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Joint range, angle and polarization estimation in polarimetric FDAMIMO radar based on Tucker tensor decomposition
EURASIP Journal on Advances in Signal Processing volume 2023, Article number: 39 (2023)
Abstract
Frequency diverse array multipleinput multipleoutput (FDAMIMO) radar is an emerging technology to offer rangeangledependent beampattern. Polarimetric FDAMIMO radar can sense additional polarization information to improve target identification capability. In this article, we investigate the problem of joint range, angle and polarization parameter estimation in a monostatic polarimetric FDAMIMO radar with an FDA at transmitter and a crossdipole array at receiver. Unlike the conventional methods in which the multidimensional data structure is rearranged into vectors or matrices by stacking operation, we propose a Tucker tensor decompositionbased scheme, which can reserve the original data structure and avoid spoiling the inherent characteristics of interest, especially when the number of snapshots is small. The thirdorder tensor model of the observed data is constructed. Two approaches named as Tucker covariance reconstruction and Tucker signal subspace are presented using the fourthorder covariance tensor decomposition. The Cramér–Rao bound for range, angle and polarization estimation is also provided. Numerical experiments demonstrate the superiorities of the proposed approaches. Specifically, two targets with identical range and close angles are effectively distinguished.
1 Introduction
Multipleinput multipleoutput (MIMO) radar [1] is a powerful technique for target detection, localization and imaging. MIMO radar array with colocated sensors has strong capability of estimating the angles of targets [2, 3]. However, with conventional MIMO radar, two targets having different angles but same range cannot be directly identified. Frequency diverse array (FDA), as an emerging scheme which was first proposed in [4], has received much attention in recent years. The carrier frequency of FDA has a small frequency increment between contiguous elements. Base on this design, it can produce anglerangedependent transmit beampattern [5, 6]. In [7], the FDA radar data processing method has been introduced systematically. FDA is widely suggested for joint anglerange localization of targets. In [8], the FDA sensors are divided into multiple subarrays and the transmit beamspace matrix is optimized with the use of convex optimization. In [9], a subarray scheme on FDA radar has been devised for range and angle estimation. A simple rangeangle localization of targets by uniform linear array (ULA) doublepulse FDA radar has been proposed in [10]. FDAMIMO radar [11,12,13], as a combination of FDA technology and MIMO radar, has received much attention and been investigated in many topics such as target detection [14, 15], jammer suppression [16], parameter estimation [17] and resolution evaluation [18], etc. Typically, the researches on joint parameter estimation in FDAMIMO radar focus on some efficient solutions given by subspacebased methods, such as MUSIC, ESPRIT and combined MUSICESPRIT [19,20,21]. In [22], the ambiguity function and ESPRIT based method has been presented for FDAMIMO radar target localization. In [23], a realvalued subspace decomposition method has been proposed for joint angle and range estimation in bistatic FDAMIMO radar. A joint range, angle and Doppler estimation method for FDAMIMO radar has been introduced [24], in which 3D search problem is simplified into three 1D search problems which reduces the computation complexity. In [25], a spatial smoothing technique combined with ESPRIT algorithm has been proposed to obtain angle, range and velocity in FDAMIMO radar. Different from these methods, atomic normbased method has also been proposed to solve the rangeangle estimation problem [26,27,28].
In practice, due to the reflected signals composed of electric and magnetic components, polarization diversity is of great significance in target identification when multiple targets are so closely spaced that they cannot be distinguished well in spatial domain. The multiparameter estimation problem in the polarimetric MIMO radar have been investigated, e.g., in [29,30,31,32,33]. More recently, the FDAMIMO radar with polarization sensitive array, which consider polarization information combined with range and angle information, has been studied in [34,35,36], which can improve the performance of target identification and increase the anglerangepolarization resolution. A sparse reconstruction beamforming method has been presented in [34]. A successive ESPRIT algorithm has been developed to estimate the anglerangepolarization parameters in [35]. In [36], a sparse polarization sensitive FDAMIMO radar with coprime frequency offsets has been explored based on successive propagator method.
Despite the fact that the performance of FDAMIMO radar multiple parameter estimation has been improved in the aforementioned approaches, the multidimensional structure is rearranged into matrices by means of stacking operation. Thus, the original data structure cannot be reserved well, resulting in the damage of the inherent characteristics of interest, especially when the number of snapshots is small. Recently, the tensorbased algorithm has received much attention as a more natural approach to store and manipulate multidimensional data. In [37], a broad overview of tensor analysis has been provided in wireless communications and MIMO radar including basic tensor operations, common tensor decompositions. The higherorder extensions of the singular value decomposition (SVD), i.e., HOSVD [38], as a Tucker tensor model [39], has played an important role in tensor signal processing. Several Tucker tensorbased methods have been proposed such as in [40,41,42,43,44] to obtain the directionofdeparture and directionofarrival estimation in bistatic MIMO radar using the HOSVD and covariance matrix reconstruction. The use of tensor representation allows us to further exploit the structure inherent in the array data.
In this article, we investigate the problem of joint range, angle and polarization parameter estimation based on Tucker tensor decomposition in a monostatic polarimetric FDAMIMO radar system. First, we develop a thirdorder tensor signal model for polarimetric FDAMIMO radar. Then, we, respectively, present two approaches named as Tucker covariance reconstruction (TCR) and Tucker signal subspace (TSS) to jointly estimate the range, angle and polarization parameters of targets. We construct the fourthorder covariance tensor, perform its truncated HOSVD for eigenstructure extraction, and utilize multiple shiftinvariance operations to achieve the rangeanglepolarization parameters with proper pairing. The detailed analyses concerning complexity as well as Cramér–Rao bound (CRB) for range, angle and polarization estimation are also provided. The performance of the two proposed algorithms are compared with that of the existing ESPRIT method in numerical experiments. The proposed tensorbased algorithms utilize the inherent multidimensional properties and avoid the matrix stacking operation that may bring error accumulation. Thus, the estimation accuracies can be enhanced especially when the number of snapshots is small. Furthermore, polarization diversity is exploited in FDAMIMO radar to distinguish multiple targets with close angles or close ranges, which can improve the capability of multitarget identification in FDAMIMO radar.
The rest of this article is organized as follows. The preliminaries of tensor are introduced in Sect. 2. The methods are described in Sect. 3, in which a tensor signal model for polarimetric FDAMIMO radar is developed, the Tucker tensor decompositionbased algorithms for joint angle, range and polarization estimation are presented, and the performance analysis is given including the algorithm complexity and CRB. In Sect. 4, the simulation results and discussion are given to verify the effectiveness of the proposed tensorbased algorithms. Finally, a conclusion is drawn in Sect. 5.
Notations: To facilitate the distinction between scalars, matrices, and tensors, the following notation is used: Scalars are denoted as italic letters, column vectors as lowercase boldface letters, matrices as boldface capitals, and tensors are written as boldface calligraphic letters. The superscript\(\left( \cdot \right) ^T\), \(\left( \cdot \right) ^H\) and \(\left( \cdot \right) ^{ *}\) denote transpose, Hermitian transpose and conjugate operators, respectively; diag\(\left( \cdot \right)\) denotes the diagonalization operation, rank \(\left( \cdot \right)\) and det\(\left( \cdot \right)\) denote rank operator and the determinant of a matrix, respectively. vec\(\left( \cdot \right)\) denotes the vectorization operator, tr\(\left( \cdot \right)\) is the trace operator. ang\(\left( \cdot \right)\) denotes the angle operator and Re\(\left( \cdot \right)\) is the real part operator of a complex number.
2 Preliminaries
A tensor is a multidimensional array [45]. An Nth order tensor can be expressed as \({\varvec{\mathcal {X}}}\in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_N}\). The fibers are the higherorder analog of matrix rows and columns. The tensor operations in our paper are consistent with [46] which are listed as follows:
Unfolding or Matricization of the tensors. The moden unfolding of an Nth order tensor \({\varvec{\mathcal {X}}} \in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_N}\) is denoted by \([{\varvec{\mathcal {X}}}]_{\left( n \right) }\). The \((i_1,i_2, \cdots ,i_N )\)th element of \({\varvec{\mathcal {X}}}\) maps to the \((i_n,j)\)th element of \([{\varvec{\mathcal {X}}}]_{\left( n \right) }\), where \(j = 1 + \sum \nolimits _{k = 1, k \ne n}^N {(i_k  1)J_k}\) with \(J_k = \prod \nolimits _{m = 1,m \ne n}^{k  1} {I_m}\).
Moreover, a matrix unfolding of the tensor \({\varvec{\mathcal {X}}}\) along the nth mode is denoted by \([{\varvec{\mathcal {X}}}]_{\left( n \right) }\) and can be understood as a matrix containing all the nmode vectors of the tensor \({\varvec{\mathcal {X}}}\). The order of the columns is chosen in accordance with [46].
The nmode inner product of a tensor \({\varvec{\mathcal {X}}}\in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_N }\) and a tensor \({\varvec{\mathcal {Y}}}\in {\mathbb {C}}^{J_1 \times J_2 \times \cdots \times J_M }\) is defined, when \(I_n=J_n\), \(n \le \min \{ M,N\}\), it is given by \(\varvec{\mathcal {Z}}=\varvec{\mathcal {X}}\bullet _n\varvec{\mathcal {Y}}\in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_{n1} \times I_{n+1} \times \cdots \times I_N \times J_1 \times J_2 \times \cdots \times J_{n1} \times J_{n+1} \times \cdots \times J_M }\), where
The nmode product of a tensor \({\varvec{\mathcal {X}}}\in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_N }\) and a matrix \({{\textbf {A}}} \in {\mathbb {C}}^{J_n \times I_n}\) along the nth mode is denoted as \({\varvec{\mathcal {Y}}} = {\varvec{\mathcal {X}}}\times _{n} {{\textbf {A}}} \in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_{n  1}\times J_n \times I_{n + 1}\times \cdots \times I_N }\), which is calculated by
It may be visualized by multiplying all nmode vectors of \({\varvec{\mathcal {X}}}\) from the lefthand side by the matrix \({{\textbf {A}}}\). The moden product admits the following properties:
The higherorder SVD (HOSVD) or Tucker decomposition of a tensor \({\varvec{\mathcal {X}}}\in {\mathbb {C}}^{I_1 \times I_2 \times \cdots \times I_N }\) is given by
where \({\varvec{\varsigma }} \in {\mathbb {C}}^{J_1 \times J_2 \times \cdots \times J_N }\) is the core tensor which satisfies the allorthogonality conditions [46] and \({\textbf {A}}_n \in {\mathbb {C}}^{I_n \times J_n}, n = 1,2, \cdots N\), are the unitary matrices of nmode singular vectors.
3 Methods
3.1 Tensor signal model of polarimetric FDAMIMO radar
As illustrated in Fig. 1, we consider a monostatic polarimetric FDAMIMO radar equipped with a transmit FDA having M sensors and a receive array having N crossdipole sensors, both are uniform linear arrays. Assume that the antennas are of ideal, identical isotropic sensors, and the range of targets is much larger than the apertures of transmit and receive arrays. The transmit array with M sensors simultaneously send M waveforms with identical bandwidth but different frequencies. Taking the first transmit sensor as the reference sensor, the carrier frequency of the mth sensor is
where \(f_0\) is the reference frequency, and \(\Delta f\) denotes the frequency increment between adjacent array sensors which is far less than the frequency \(f_0\). The transmitting signal of the mth sensor can be expressed as
where \(\psi _m \left( t \right)\) is the baseband waveform envelope, and T is the radar pulse duration. It is assumed that the transmitted M waveforms of each sensor are orthogonal to each other.
For the pth farfield target, \(p = 1,2, \cdots ,P\), assume that it is localized at angle \(\theta _p\) and range \(r_p\). The propagation time from the mth transmit sensor to the nth receive sensor can be written as
where \(d_t\) and \(d_r\), respectively, denote the intersensor spacing at the transmitter and receiver, which are no more than half a wavelength \(\lambda _0=c/f_0\), with c being the propagation speed. \({2r_p}/{c}\) is the common propagation time delay, and the next two terms represent the time shifts brought by the transmitter and receiver arrays, respectively.
At the receiver array of the polarimetric FDAMIMO radar, the crossdipoles are used to measure the polarization states of a transverse electromagnetic (TEM) wave and improve target identification capability by exploiting polarization diversity. The polarization state of TEM wave can be represented by parameters \((\gamma , \eta )\), which are two phase descriptors of a polarized signal [47, 48]. Thus the complex electric field vector is
where A is the electric field amplitude. \(\tan \gamma\) represents the ratio of the electric field amplitude in the X direction to that in the Y direction, \(\gamma \in \left[ {0,\frac{\pi }{2}} \right]\), and \(\eta\) signifies the phase difference between the electric fields in the Y direction and X direction, \(\eta \in \left[ {\pi , \pi } \right]\).
Under the assumption of narrowband and farfield signals, the signals are reflected by P farfield targets. With a crossdipole uniform linear array used at the receiver, the polarization vector for the pth target can be written as
where \(0 \le \gamma _p \le {\pi / 2}\), \({\pi } \le \eta _p < {\pi }\). Thus, the received signal from the mth transmit sensor to the nth crossdipole sensor can be expressed as
In noisy environment, after sampling, recombination and matched filtering with the orthogonal transmit waveform \(\psi _m(t)\), the output of a polarimetric FDAMIMO radar can be reformulated as
for \(m = 1, 2, \cdots , M\); \(n = 1, 2, \cdots , N\), and \(l = 1, 2, \cdots , L\), where L is the number of snapshots. \(\beta _p^{(l)}\) is the complex scattering coefficient of the pth target in the lth snapshot. \(\textbf{y}_{m,n,l}\) and \(\textbf{z}_{m,n,l}\) denote the output data and noise with respect to the mth transmit sensor, nth receive sensor and the lth snapshot, respectively.
In a conventional way, \({\textbf {y}}_{m,n,l}\) is first arranged into a matrix \(\textbf{Y}^{(l)}\in \mathbb {C}^{2N\times M}\) as
where \({\textbf {b}}^{(l)}=[\beta _1^{(l)}, \cdots , \beta _P^{(l)}]\) denotes the complex scattering coefficient vector related to the radar cross section (RCS). \({\textbf {Z}}^{(l)}\) represents a Gaussian white noise matrix. \({{\textbf {V}}}({\theta }, {\gamma }, {\eta })=[\textbf{v}(\theta _1, \gamma _1, \eta _1), \cdots , \textbf{v}(\theta _P, \gamma _P, \eta _P)]\in \mathbb {C}^{2 \times P}\). \(\textbf{A}_t({\theta }, {r})=[\textbf{a}_t(\theta _1, r_1),\textbf{a}_t(\theta _2, r_2),\cdots ,\textbf{a}_t(\theta _P, r_P)]\in \mathbb {C}^{M \times P}\). \(\textbf{A}_r({\theta })=[\textbf{a}_r(\theta _1),\textbf{a}_r(\theta _2),\cdots ,\textbf{a}_r(\theta _P)]\in \mathbb {C}^{N \times P}\). The pth steering vector \({\textbf {a}}_t (\theta _p, r _p)\) of the transmit array can be written as
and the pth steering vector \({\textbf {a}}_r (\theta _p )\) of the receive array can be written as
By the vectorization of \(\textbf{Y}^{(l)}\), the observed matrix can be rewritten as a \(2MN \times L\) vector \({{\textbf {y}}}^{\left( l \right) } = {\textbf {A}}(\theta , \phi ,\gamma ,\eta ){\textbf {b}}^{(l)} + {\textbf {z}}^{\left( l \right) }\), where \({\textbf {z}}^{(l)}\) denotes the noise vector, whose elements are white Gaussian distributed with mean zero and variance \(\sigma\). \({\textbf {A}}(\theta , r, \gamma , \eta )=\left[ {\textbf {a}}_1 (\theta , r, \gamma , \eta ), \cdots , {\textbf {a}}_P (\theta , r, \gamma , \eta ) \right] \in \mathbb {C}^{2MN \times P}\) represents the joint transmitreceivepolarization steering matrix, with
where \(\otimes\) denotes the Kronecker product. Then, the data in L snapshots for \({{\textbf {y}}}^{\left( l \right) }\) are collected to form a matrix \({{\textbf {Y}}} \in \mathbb {C}^{2MN \times L}\),
where \({{\textbf {Y}}}=\left[ {{{\textbf {y}}}^{(1)},\cdots ,{{\textbf {y}}}^{(L)} } \right]\), \({{\textbf {B}}} = \left[ {{{\textbf {b}}}^{(1)},\cdots ,{{\textbf {b}}}^{(L)} } \right]\) and \({{\textbf {Z}}} = \left[ {{{\textbf {Z}}}^{(1)},\cdots , {{\textbf {Z}}}^{(L)} } \right]\) represent the observation matrix, target scattering matrix and noise matrix, respectively, and
where \(\diamond\) represents the Khatri–Rao product.
Note that the signal model in (18) is expressed as the form of a matrix via a stacking operation. Such a highly structured matrix cannot completely capture the inherent multidimensional structure. In our paper, \({\textbf {y}}_{m,n,l}\) in (13) is directly arranged into a thirdorder tensor \({\varvec{\mathcal {Y}}}\in {\mathbb {C}}^{M\times 2N\times L}\), then
where \({\varvec{\mathcal {\iota }}}\in {\mathbb {C}}^{P \times P \times P}\) is an identity tensor. Obviously, we have \({\textbf {Y}} = [{\varvec{\mathcal {Y}}}]_{(3)}^T\) and \({\textbf {Z}} = [{\varvec{\mathcal {Z}}}]_{(3)}^T\). The thirdorder tensor of the polarimetric FDAMIMO radar can be shown in Fig. 2.
Based on the constructed tensor \({\varvec{\mathcal {Y}}}\) in (20), the problem of interest is to jointly estimate the target parameters in polarimetric FDAMIMO radar, including the angle \(\theta _p\), range \(r _p\) and two polarization parameters \(\gamma _p\) and \(\eta _p\), for \(p=1 \cdots , P\).
3.2 Tucker tensor decomposition for joint angle, range and polarization estimation
In this section, Tucker tensor decomposition methods are explored for anglerangepolarization estimation in FDAMIMO radar. We present two methods: Tucker covariance reconstruction (TCR) algorithm and Tucker signal subspace (TSS) algorithm.
3.2.1 Tucker covariance reconstruction (TCR) algorithm
According to the definition of the product of two tensors, a fourthorder tensor \({\varvec{\mathcal {R}}}\in {\mathbb {C}}^{M \times 2N \times M \times 2N}\) for the observation tensor \({\varvec{\mathcal {Y}}}\) can be calculated by
Then the HOSVD of the tensor \({\varvec{\mathcal {R}}}\) is calculated as
where \({\varvec{\varsigma }}\in {\mathbb {C}}^{M \times 2N \times M \times 2N}\) is the core tensor, and \({{\textbf {U}}}_i\) for \(i = 1,2,3,4\) denote the left singular vector matrices of the imodule expansion of \({\varvec{\mathcal {R}}}\), expressed as
with \({{\textbf {U}}}_1 = {{\textbf {U}}}_3^*\) and \({{\textbf {U}}}_2 = {{\textbf {U}}}_4^ *\).
To eliminate the noise, a truncated HOSVD [38] of the tensor \({\varvec{\mathcal {R}}}\), which only contain the signal subspace component, can be obtained as
where \({\varvec{\varsigma }}_{s} \in {\mathbb {C}}^{P \times P \times P \times P}\) is the core tensor. \({{\textbf {U}}}_{1\,s} \in {\mathbb {C}}^{M \times P}\) and \({{\textbf {U}}}_{2\,s}\in {\mathbb {C}}^{2N \times P}\) contain the column vectors of \({{\textbf {U}}}_1\) and \({{\textbf {U}}}_2\) corresponding to the P dominant singular values, respectively.
First, a reconstructed covariance matrix \({{\textbf {R}}}_s \in {\mathbb {C}}^{2MN \times 2MN}\) is formed based on the tensor \({\varvec{\mathcal {R}}}_s \in {\mathbb {C}}^{M \times 2N \times M \times 2N}\) in (24) by Hermitian unfolding, which is given by
where \({\textbf {{R}}}\) is the covariance matrix of the observation matrix \({\textbf {{Y}}}\), which can be estimated from finite snapshots in practice, given by \({{\hat{{\textbf {R}}}}} = \frac{1}{L} {{{\textbf {YY}}}^H }\). Actually, \({{\textbf {U}}}_{is}\) (\(i=1, 2\)) is the left singular vector matrix of \([{\varvec{\mathcal {R}}}_s]_{(i)}\) corresponding to the first P largest singular values on the diagonal element of \([{\varvec{\mathcal {R}}}_s]_{(i)}\). Both \({{\textbf {U}}}_{1\,s} {{\textbf {U}}}_{1\,s}^H\) and \({{\textbf {U}}}_{2\,s} {{\textbf {U}}}_{2\,s}^H\) are unitary matrices.
Then, the eigenvalue decomposition (EVD) of \({{\textbf {R}}}_s\) is performed which can be written as \({{\textbf {R}}}_s = {{\textbf {E}}}_s {\varvec{\Lambda }} _s {{\textbf {E}}}_s^H + {{\textbf {E}}}_n {\varvec{\Lambda }} _n {{\textbf {E}}}_n^H\), where \({\varvec{\Lambda }} _s\) is a diagonal matrix whose elements on the diagonal line are the P dominant eigenvalues. \({{\textbf {E}}}_s \in {\mathbb {C}}^{2MN \times P}\) denotes the signal subspace containing the eigenvectors corresponding to the P eigenvalues. Also, \({\varvec{\Lambda }} _n\) is a diagonal matrix with the remaining \(2MN  P\) eigenvalues on its diagonal line, and \({{\textbf {E}}}_n \in {\mathbb {C}}^{2MN \times (2MN  P)}\) represents the noise subspace which contains the eigenvectors corresponding to the \(2MN  P\) eigenvalues. Therefore, the rangle, angle and polarization parameters of each target can be estimated from \({{\textbf {E}}}_s\). Obviously, \({{\textbf {E}}}_s\) spans the same signal subspace with \({{\textbf {A}}}(\theta , r, \gamma ,\eta )\) in (19). As a result, there exists a unique nonsingular fullrank matrix \({\textbf {T}}\) such that \({{\textbf {E}}}_s = {{\textbf {A}}}(\theta , r, \gamma , \eta ){\textbf {T}}\).
Next, some submatrices can be obtained by proper processing of \({\textbf {{E}}}_s\), which is based on the shift invariance of \({{\textbf {A}}}(\theta , r, \gamma ,\eta )\) maintained by \({{\textbf {E}}}_s\). Firstly, \({{\textbf {E}}}_s \in {\mathbb {C}}^{2MN \times P}\) is divided into M blocks, each of which is a matrix of \(2N \times P\). Since there exists shift invariance for both transmit and receive arrays, different parameters can be estimated by choosing the following different submatrices:
For angle estimation, the shiftinvariance property of the receive array is exploited. Define two selection matrices \({\textbf {J}}_{r1}=[{\textbf {I}}_{N1},{\textbf {O}}_{(N1)\times 1}]\) and \({\textbf {J}}_{r2}=[{\textbf {O}}_{(N1)\times 1}, {\textbf {I}}_{N1}]\), then we have
where \({\varvec{\Phi }} _r= \textrm{diag}\left( \phi _{r1}, \phi _{r2}, \cdots , \phi _{rP} \right)\) is the rotational matrix associated with the receive array, \(\phi _{rp} = e^{ j2\pi (d_r/\lambda _0 )\sin \theta _p}\). Thus we can construct two submatrices \({{\textbf {E}}}_{r1} \in {\mathbb {C}}^{2M(N  1) \times P}\) and \({{\textbf {E}}}_{r2} \in {\mathbb {C}}^{2M(N  1) \times P}\) with the first and last \(2(N  1)\) rows of each block of \({{\textbf {E}}}_s\), respectively, i.e.,
then
where \({\varvec{\Psi }} _r = {\textbf {T}}^{  1} {\varvec{\Phi }} _r {\textbf {T}}\) denotes the EVD of \({\varvec{\Psi }} _r\), with \({\varvec{\Phi }} _r\) and \({\textbf {T}}^{  1}\) representing the eigenvalue and eigenvector matrices of \(\varvec{\Psi } _r\), respectively. Thus the estimate of the angle \(\theta _p\) can be obtained from \(\varvec{\Phi } _r\).
For range estimation, the shiftinvariance property of the transmit array is used. Define two selection matrices \({\textbf {J}}_{t1}=[{\textbf {I}}_{M1},{\textbf {O}}_{(M1)\times 1}]\) and \({\textbf {J}}_{t2}=[{\textbf {O}}_{(M1)\times 1}, {\textbf {I}}_{M1}]\), then
where \({\varvec{\Phi }} _t= \textrm{diag}\left( \phi _{t1}, \phi _{t2}, \cdots , \phi _{tP} \right)\) is the rotational matrix with respect to the transmit array, where \(\phi _{tp}\) satisfies
which contains both the range \(r_p\) and angle \(\theta _p\). Therefore, we, respectively, generate two submatrices \({{\textbf {E}}}_{t1} \in {\mathbb {C}}^{2(M  1)N \times P}\) and \({{\textbf {E}}}_{t2} \in {\mathbb {C}}^{2(M  1)N \times P}\) using the first and last \(M1\) blocks of \({{\textbf {E}}}_{s}\), i.e.,
Thus we have
where \({\varvec{\Psi }} _t = {\textbf {T}}^{  1} {\varvec{\Phi }} _t {\textbf {T}}\) stands for the EVD of \(\varvec{\Psi } _t\), and \({\varvec{\Phi }} _t\), with \({\textbf {T}}^{  1}\) being the eigenvalue and eigenvector matrices of \(\varvec{\Psi } _t\), respectively. Upon the angle \(\theta _p\) has been obtained from \(\varvec{\Phi } _r\), we can further calculate the estimate of the range \(r _p\) from \(\varvec{\Phi } _t\).
For polarization parameter estimation, let us define two selection vectors \({\textbf {J}}_{v1}=[0, 1]\) and \({\textbf {J}}_{v2}=[1, 0]\), then we have
where \({\varvec{\Phi }} _v= \textrm{diag}\left( \phi _{v1}, \phi _{v2}, \cdots , \phi _{vP} \right)\), and \(\phi _{vp}\) is the ratio of the two elements of the pth polarization vector,
which contains the angle \(\theta _p\) and two polarization parameters \(\gamma _p\) and \(\eta _p\). Thus, we form two submatrices \({{\textbf {E}}}_{v1} \in {\mathbb {C}}^{MN \times P}\) and \({{\textbf {E}}}_{v2} \in {\mathbb {C}}^{MN \times P}\) from the even and odd rows of \({{\textbf {E}}}_{s}\), respectively, i.e.,
Then
where \({\varvec{\Psi }} _v = {\textbf {T}}^{  1} {\varvec{\Phi }} _v {\textbf {T}}\), with \({\varvec{\Phi }} _v\) and \({\textbf {T}}^{  1}\) being the eigenvalue and eigenvector matrices of \(\varvec{\Psi } _v\), respectively. Since the angle \(\theta _p\) has been obtained from \(\varvec{\Phi } _r\), we can further achieve the estimates of the two polarization parameters \(\gamma _p\) and \(\eta _p\) from \(\varvec{\Phi } _v\).
In the above analysis, \({\varvec{\Psi }} _r\), \({\varvec{\Psi }}_t\) and \({\varvec{\Psi }}_v\) can be calculated from (28), (32) and (36), respectively, by
where \(\dag\) denotes the pseudoinverse operation of a matrix.
Finally, the estimates of angle \(\theta _p\), range \(r _p\), polarization parameters \(\gamma _p\) and \(\eta _p\) for \(p=1,\cdots ,P\), are calculated by
Parameter pairing: In the case of multiple targets, we observe that the EVDs of \({\varvec{\Psi }}_r\), \({\varvec{\Psi }} _t\) and \({\varvec{\Psi }}_v\) share the same eigenvector matrix \({\textbf {T}}^{  1}\) but may have different sort in columns due to the existence of noise. To correctly estimate the multitarget parameters, the following pairing procedure is proposed: First, perform the EVD of \({\varvec{\Psi }}_r\) to obtain its eigenvector matrix \({\textbf {Q}}\) which corresponds to the eigenvalue matrix \({\varvec{\Phi }}_r\). Then, construct a new signal subspace matrix \({\tilde{{\textbf {E}}}}_{s} = {{\textbf {E}}}_{s}{} {\textbf {Q}}\). Next, obtain the submatrices \({\tilde{{\textbf {E}}}}_{t1}\), \({\tilde{{\textbf {E}}}} _{t2}\) and \({\tilde{{\textbf {E}}}} _{v1}\), \({\tilde{{\textbf {E}}}} _{v2}\) from \({\tilde{{\textbf {E}}}}_{s}\), respectively, and similarly calculate \({\tilde{\varvec{\Psi }}}_t\) and \({\tilde{\varvec{\Psi }}}_v\) according to (37). The diagonal elements of \({\tilde{\varvec{\Psi }}}_t\) and \({\tilde{\varvec{\Psi }}}_v\) contain the paired range and polarization parameters, which correspond to the angle parameter of the pth target.
The steps of the proposed TCR algorithm for range, angle and polarization estimation is given in Alg. 1.
3.2.2 Tucker signal subspace (TSS) algorithm
Compared with the reconstruction of covariance matrix \({{\textbf {R}}}_s \in {\mathbb {C}}^{2MN \times 2MN}\), the direct utilization of the tensor \({\varvec{\mathcal {R}}}_s \in {\mathbb {C}}^{ M \times 2N \times M \times 2N}\) can make better use of the multidimensional structure of tensors. From the structure of the Tucker signal subspace tensor \({\varvec{\mathcal {R}}}_s\), we can see that it contains the signal subspaces corresponding to the transmit and receive steering matrices, from which the multiple parameters can be obtained by using the properties of shift invariance in both transmit and receive arrays, respectively.
At first, like the TCR algorithm, we calculate the fourthorder tensor \({\varvec{\mathcal {R}}}\in {\mathbb {C}}^{M \times 2N \times M \times 2N}\) for \({\varvec{\mathcal {Y}}}\) as \({\varvec{\mathcal {R}}} = \frac{1}{L}{\varvec{\mathcal {Y}}}{\bullet }_{3}{\varvec{\mathcal {Y}^*}}\). Recall that the truncated HOSVD of \({\varvec{\mathcal {R}}}\) is performed as
where \({{\textbf {U}}}_{1s} \in {\mathbb {C}}^{M \times P}\) and \({{\textbf {U}}}_{2s} \in {\mathbb {C}}^{2N \times P}\) consist of the P dominant eigenvectors of \({{\textbf {U}}}_1\) and \({{\textbf {U}}}_2\), respectively. Obviously, \({{\textbf {U}}}_{1\,s}\) and \({{\textbf {A}}}_t(\theta , r)\) span the same signal subspace, so there exists a unique nonsingular fullrank matrix \({\textbf {T}}_1\) satisfying \({{\textbf {U}}}_{1\,s} = {{\textbf {A}}}_t(\theta , r){\textbf {T}}_1\). Similarly, \({{\textbf {U}}}_{2\,s}\) and \({\tilde{{\textbf {A}}}}_{r}(\theta ,\gamma ,\eta )\) span the same signal subspace, and there also exists a unique nonsingular fullrank matrix \({\textbf {T}}_2\) satisfying \({{\textbf {U}}}_{2\,s} = {\tilde{{\textbf {A}}}}_{r}(\theta , \gamma , \eta ){\textbf {T}}_2\), where \({\tilde{{\textbf {A}}}}_{r}(\theta , \gamma , \eta )={{\textbf {A}}_r} (\theta ) \diamond {{\textbf {V}}}(\theta , \gamma , \eta )\). Then we can, respectively, use the shift invariance properties to both transmit and receive arrays to estimate the angle, range and polarization parameters.
For angle estimation, according to the shiftinvariance property of the receiving array, we have
where \({\textbf {J}}_{r1}=[{\textbf {I}}_{N1},{\textbf {O}}_{(N1)\times 1}]\) and \({\textbf {J}}_{r2}=[{\textbf {O}}_{(N1)\times 1}, {\textbf {I}}_{N1}]\), and the rotational matrix is \({\varvec{\Phi }}_r=\textrm{diag}(\phi _{r1}, \phi _{r2}, \cdots , \phi _{rP})\) with \(\phi _{rp}=e^{ j2\pi (d_r/\lambda _0)\sin \theta _p}\). Thus, the submatrices \({{\textbf {U}}}_{r1}\in {\mathbb {C}}^{2(N  1) \times P}\) and \({{\textbf {U}}}_{r2} \in {\mathbb {C}}^{2(N  1) \times P}\) are, respectively, constructed by selecting the first \(2(N  1)\) and last \(2(N  1)\) rows of \({{\textbf {U}}}_{2s}\), i.e.,
then
where \({\textbf {D}} _r= {\textbf {T}}_2^{  1} {\varvec{\Phi }} _r{\textbf {T}}_2\). We see that \({\varvec{\Phi }}_r\) is the eigenvalue matrix of \({\textbf {D}} _r\). From the diagonal elements of \({\varvec{\Phi }}_r\), we can estimate the angle \(\theta _p\) for the pth target.
For range estimation, according to the shiftinvariance property of the transmit array, we have
where \({\textbf {J}}_{t1}=[{\textbf {I}}_{M1},{\textbf {O}}_{(M1)\times 1}]\) and \({\textbf {J}}_{t2}=[{\textbf {O}}_{(M1)\times 1}, {\textbf {I}}_{M1}]\). The rotational matrix is \({\varvec{\Phi }} _t=\mathrm{{diag}}( \phi _{t1}, \phi _{t2}, \cdots , \phi _{tp})\), where \(\phi _{tp} = e^{ j2\pi ((d_t/\lambda _0 )\sin \theta _p ({\Delta f}/c)2r_p)}\). Therefore, two submatrix \({{\textbf {U}}}_{t1} \in {\mathbb {C}}^{(M  1) \times P}\) and \({{\textbf {U}}}_{t2} \in {\mathbb {C}}^{(M  1) \times P}\) are constructed by using the first \(M1\) and last \(M1\) rows of \({{\textbf {U}}}_{1s}\), respectively, i.e.,
then
where \({\textbf {D}} _t = {\textbf {T}}_1^{  1} {\varvec{\Phi }} _t {\textbf {T}}_1\). We observe that \({\varvec{\Phi }}_t\) is the eigenvalue matrix of \({\textbf {D}} _t\). Since \({\varvec{\Phi }} _t\) contains both the angle and range parameters, the range \(r _p\) can be calculated when \(\theta _p\) has been estimated.
For polarization parameter estimation, the shiftinvariance property of the receive array is utilized again, then we have
where \({\textbf {J}}_{v1}=[0, 1]\) and \({\textbf {J}}_{v2}=[1, 0]\), and \({\varvec{\Phi }} _v= \textrm{diag}(\phi _{v1}, \phi _{v2}, \cdots , \phi _{vP})\), with \(\phi _{vp} = \frac{{  \cos \gamma _p }}{{\sin \gamma _p \cos \theta _p e^{j\eta _p} }}\) being the ratio of the first and second elements of the polarization vector \({\textbf {v}}(\theta _p, \gamma _p, \eta _p)\). We select the even rows from \({{\textbf {U}}}_{2\,s}\) to form the submatrix \({{\textbf {U}}}_{v1} \in {\mathbb {C}}^{N \times P}\), and odd rows to generate the submatrices \({{\textbf {U}}}_{v2}\in {\mathbb {C}}^{N \times P}\), i.e.,
then
where \({\textbf {D}} _v = {\textbf {T}}_2^{  1} {\varvec{\Phi }} _v{\textbf {T}}_2\), and \({\varvec{\Phi }} _v\) is the eigenvalue matrix of \({\textbf {D}}_v\). Thus, \(\gamma _p\) and \(\eta _p\) can be calculated if \(\theta _p\) has been obtained.
Therefore, the estimates of \(\theta _p\), \(r _p\), \(\gamma _p\) and \(\eta _p\) for \(p=1,\cdots ,P\), are obtained from the eigenvalues of \({\textbf {D}} _r\), \({\textbf {D}} _t\) and \({\textbf {D}} _v\), and finally calculated by the abovegiven (38)(41).
Parameter pairing: It is necessary to match the parameters of angle, range and polarizations for multiple targets in the proposed TSS algorithm. We note that \({\textbf {D}}_t\) and \({\textbf {D}} _r\) own different eigenvector matrices \({\textbf {T}}_1^{  1}\) and \({\textbf {T}}_2^{  1}\), so we cannot match them according to eigenvectors. Therefore, the pairing procedure is based on sorting the diagonal elements of \({\varvec{\Phi }} _r\), \({\varvec{\Phi }} _t\) and \({\varvec{\Phi }} _v\) according to the criterion of
for \(p = 1,2, \cdots , P\).
The steps of the proposed TSS algorithm for range, angle and polarization estimation is given in Alg. 2.
3.3 Performance analysis
In this section, we analyze the computational complexity of the proposed algorithms, and provide the CRB of joint range, angle and polarization estimation in polarimetric FDAMIMO radar.
3.3.1 Computational complexity
According to some basic computational complexities of the SVD algorithm and the subspace methods in [35, 41], we analyze the complexities of the two proposed algorithms. The main computational burden of the proposed TCR algorithm comes from the calculation of the covariance tensor, HOSVD, covariance reconstruction, eigenvalue decomposition and multiparameter estimation. Among them, \(\mathcal {O}\{(2MN)^2 L\}\) results from the fourthorder covariance tensor calculation. \(\mathcal {O}\{4(2\,M)^2 N^2 P\}\) results from the truncated HOSVD in tensor decomposition method. \(\mathcal {O}\{(2MN)^3\}\) results from the covariance reconstruction. \(\mathcal {O}\{(2MN)^2 P\}\) results from the eigenvalue decomposition of the signal covariance matrix. \(\mathcal {O}\{2M(N1)(2P)^2+(2P)^3+P^3\}\), \(\mathcal {O}\{2(M1)N(2P)^2+(2P)^3+P^3\}\) and \(\mathcal {O}\{MN(2P)^2+(2P)^3+P^3\}\) result from the estimation of angle, range and polarizations using shiftinvariance method, respectively. Thus, the complexity of the proposed Tucker covariance reconstruction algorithm is \(\mathcal {O}\{8\,M^3N^3+20\,M^2 N^2 P+4\,M^2 N^2\,L+4(5MN2\,M2N)P^2+27P^3\}\). Similarly, the main computational burden of the proposed TSS algorithm is the calculation of the covariance tensor, HOSVD, eigenvalue decomposition, multiparameter estimation and parameter pairing, in which \(\mathcal {O}\{(N1)(2P)^2+(2P)^3+P^3\}\), \(\mathcal {O}[2(M1)(2P)^2+(2P)^3+P^3\}\) and \(\mathcal {O}\{M(2P)^2+(2P)^3+P^3\}\), respectively, result from the angle, range and polarization estimation exploiting the shiftinvariance method. \(\mathcal {O}\{P^3\}\) results from the pairing between the parameters. Thus, the complexity of the proposed Tucker signal subspace algorithm is \(\mathcal {O}\{16\,M^2 N^2 P+4\,M^2 N^2\,L+4(3\,M+N3)P^2+28P^3\}\).
3.3.2 Cramér–Rao bound
Under the hypothesis of target scattering signals stochastic and unknown to the receiver, the performance bound of multiparameter estimation can be provided by deriving the expression of stochastic CRB [49,50,51]. The CRB of joint range, angle and polarization estimation in polarimetric FDAMIMO radar can be given by
where \(\odot\) denotes the Hadamard product, and \({\textbf {1}}_{4 \times 4}\) is a \(4 \times 4\) matrix with all the entities are ones. \({\varvec{\Pi }}_{{\textbf {A}}}^ \bot = {\textbf {I}}_{2MN}  {\textbf {A}}({\textbf {A}}^H {\textbf {A}})^{  1} {\textbf {A}}^H\) and \({\textbf {R}} = {\textbf {APA}}^H + \sigma {\textbf {I}}_{2MN}\), where \({\textbf {P}} = E\left[ {{\textbf {B}}{} {\textbf {B}}^H }\right]\) is the covariance matrix of target signals, and
is the partial derivative matrix of \({\textbf {A}}\) with respect to \({\varvec{\theta }}, {\varvec{r}}, {\varvec{\gamma }}\) and \({\varvec{\eta }}\), with \({\varvec{\theta }}=[{\theta }_1, {\theta }_2,\cdots , {\theta }_P]^T\), \({\varvec{r}}=[{r}_1, {r}_2,\cdots , {r}_P]^T\), \({\varvec{\gamma }}=[{\gamma }_1, {\gamma }_2,\cdots , {\gamma }_P]^T\), \({\varvec{\eta }}=[{\eta }_1, {\eta }_2,\cdots , {\eta }_P]^T\). \(\frac{{\partial {\textbf {A}}}}{{\partial {\varvec{\theta }} }}= \left[ \frac{{\partial {\textbf {a}}_1}}{{\partial \theta _1 }}, \cdots , \frac{{\partial {\textbf {a}}_P}}{{\partial \theta _P}} \right]\), \(\frac{{\partial {\textbf {A}}}}{{\partial {\varvec{r}} }}= \left[ \frac{{\partial {\textbf {a}}_1}}{{\partial r_1 }}, \cdots , \frac{{\partial {\textbf {a}}_P}}{{\partial r_P}} \right]\), \(\frac{{\partial {\textbf {A}}}}{{\partial {\varvec{\gamma }} }}= \left[ \frac{{\partial {\textbf {a}}_1}}{{\partial \gamma _1 }}, \cdots , \frac{{\partial {\textbf {a}}_P}}{{\partial \gamma _P}} \right]\), \(\frac{{\partial {\textbf {A}}}}{{\partial {\varvec{\eta }} }}= \left[ \frac{{\partial {\textbf {a}}_1}}{{\partial \eta _1 }}, \cdots , \frac{{\partial {\textbf {a}}_P}}{{\partial \eta _P}} \right]\), where
The diagonal elements of the CRB matrix give the CRBs of the estimates of angle \(\varvec{\theta }\), range \(\varvec{r}\) and polarization parameters \(\varvec{\gamma }\) and \(\varvec{\eta }\).
4 Results and discussion
In this section, we present the numerical experiment results to illustrate the performance of the proposed Tucker tensor decompositionbased algorithms for joint anglerangepolarization estimation. Consider a polarimetric FDAMIMO radar system with intersensor spacing \(d_t = d_r = \lambda _0 /2\). The carrier frequency of the first sensor is \(f_0 = 10GHz\), and the linear frequency increment is \(\Delta f = 5kHz\). The polarization mode is assumed to be elliptical polarized, and the polarization state of each target echo is different.
First, we verify the effectiveness of the proposed TCR and TSS algorithms. Assume that the polarimetric bistatic MIMO radar has \(M = 8\) transmit sensors and \(N = 8\) crossdipole receive sensors. There are \(P = 3\) uncorrelated targets which are distributed dispersedly, whose anglerange parameters are \((\theta , r) = (40^\circ ,\,5\,{\text {km}}), (0^\circ ,\,2\,{\text {km}}), (50^\circ ,\,7\,{\text {km}})\) and the polarization parameters are \((\gamma , \eta ) = (\pi /10,\,\pi /5), (2\pi /5,\,2\pi /5), (\pi /4,\,7\pi /10)\). The signaltonoise ratio (SNR) is 10dB and the number of snapshots is \(L = 100\).
The simulation results of the TCR and TSS algorithms are shown in Figs. 3 and 4, respectively, with 100 Monte Carlo trials. From Figs. 3 and 4, it is observed that the estimates of angle, range and polarization parameters are close to their true values, and both the pairing methods given in Alg. 1 and Alg. 2 work well. The effectiveness and accuracies of the proposed algorithms can be verified from the simulation results.
Next, we compare the proposed TCR and TSS algorithms with the successive ESPRIT algorithm [35] and the successive propagator method [36] to evaluate the performance of range, angle and polarization estimation in FDAMIMO radar. Here the root mean square error (RMSE) in estimating the parameter \(\theta _p\) for the pth target is calculated by
where t denotes the number of Monte Carlo experiments. The RMSEs of \(r_p\), \(\gamma _p\) and \(\eta _p\) have the same calculation method as that of \(\theta _p\). The number of transmit sensors is \(M = 6\), and the number of crossdipole receive sensors is \(N = 6\). The number of snapshots is \(L= 5\). Assume that the SNR changes from −5 to 20dB, and three uncorrelated targets are distributed dispersedly with \((\theta , r) = (40^\circ , 4\,{\text {km}}), (30^\circ , 2\,{\text {km}}), (10^\circ , 5\,{\text {km}})\) and \((\gamma , \eta ) = (\pi /10,\pi /5)(\pi /5, 3\pi /5)(\pi /4, 2\pi /5)\). 500 Monte Carlo experiments are performed. The RMSEs of range, angle and two polarization parameters for different algorithms are shown in Fig. 5. It reveals that with the increase in SNR, the performance of the four algorithms is improved. The estimation accuracies of the two proposed tensor based algorithms are higher than that of the matrixbased successive ESPRIT algorithm and the successive propagator method, especially when the number of snapshots is small (\(L=5\)). It implies that in the proposed tensorbased methods, the inherent multidimensional structure of the array data is well reserved. However, the multidimensional data structure in the successive ESPRIT algorithm and successive propagator method is rearranged into a matrix by stacking operation, thus the original structural characteristics is destroyed, especially in a small number of snapshots. Therefore, the TSS and TCR algorithms achieve more accuracies than the successive ESPRIT algorithm and successive propagator method. Besides, the TSS algorithm is slightly superior to the TCR algorithm in accuracy by directly utilizing the signal subspace of Tucker tensor decomposition.
Then, we compare the performance of different algorithms in single snapshot environment. Assume that we conduct the simulation under the same conditions with the above experiment, except that the number of snapshots is \(L=1\). The RMSEs of the range, angle and polarization parameters with a single snapshot are shown in Fig. 6. We observe that the proposed TSS algorithm still works well even if the number of snapshots is one. In this case, the successive ESPRIT algorithm, the successive propagator method and the proposed TCR algorithm, which more or less depend on the calculation of covariance matrix, have lost the ability of parameter estimation when \(L<P\). This shows the advantage of the proposed TSS algorithm by fully capturing the tensor nature for multiple parameter estimation in FDAMIMO radar.
Next, we investigate the relationship between RMSE and the number of snapshots for different algorithms. The simulation conditions are identical with that of the above experiment except that SNR = 0 dB and the number of snapshots ranges from 1 to 29. The performance comparison of the angle, range and polarization parameters among the four algorithms is shown in Fig. 7. It is revealed that with the increase in the number of snapshots, the estimation accuracies of the four algorithms is improved. Also, when the number of snapshots is relatively small, the estimation accuracies of both the proposed TSS and TCR algorithms are higher than the successive ESPRIT algorithm and the successive propagator method. The TSS algorithm owns the highest accuracy among the four methods since it directly achieves the signal subspace from the fourthorder covariance tensor.
From the above several experiments we can observe that, by stacking the original multidimensional data into highly structured matrices may bring error accumulation and degrade the performance of multiparameter estimation, especially when the number of snapshots is small. The proposed Tucker tensor decomposition based algorithms can utilize the inherent multidimensional structure characteristics in polarimetric FDAMIMO radar, and have little damage to the original data structure. The TSS algorithm can better fit for the signal subspace estimation of multiple parameters in FDAMIMO radar, so it can locate targets more accurately.
Moreover, we compare the running time of the TCR, TSS, successive ESPRIT and successive propagator method under two different array configurations: \(M=6, N=6\), and \(M=30, N=30\). Other simulation conditions are the same as those in the second experiment. From Table 1 we observe that when M and N are relatively small, the TCR algorithm has slightly larger running time than the other three algorithms, the TSS and successive ESPRIT algorithms own similar and mediate running time, and the successive propagator method has the smallest running time. When M and N become large, the running time of the TCR algorithm significantly increases and is larger than the other three algorithms. The successive propagator method has the smallest running time but the lowest estimation accuracy. The TSS algorithm has a little larger running time than the successive propagator method but smaller running time than the TCR and successive ESPRIT algorithms. As a whole, the TSS algorithm can bring mediate complexity and higher estimation accuracy compared to the other methods.
Finally, we examine the estimates of range and angle when there are \(P = 2\) closely spaced targets. Assume that \(M = 12\), \(N = 12\), SNR = 20dB and \(L=100\). The two targets are diversely polarized with polarization parameters \((\gamma , \eta ) = (\pi /10, \pi /5)(2\pi /5, 2\pi /5)\). Also, the two targets have the same range and close angles \((\theta , r) = (0^\circ , 5\,{\text {km}})(2^\circ , 5\,{\text {km}})\). The estimation result of 100 Monte Carlo trials for range and angle estimation is shown in Fig. 8. It indicates that the proposed TSS and TCR algorithms can distinguish two targets having the same range and the close angles. Polarimetric FDAMIMO radar can sense additional polarization information to improve target identification capability. By the use of polarization diversity in FDAMIMO radar receiver, the angles and ranges of multiple closely spaced targets can be effectively identified with highresolution.
Discussion: From the above experiment results, it is also note that there is still further improvement in accuracy to get closer to the CRB. For instance, based on the presented Tucker tensor model of polarimetric FDAMIMO radar, some effective noise suppression approach such as Tensor completion can be investigated, which we will conduct an indepth research in future work.
5 Conclusion
In this article, we estimate range, angle and polarization parameters in polarimetric FDAMIMO radar based on Tucker tensor decomposition. The threeorder tensor signal model is constructed, and two tensorbased algorithms are, respectively, presented using the fourthorder covariance tensor decomposition. The complexity analysis and CRB for range, angle and polarizations are also provided. The proposed algorithms can effectively estimate the rangeanglepolarization parameters with proper pairing. The tensorbased algorithms improve the performance compared with the matrixbased methods, especially under a small number of snapshots. This improvement results from the inherent multidimensional structure of FDAMIMO radar data that is well reserved, which enables us to effectively reduce the accumulate error. In addition, by the use of the polarimetric FDAMIMO radar that can sense additional polarization information, two targets with identical range and close angles can be effectively distinguished.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Abbreviations
 FDA:

Frequency diverse array
 MIMO:

Multipleinput multipleoutput
 TCR:

Tucker covariance reconstruction
 TSS:

Tucker signal subspace
 CRB:

Cramér–Rao bound
 ULA:

Uniform linear array
 MUSIC:

Multiple signal classification
 ESPRIT:

Estimation of signal parameters via rotation invariance technique
 HOSVD:

Higherorder singular value decomposition (SVD)
 TEM:

Transverse electromagnetic mode
 EVD:

Eigenvalue decomposition
 SNR:

Signaltonoise ratio
 RMSE:

Root mean square error
References
E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, R. Valenzuela, MIMO radar: an idea whose time has come. IEEE Radar conf. (Hawaii, USA, 2004), pp. 71–78
Y.D. Zhang, M.G. Amin, B. Himed, Joint DOD/DOA estimation in MIMO radar exploiting timefrequency signal representations. EURASIP J. Adv. Signal Process. 2012, 102 (2012)
F. Zhang, C. Ji, Z. Zhang et al., Noncircular signal DOA estimation based on coprime array MIMO radar. EURASIP J. Adv. Signal Process. 2021, 99 (2021)
P. Antonik, M.C. Wicks, H.D. Griffiths, C.J. Baker, Frequency diverse array radars. In 2006 IEEE Conference on Radar, 3 (2006)
W.Q. Wang, Overview of frequency diverse array in radar and navigation applications. IET Radar, Sonar Navig. 10(6), 1001–1012 (2016)
H. Chen, H. Shao, H. Chen, Anglerangepolarizationdependent beamforming for polarization sensitive frequency diverse array. EURASIP J. Adv. Signal Process. 2019, 23 (2019)
Y. Dong, Frequency diverse array radar data processing. In: 2018 International Conference on Radar (RADAR), 1–5 (2018)
W.Q. Wang, H.C. So, Transmit subaperturing for range and angle estimation in frequency diverse array radar. IEEE Trans. Signal Process. 62(8), 2000–2011 (2014)
L. Huang, X. Li, P. C. Gong, et al. Frequency diverse array radar for target rangeangle estimation. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. (2016)
W.Q. Wang, H. Shao, Rangeangle localization of targets by a doublepulse frequency diverse array radar. IEEE J. Sel. Top. Signal Process. 8(1), 106–114 (2014)
P.F. Sammartino, C.J. Baker, H.D. Griffiths, Frequency diverse MIMO techniques for radar. IEEE Trans. Aerosp. Electron. Syst. 49(1), 201–222 (2013)
J. Xu, G. Liao, S. Zhu, L. Huang, H.C. So, Joint range and angle estimation using MIMO radar with frequency diverse array. IEEE Trans. Signal Process. 63(13), 3396–3410 (2015)
J. Xu, G. Liao, Y. Zhang, H. Ji, L. Huang, An adaptive rangeangleDoppler processing approach for FDAMIMO radar using threedimensional localization. IEEE J. Sel. Top. Signal Process. 11(2), 309–320 (2017)
B. Huang, A. Basit, R. Gui, W.Q. Wang, Adaptive moving target detection without training data for FDAMIMO radar. IEEE Trans. Veh. Technol. 71(1), 220–232 (2022)
L. Lan, A. Marino, A. Aubry, A.D. Maio, G. Liao, J. Xu, Y. Zhang, GLRTbased adaptive target detection in FDAMIMO radar. IEEE Trans. Aerosp. Electron. Syst. 57(1), 597–613 (2021)
L. Lan, J. Xu, G. Liao, Y. Zhang, F. Fioranelli, H.C. So, Suppression of mainbeam deceptive jammer with FDAMIMO radar. IEEE Trans. Veh. Technol. 69(10), 11584–11598 (2020)
L. Lan, M. Rosamilia, A. Aubry, A.D. Maio, G. Liao, Singlesnapshot angle and incremental range estimation for FDAMIMO radar. IEEE Trans. Aerosp. Electron. Syst. 57(6), 3705–3718 (2021)
R. Gui, B. Huang, W.Q. Wang, Y. Sun, Generalized ambiguity function for FDA radar joint range, angle and Doppler resolution evaluation. IEEE Geosci. Remote Sens. Lett. 19, 1–5 (2020)
J. Xiong, W.Q. Wang, K. Gao, FDAMIMO radar rangeangle estimation: CRLB, MSE, and resolution analysis. IEEE Trans. Aerosp. Electron. Syst. 54(1), 284–294 (2018)
Y. Yan, J. Cai, W.Q. Wang, Twostage ESPRIT for unambiguous angle and range estimation in FDAMIMO radar. Digit. Signal Process. 92, 151–165 (2019)
Y. Song, G. Zheng, G. Hu, A combined ESPRITMUSIC method for FDAMIMO radar with extended range ambiguity using staggered frequency increment. Int. J. Antennas Propag. 2019, (2019)
Z. Xu, B. Huang, H. Hu, H. Chen and W. Q. Wang, Ambiguity functionbased ESPRIT algorithm for FDAMIMO radar target localization. 2020 IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM). 1–4 (2020)
F. Liu, X. Wang, M. Huang et al., Joint angle and range estimation for bistatic FDAMIMO radar via realvalued subspace decomposition. Signal Process. 185(4), 108065 (2021)
J. Xu, W. Q. Wang, C. Cui, R. Gui, Joint range, angle and Doppler estimation for FDAMIMO radar. In: 2018 IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM). 499–503 (2018)
Z. Zhao, Z. Wang, Y. Sun, Joint angle, range and velocity estimation for bistatic FDAMIMO radar. In: IEEE 2nd Advanced Information Technology, Electronic and Automation Control Conf. (IAEAC). 818–824 (2017)
W. Wu, F. Xi, Target localization for FDAMIMO radar with random frequency increment via atomic norm minimization. In: 2019 IEEE MTTS International Microwave Biomedical Conference (IMBioC). 1–4 (2019)
W. G. Tang, H. Jiang, Q. Zhang, Rangeangle decoupling and estimation for FDAMIMO radar via atomic norm minimization and accelerated proximal gradient. IEEE Signal Process. Lett. 27, 366–370 (2020)
W. G. Tang, H. Jiang, S. Pang, Gridless angle and range estimation for FDAMIMO radar based on decoupled atomic norm minimization. In: ICASSP 2019  2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 4305–4309 (2019)
H. Jiang, D.F. Wang, C. Liu, Joint parameter estimation of DOD/DOA/polarization for bistatic MIMO radar. J. China Univ. Posts Telecommun. 17(5), 32–37 (2010)
H. Jiang, D. F. Wang, C. Liu, Estimation of DOD and 2DDOA and polarizations for bistatic MIMO radar. The 19th Annual Wireless and Optical Communications Conference (WOCC 2010). 1–5 (2010)
G. Zheng, M. Yang, B. Chen, W. Guo, Angle and polarization estimation using ESPRIT with polarimetric MIMO radar. In: IET International Conference on Radar Systems (Radar 2012). 1–4 (2012)
W. Guo, M. Yang, B. Chen, G. Zheng, Joint DOA and polarization estimation using MUSIC method in polarimetric MIMO radar. IET International Conference on Radar Systems (Radar 2012). 1–4 (2012)
H. Jiang, Y. Zhang, J. Li et al., A PARAFACbased algorithm for multidimensional parameter estimation in polarimetric bistatic MIMO radar. EURASIP J. Adv. Signal Process. 2013, 133 (2013)
H. Chen, H. Shao, W. Wang, Sparse reconstructionbased anglerangepolarizationdependent beamforming with polarization sensitive frequency diverse array. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 2891–2895 (2016)
B. Li, W. Bai, G. Zheng, Successive ESPRIT algorithm for joint DOArangepolarization estimation with polarization sensitive FDAMIMO radar. IEEE Access. 6, 36376–36382 (2018)
J. Wang, S. Jiang, Anglepolarizationrange estimation using sparse polarization sensitive FDAMIMO radar with coprime frequency offsets. IEEE Access. 7, 146759–146771 (2019)
H. Chen, F. Ahmad, S. Vorobyov, F. Porikli, Tensor decompositions in wireless communications and MIMO radar. IEEE J. Sel. Top. Signal Process. 15(3), 438–453 (2021)
M. Haardt, F. Roemer, G.D. Galdo, Higherorder SVDbased subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems. IEEE Trans. Signal Process. 56(7), 3198–3213 (2008)
L.R. Tucker, Some mathematical notes on threemode factor analysis. Psychometrika. 31(3), 279–311 (1966)
Y. Cheng, R. Yu, H. Gu et al., MultiSVD based subspace estimation to improve angle estimation accuracy in bistatic MIMO radar. Signal Process. 93(7), 2003–2009 (2013)
X. Wang, W. Wang, X. Li et al., A tensorbased subspace approach for bistatic MIMO radar in spatial colored noise. Sensors. 14(3), 3897–3907 (2014)
F. Wen, X. Xiong, J. Su et al., Angle estimation for bistatic MIMO radar in the presence of spatial colored noise. Signal Process. 134, 261–267 (2017)
F. Wen, Z. Zhang, G. Zhang, Y. Zhang, X. Wang, X. Zhang, A tensorbased covariance differencing method for direction estimation in bistatic MIMO radar with unknown spatial colored noise. IEEE Access. 5, 18451–18458 (2017)
C. Mao, J. Shi, F. Wen, Target localization in bistatic EMVSMIMO radar using tensor subspace method. IEEE Access. 7, 163119–163127 (2019)
T.G. Kolda, B.W. Bader, Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
L.D. Lathauwer, B.D. Moor, J. Vandewalle, A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)
A. Nehorai, E. Paldi, Vectorsensor array processing for electromagnetic source localization. IEEE Trans. Signal Process. 42(2), 376–398 (1994)
K.C. Tan, K.C. Ho, A. Nehorai, Linear independence of steering vectors of an electromagnetic vector sensor. IEEE Trans. Signal Process. 44(12), 3099–3107 (1996)
P. Stoica, A. Nehorai, Performance study of conditional and unconditional directionofarrival estimation. IEEE Trans. Acoust. Speech Signal Process. 38(10), 1783–1795 (1990)
M. Pesavento, A.B. Gershman, Maximumlikelihood directionofarrival estimation in the presence of unknown nonuniform noise. IEEE Trans. Signal Process. 49(7), 1310–1324 (2001)
P. Stoica, E.G. Larsson, A.B. Gershman, The stochastic CRB for array processing: a textbook derivation. IEEE Signal Process. Lett. 8(5), 148–150 (2001)
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This document is the results of the research funded by the National Natural Science Foundation of China under Grants 61371158 and 61771217, and the Natural Science Foundation of Jilin Province (CN) under Grant 20220101100JC and 20180101329JC.
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ZQ, JH and LYC proposed the tensor model and theoretical algorithms. ZQ carried out the simulation experiments. LYC analyzed the estimation performance. JH improved the writing and edited the revised version. All authors read and approved the final manuscript.
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Zhang, Q., Jiang, H. & Liu, Y. Joint range, angle and polarization estimation in polarimetric FDAMIMO radar based on Tucker tensor decomposition. EURASIP J. Adv. Signal Process. 2023, 39 (2023). https://doi.org/10.1186/s13634023009971
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DOI: https://doi.org/10.1186/s13634023009971
Keywords
 FDAMIMO radar
 Tucker tensor decomposition
 Rangeanglepolarization estimation
 Polarization sensitive array
 Signal subspace