A PARAFAC-based algorithm for multidimensional parameter estimation in polarimetric bistatic MIMO radar
© Jiang et al.; licensee Springer. 2013
Received: 6 July 2012
Accepted: 19 July 2013
Published: 2 August 2013
In this article, we investigate the problem of applying the parallel factor quadrilinear decomposition technique to multidimensional target parameter estimation in a polarimetric bistatic multiple-input multiple-output (MIMO) radar system with a uniform rectangular array at the transmitter and a cross-dipole-based uniform rectangular array at the receiver. The signal model is developed, and a novel algorithm is proposed exploiting the quadrilinear alternating least squares to jointly estimate the two-dimensional direction of departure (2D-DOD), two-dimensional direction of arrival (2D-DOA), polarization parameters and Doppler frequency. Multidimensional parameters can be automatically paired by this algorithm to avoid the performance degradation resulting from wrong pairing. The developed algorithm requires neither multidimensional spectral peak searching nor covariance matrix estimation and several eigen-value decompositions that may bring error accumulation. Furthermore, multiple targets having close 2D-DODs and close 2D-DOAs or even the same 2D-DOD or 2D-DOA are distinguishable by means of polarization diversity. The algorithm improves the performance of multi-target identification and three-dimensional localization. Numerical simulations demonstrate the effectiveness of the proposed algorithm.
In recent years, joint parameter estimation in multiple-input multiple-output (MIMO) radar [1–3] has drawn considerable attention for target identification, localization, imaging, etc. Specifically, bistatic MIMO radar with respectively colocated transmitter and receiver array antennas is widely investigated by many researchers for its capability of jointly estimating the direction of departure (DOD) and direction of arrival (DOA) of targets. Many spatial spectrum estimation approaches have been developed for joint DOD and DOA estimation in a bistatic MIMO radar [4–11]. In , a two-dimensional (2D) Capon-based method was developed by searching through all the 2D space to find the DOA and DOD of a target. By exploiting the invariance property of both the transmitter and receiver arrays, some approaches using the estimation of signal parameter via rotational invariance techniques (ESPRIT) [5–7] have been presented for joint DOA and DOD estimation, avoiding the exhaustive peak searching. However, an additional pair matching between the DOAs and DODs is required for ESPRIT-like methods. Besides, polynomial root finding  and combined ESPRIT-multiple signal classification (MUSIC) approach  were investigated by decomposing the 2D angle estimation into double one-dimensional (1D) angle estimation, allowing an automatic pairing between the DOAs and DODs. To reduce computation load, a reduced-dimension MUSIC method was proposed in . Considering the characteristics of non-stationary target signals, joint estimation of DOD and DOA information of maneuvering targets was examined in  exploiting spatial time-frequency distribution. For moving targets, the literatures in [12–14] addressed to joint DOD, DOA and Doppler frequency estimation.
Notably, the polarization state of a target will change upon reflection. Polarization diversity has been proved to be important in target identification especially when multiple targets are so closely spaced that they cannot be distinguished well in spatial domain. In [15, 16], joint DOD-DOA-polarization estimation and joint DOD, 2D-DOA and polarization estimation exploiting the ESPRIT technique were proposed for bistatic MIMO radar. The simulations have shown that multiple targets having close DODs or DOAs but different polarizations own different array manifolds and are distinguishable based on their polarization diversity.
Recently, parallel factor (PARAFAC) analysis, as an analysis method of high-dimensional data, has become a new technology applied to signal processing and communication field [17, 18]. The parallel factor model is a generalization of low-rank decomposition to three- or multi-way arrays, which was first introduced as a data analysis tool in psychometrics. Over the past decades, PARAFAC ideas have been applied in multiple-invariance sensor array processing with emphasis on identifiability results . In recent years, PARAFAC has become a new research means in MIMO radar [19–21]. The PARAFAC analysis algorithms [19, 20] and adaptive PARAFAC algorithm  have been developed for the estimation of DOAs and DODs of multiple targets.
Despite the fact that several PARAFAC-based methods have been proposed for DOA and DOD estimation in bistatic MIMO radars, trilinear decomposition algorithms [17–21] with trilinear alternating least square (TALS) are commonly used to fit PARAFAC model. In this article, we investigate joint estimation of DOD, DOA, Doppler frequency and polarization. The polarization can be considered as the fourth dimension and hence a four-way description happens to fit better as long as the targets have diverse polarizations. Thus, PARAFAC quadrilinear decomposition [22, 23] with quadrilinear alternating least square (QALS) is applied to multidimensional parameter estimation in polarimetric bistatic MIMO radar.
With the restriction that targets can only be located on a 2D plane in previous algorithms based on 1D-DOD and 1D-DOA estimation using uniform linear arrays, the estimation of 2D-DOD and 2D-DOA is investigated in this article exploiting a uniform rectangular array at the transmitter and a cross-dipole-based uniform rectangular array at the receiver. The signal model for polarimetric bistatic MIMO radar is developed, and a novel PARAFAC QALS-based algorithm is presented for joint estimation of seven target parameters, including the 2D-DOA, 2D-DOD, two polarization parameters and Doppler frequency. The proposed algorithm avoids multidimensional spectral peak searching, covariance matrix estimation and several eigen-decompositions that may bring error accumulation, which can enhances the accuracy of estimation. Multidimensional parameter pairing is obtained automatically by this algorithm, which can avoid the performance degradation resulting from wrong pairing. Furthermore, polarization diversity of multiple target characteristics is exploited to distinguish multiple targets having close 2D-DODs and 2D-DOAs or even the same 2D-DOD or 2D-DOA, which can improve the resolution of multi-target identification and 3D localization. The merits of the algorithm are investigated via numerical simulations.
This paper is organized as follows: in the next section, the signal model for polarimetric bistatic MIMO radar is developed. A novel PARAFAC quadrilinear decomposition-based algorithm for 2D-DOA, 2D-DOD, polarizations and Doppler frequency estimation is presented in section 3. In section 4, the results of simulation are given to verify the performance of the proposed method. Finally, a conclusion is drawn in section 5.
2 Signal model for polarimetric MIMO radar
2.1 Received signal
where ◇ represents the Khatri-Rao product (columnwise Kronecker product). A t (θ t ,ϕ t ), A r (θ r ,ϕ r ) and G(θ r ,ϕ r ,γ,η) denote the transmit steering matrix, receive steering matrix and polarization matrix, respectively. A t (θ t ,ϕ t ) = [a t (θ t 1,ϕ t 1),⋯,a t (θ tP ,ϕ tP )], A r (θ r ,ϕ r ) = [a r (θ r 1,ϕ r 1),⋯,a r (θ rP ,ϕ rP )], and G(θ r ,ϕ r ,γ,η) = [g(θ r 1,ϕ r 1,γ 1,η 1),⋯,g(θ rP ,ϕ rP ,γ P ,η P )]. The matrix contains M transmit waveforms, , and S S H = I M due to waveform orthogonality and normalization. denotes the noise matrix, and b (l) is the target vector, .
2.2 Pulse compression and vectorization
with Y = [y (1),⋯,y (L)] and N = [n (1),⋯,n (L)] representing the observed matrix and noise matrix. B = [b (1),⋯,b (L)] T denotes the target matrix, which contains the estimated parameter of Doppler frequency f d .
Based on the signal model of polarimetric MIMO radar developed in (12), the problem of interest is to jointly estimate the multidimensional parameters θ t ,ϕ t ,θ r ,ϕ r ,γ,η and f d .
3 PARAFAC quadrilinear decomposition-based algorithm for multidimensional parameter estimation
In this section, we present the multidimensional parameter estimation algorithm in polarimetric MIMO radar using PARAFAC quadrilinear decomposition.
3.1 PARAFAC quadrilinear decomposition model and uniqueness theorem
Definition 1. (Quadrilinear decomposition in tensor format) A quadrilinear decomposition of a four-order tensor is a decomposition of the type , where a m ,b m ,c m ,d m are the m th columns of the matrices , and , respectively, and ∘ is the outer product, i.e., (a m ∘ b m ∘ c m ∘ d m ) ijkl = a im b jm c km d lm , for all values of the indices i,j,k and l.
Definition 2. (Quadrilinear decomposition in matrix format) Let , and be the four matrix representations of a four-way array. The quadrilinear decomposition of can be written as four equivalent matrices X (1) = (A ◇ B ◇ C)D T , X (2) = (D ◇ A ◇ B)C T , X (3) = (C ◇ D ◇A)B T and X (4) = (B ◇ C ◇ D)A T .
Definition 3. (Kruskal-rank or k-rank)  Consider a matrix . If rank (A) = r, then A contains a collection of r linearly independent columns. Moreover, if every k < M columns of A are linearly independent, but this does not hold for every k+1 columns, then A has k-rank k A = k. Note that k A < rank (A),∀ A.
then A , B, C and D are unique up to permutation and scaling of columns, meaning that any other quadruple , and that gives rise to X is related to A , B , C , D via , , , where Π is a permutation matrix, Λ 1,Λ 2, Λ 3 and Λ 4 are diagonal scaling matrices satisfying Λ 1 Λ 2 Λ 3 Λ 4 = I M .
3.2 Parameter matrix estimation using QALS
where Y (1), Y (2), Y (3) and Y (4) denote the slice sets of the four-way array along the snapshot, polarization, reception and transmission ways, respectively.
which means that the four-way array satisfies the Kruskal-rank Theorem. Thus, A t , A r , G and B can be recovered uniquely up to permutation and scaling ambiguity.
Initialize the estimation , , and with random matrices, denoting them as , , and . The iteration number is k = 1,2,3,⋯.
- ii)Given , and , obtain the least squares solution of the k th iteration of according to(19)
where ∥∥ F denote the Frobenius norm of its matrix argument.
- iii)Update the k th iteration solution based on , and , such that(20)
- iv)Update the k th iteration solution based on , and , such that(21)
- v)Substitute the estimated , and into(22)
acquiring the least squares solution of the k th iteration .
Compute the error at the k th iteration. The algorithm has converged when , where ε is an error threshold.
3.3 Multidimensional parameter estimation
Upon obtaining the estimates of , , and , the following problem is to estimate multidimensional parameters of P targets from these parameter matrices. The algorithm is based on available Vandermonde structure  and Kronecker product structure in the matrix blocks, also uses the polarization-sensitive array processing technology .
3.3.1 2D-DOD estimation
3.3.2 2D-DOA estimation
3.3.3 Polarization parameter estimation
3.3.4 Doppler frequency estimation
Upon the above analysis, multidimensional parameter θ t ,ϕ t ,θ r ,ϕ r ,γ,η and f d for the p th target can be effectively calculated, p = 1,⋯,P.
From the Kruskal-rank Theorem in section 3.1, the sufficient condition is derived herein to guarantee that the quadrilinear decomposition model is unique. Therefore, with polarization diversity, the 2D-DODs and 2D-DOAs of multiple closely spaced targets in MIMO radar can be uniquely identified, and high-resolution estimation can be achieved. It is mentioned that, based on the signal model described in (14), the trilinear decomposition-based algorithm with TALS can also be applied to identify multidimensional parameters using three matrices A t , A r ◇ G and B as the slice sets of three-way array . However, for trilinear decomposition, the Kruskal-rank condition can not be satisfied in the situation that P targets have the same 2D-DOD, since the inequality is needed to uniquely identify the three parameter matrices.
Besides, robust iterative fitting of multilinear approaches such as linear programming (LP) or weighted median filtering iteration  can also be applied here to further yield a better solution for robust estimation in non-Gaussian noise.
The following assumption is made in the article: (i) The antennas are assumed to be ideal, identical isotropic scatterers. In practical MIMO radar systems, the effect of mutual coupling and array self-calibration methods should be considered. (ii) The target amplitude is assumed to remain constant during snapshot collection (Swerling I model). The condition can be relaxed to Swerling II model  when Doppler shift is known. (iii) The number of targets is assumed to be known. In practical situation, the detection of the number of targets, such as the minimum description length (MDL) criterion, should be investigated to obtain the rank of the four-way array . (iv) The targets are assumed to be point sources. In practical MIMO radar implementation, distributed target models and methods can be further considered.
The simulations are conducted to verify the effectiveness of the proposed method in this section. Consider a polarimetric bistatic MIMO radar system with M 1 = M 2 = 3 and N 1 = N 2 = 3. The inter-sensor spacings are d t = d r = λ/2. The duration of a snapshot is T s = 50μ s, and the length of transmit codes is K = 1024. The amplitudes of P targets are β 1 = β 2 = ⋯ β P = 1. The number of snapshots is L = 100, and Monte Carlo trials are 100.
Simulation 1. Consider P = 3 targets with 2D-DOD, 2D-DOA, polarization parameters and Doppler frequency respectively being (θ t 1,θ t 2,θ t 3) = (10°,30°,70°),(ϕ t 1,ϕ t 2,ϕ t 3) = (40°,-60°,-10°), (θ r 1,θ r 2,θ r 3) = (50°,80°,20°),(ϕ r 1,ϕ r 2,ϕ r 3) = (10°,30°,60°),(γ 1,γ 2,γ 3) = (π/10,π/4,9π/20),(η 1,η 2,η 3) = (π/5,4π/5,2π/5) and (f d 1,f d 2,f d 3) = (1,000,3,000,5,000) Hz.
Multidimensional parameters estimation result for three targets (SNR = 10 dB)
Simulation 2. This simulation examines the estimates of 2D-DOD and 2D-DOA when there are P = 2 closely spaced targets. SNR = 10 dB. The two targets are diversely polarized with polarization parameters (γ 1,γ 2) = (π/10,9π/20), (η 1,η 2) = (π/5,4π/5). The Doppler frequency is (f d 1,f d 2) = (1,000,3,000) Hz. The following three scenarios are respectively considered.
2D angle estimation result when two targets have close 2D-DODs and close 2D-DOAs (SNR = 10 dB)
2D angle estimation result when two targets have the same 2D-DOD and close 2D-DOAs (SNR = 10 dB)
2D angle estimation result when two targets have the same 2D-DOA and close 2D-DODs (SNR = 10 dB)
The simulations of the above scenarios indicate that the proposed algorithm can distinguish multiple targets having close 2D-DODs and 2D-DOAs, or even the same 2D-DOD or 2D-DOA by polarization diversity. Therefore, it can uniquely identify the 2D transmit/receive angles of multiple closely spaced targets with high-resolution. However, traditional methods only contain angle parameter estimation without polarization information, leading to their performance degradation in locating multiple closely spaced targets.
where . From (44), it is shown that Y is a PARAFAC trilinear decomposition model. Thus, TALS-based algorithm in  can be applied to recover the three parameter matrices A t (θ t ,ϕ t ), and B(f d ). Following this, 2D-DOD, 2D-DOA as well as polarization and Doppler frequency can be estimated exploiting the Vandermonde structure and Kronecker matrix products in the three parameter matrices. The extended ESPRIT-based algorithm employs the shift invariance property of transmit and receive arrays, in which both the steering vectors of transmit and receive arrays have the form of kronecker product for the rectangular arrays, as shown in (6). The signal subspace can be obtained by eigen-value decomposition of covariance matrix of the observed matrix Y in (12). Then, multidimensional parameters can be estimated by properly partitioning the signal subspace matrix into different sub-matrices . The procedure of pairing 2D-DODs and 2D-DOAs for multiple targets is performed similarly to .
In this article, we investigate multidimensional target parameter estimation in a polarimetric bistatic MIMO radar system using PARAFAC quadrilinear decomposition. The signal model has been developed and a novel algorithm for joint estimation of 2D-DOA, 2D-DOD, polarization parameters and Doppler frequency has been presented based on QALS. The algorithm has many merits: (i) it requires neither multidimensional spectral peak searching nor covariance matrix estimation and several eigen-decompositions that may bring error accumulation, which enhances the accuracy of estimation; (ii) multidimensional parameters can be well paired automatically, which reduces the complexity of additional pairing; (iii) it can distinguish multiple targets having close 2D-DODs and 2D-DOAs or even the same 2D-DOD or 2D-DOA by exploiting polarization diversity and the uniqueness of quadrilinear decomposition. Thus, the performance of multi-target 2D-DOD and 2D-DOA estimation in polarimetric MIMO radar has been greatly improved; (iv) unlike the previous algorithms with only respect to 1D-DOD and 1D-DOA estimation in bistatic MIMO radar, the proposed algorithm can obtain 2D-DOA and 2D-DOD estimation with high-resolution, which is of importance for accurate identification and 3D localization of multiple targets in MIMO radar.
The work is supported by the National Natural Science Foundation of China (No. 61071140). The authors wish to thank the anonymous reviewers for their valuable comments and suggestions which greatly improved the manuscript.
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