm can obtain good estimation performance without any information of the GPU. Then, taking into account the simultaneous problems of GPU and MC, we revise the C-SPD ESPRIT-algorithm and apply it to a class of separated EMVSs in Section 3.2. This can solve the problem of GPU and MC efficiently.