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An effective method via tensor decomposition is proposed to deal with the joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation of noncircular sources in colocated coprime MIMO radar. By decomposing the transmitter and receiver into two sparse subarrays, noncircular property of source can be used to construct new extended received signal model for two sparse subarrays. The new received model can double the virtual array aperture due to the elliptic covariance of imping sources is nonzero. To further exploit the multidimensional structure of the noncircular received model, we stack the subarray output and its conjugation according to mode-1 unfolding and mode-2 unfolding of a third-order tensor, respectively. Thus, the corresponding extended tensor model consisted of noncircular information for DOA and DOD can be obtained. Then, the higher-order singular value decomposition technique is utilized to estimate the accurate signal subspace and angular parameter can be automatically paired via the rotational invariance relationship. Specifically, the ambiguous angle can be eliminated and the true targets can be achieved with the aid of the coprime property. Furthermore, a closed-form expression for the deterministic CRB under the NC sources scenario is also derived. Simulation results verify the superiority of the proposed estimator.
In recent years, the issue of angle parameter estimation in multiple-input multiple-output (MIMO) radar has become a hot research topic due to its broad applications in wireless communications, radar, sonar and navigation.[1–6] MIMO radar together with waveform diversity can exhibit higher resolution, better parameter identifiability, greater flexibility in the beampattern design, and more degrees of freedom (DOFs) over the conventional phased-array radar associated with coherent waveform. According to the deployment of transmit array and receive array, MIMO radar is generally categorised into statistical MIMO radar and colocated MIMO radar.[7,8] Furthermore, the colocated MIMO radar can be divided into bistatic and monostatic MIMO radar. For bistatic MIMO radar system, the direction-of-departure (DOD) and direction-of-arrival (DOA) are different, while the DOD and DOA are the same for monostatic MIMO radar because the transmitter and receiver are closely enough. In this work, our task is to investigate the DOD and DOA estimation for locating multiples targets in the bistatic MIMO radar system.
In order to determine DOD and DOA estimations in bistatic MIMO radar, many excellent methods have been proposed. In Refs. [9] and [10], peak searching methods are proposed to estimate the DOD and DOA, which demand high computational complexity. In order to avoid the downside of the peak searching methods, a polynomial root finding algorithm with automatical angular pairing is proposed in Ref. [11] for jointly estimating of DOD and DOA. In Ref. [12], the rotational invariance technique is proposed for angle parameter estimation in bistatic MIMO radar. Furthermore, a modified ESPRIT algorithm in Ref. [13] is presented to avoid the angle pairing process. In Ref. [14], by using the temporal uncorrelated characteristic of the spatial colored noise, a special fourth-order tensor structure is constructed to suppress the effect of colored noise in bistatic MIMO radar system. In Ref. [15], a beamspace-based technique by focusing transmitted energy within the desired spatial sector is proposed to improve the DOD and DOA estimation performance. The Sigmoid transform method is investigated to improve the joint DOD and DOA estimation performance for wideband bistatic MIMO radar system in the condition of impulsive noise environment in Ref. [16]. In order to deal with the antennas failure problem in bistatic MIMO radar, image entropy and low rank block hankel structured matrix completion method is proposed to recover the successive missing data.[17] In Ref. [18], by exploting the multidimensional structure inherented in the received data, the PARAFAC techqnique is adopted to achieve accuracy angle estimation in the presence of gain-phase error for bistatic MIMO radar.
However, all the methods mentioned above assume that both the transmit and receive arrays in bistatic MIMO radar are uniform linear array with half-wavelength spacing. To further improve the number of degrees of freedom (DOFs) and estimated resolution, it is necessary to design sparse transmit and receive arrays in MIMO radar. In Ref. [19], a new nested array configuration in transmit and receive side is proposed to enhance the number of DOFs of difference coarray of the sum coarray (DCSC). In Ref. [20], a coprime MIMO radar is designed to deal with the DOAs estimation of mixed coherent and uncorrelated sources, where the transmit array and receive array are the first sparse subarray and the second sparse subarray of a generalized coprime array, respectively. In Ref. [21], symmetric coprime array MIMO radar configuration is used to extend the maximum consecutive lags in the differece coarray domain of symmetric sum coarray after vectorizing the sample covariance matrix. But, the designed coprime or nested array configurations in Refs. [9–21] mainly improve the DOA estimation performance for monostatic MIMO radar. Different from the monostatic MIMO radar, the DOD and DOA in bistatic MIMO radar are different, causing more angle parameters to be estimated. In order to make full use of the advantages of coprime/nested array in bistatic MIMO radar, consective difference coarray corresponding to DOD and DOA are achieved by extracting specified covariance matrix elements in Ref. [22]. Although the sparse array configurations used in Refs. [19–22] can improve the angle estimation accuracy, the methods to estimate the DOD and DOA mainly focus on the difference coarray of the nested array or coprime array. According to Refs. [23–25], there exist two subararys both for nested array and coprime array. As analyzed in Refs. [26–28], high resolution angle parameter estimation can be obtained by exploiting the results of the two sparse subarrays in coprime array. Thus, we can improve the DOD and DOA estimation perforamce for bistatic coprime MIMO radar by dividing the coprime array in transmit and receive side into two uniform sparse linear subarrays. In Ref. [29], a partial spectral search (PSS) method is proposed to realize the two-dimensional (2D) DOA estimation for coprime plannar arrays (CPA), but, a tremendous computational burden is inevitable due to the 2D partial spectrum peak searching process. In order to alleviate the computation complexity, a computationally efficient one-dimensional (1D) PSS method via reduced dimensional transform in Ref. [30] is conducted to 2D DOA estimation for CPA. In Ref. [31], a parallel factor (PARAFAC) model is constrcuted by using the multidimensional structure inherent in the received data of each subarray and the convergence speed of the proposed PARAFAC algorithm is fast via propagator method (PM) as the initialization of the angle estimation. The proposed methods in Refs. [29–31] can also be applied to joint DOD and DOA estimations in bistatic coprime MIMO radar becasue the received signal model in CPA is similar with the model in bistatic coprime MIMO radar. However, all the mentioned algorithms in Refs. [29–31] ignore the noncircular characteristic of the imping sources when estimate the angle parameter for CPA. It is well known that the angle estimaiton accuracy and resolved sources can be significant improved by using noncircular characteristic of the noncircular signals.[32] In this paper, we improve the DOD and DOA estimation performance for bistatic coprime MIMO radar with the aid of the noncircular characteristic. By combining the noncircular characteristic and bistatic coprime MIMO radar, two new extend three-order tensor models corresponding to DOD and DOA can be constructed by exploiting the multidimensional inherent structure. And, high accuracy novel signal subspace matrices can be obtained by using the multi-SVD method. Then, the rotational invaraince technique can be used and the estimated DODs and DOAs are automatically paired. Furthermore, all the DODs and DOAs can be achieved by utilising the linear relationship between the true and ambiguous angle parameter and the true estimated DODs and DOAs can be obtained by using the coprime property. The effectiveness and superiority of the proposed algorithm are verified by numerical simulation in Section
The main contributions of this work can be summarized below.
(i) Two new three order tensor signal models corresponding to DOD and DOA are constructed, which can make full use of the noncircular characteristic inherent in the impinging sources.
(ii) Accurate signal subspace matrices are obtained by performing multi-SVD on new three-order tensor signal models and automatically paired estimated DODs and DOAs are identified by using the rotational invaraince technique. Furthermore, accuracy estimated true DODs and DOAs can be obtained by eliminating the ambiguous angles with the help of the coprime property.
(iii) The proposed method can avoid spectrum peak searching processing compared with the PSS method and extend the array aperture compared with the PARAFAC method by using the noncircualr characteristic.
The rest of this paper is organized as follows. In Section
Notations: Throughout this paper, scalars, vectors, and matrices are denoted by lowercase letters, boldface lowercase letters, and boldface uppercase letters, respectively. The superscripts *, T, and H denote the complex conjugate, the transpose, and the complex conjugate transpose, respectively. The Moore–Penrose pseudoinverse is denoted by the superscript dagger †. The symbols ⊗, ⊙, and ⊕ denote the Kronecker product, Kronecker–Rao operation, and Hadamard product, respectively. The diag(⋅) or Dn denotes a diagonal matrix. Additionally, 0 and I denote the zero matrix and identity matrix with appropriate dimension.
Consider a bistatic MIMO radar system with M transmit array and N receive array to locate K targets in the 2D detection area. As illustrated in Fig.
The transmitter utilizes the M arrays to emit different narrowband orthogonal noncircular signals, which have the identifical bandwidth and center frequency. Assume there are K narrowband far-field uncorrelated sources with (θk, ϕk), k = 1,2,…,K, where θk and ϕk denotes the DOD and DOA of the k-th sources, respectively. Thus, the signal model in the receiving end can be expressed as
![](cpb_29_5_054304/cpb_29_5_054304_ieqn1.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn2.gif)
For matrices A ∈ ℂM × K, B ∈ ℂN × K, and K ∈ ℝMN × MN, using the commutative relationship of Kronecker–Rao operation, we can obtained that A ⊙ B = K(B ⊙ A). Therefore, we can find that there exists a transform relationship between the extend joint steering matrix for DOA and DOD as follows:
Firstly, some tensor basics are introduced to better understand the multidimensional structure and the detailed introduction about the concept and operations of tensor can be found in Refs. [33,34].
According to Definition
Accodring to Eqs. (
![](cpb_29_5_054304/cpb_29_5_054304_ieqn13.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn14.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn15.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn16.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn17.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn18.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn19.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn20.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn21.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn22.gif)
Thus, the extend signal subspace matrix
As
Define two selection matrix as follows:
Thus, by utilizing the rotational invariance property inherented in Anc1, we can obtain
![](cpb_29_5_054304/cpb_29_5_054304_ieqn25.gif)
Then, in order to estimate the DOA
![](cpb_29_5_054304/cpb_29_5_054304_ieqn28.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn29.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn30.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn31.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn32.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn33.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn34.gif)
Thus, the extend signal subspace matrix
Due to that
Insertation
According to the selection matrices in Eq. (
![](cpb_29_5_054304/cpb_29_5_054304_ieqn39.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn40.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn41.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn42.gif)
It is easily to show that by using the non-circular characteristic of the imping sources, the virtual aperture of coprime MIMO radar can be enhanced and the extend tensor model corresponding to DOD and DOA can make full use of the multidimensional structure information inherented in the received data. From Eqs. (
Assume that the DOD and DOA of a single noncircular target are (θ,ϕ), the phase differences for DOD and DOA in the coprime MIMO radar can be respectively denoted as
Due to the phase periodicity of complex exponential function is 2π, thus the phase differences for DOD and DOA can be redenoted as
Then, by using the coprime property, the true DOD and DOA satisfy
As M1 and M2 satisfies the coprime property, {kθ1}, {kθ2} and {kϕ1, kϕ2} are unique that satisfy Eq. (
After obtaining the estimated (sin(θa1),sin(ϕa1)), (sin(θa2), sin(ϕa2)) for the first subarray and the second subarray, the finally true DOD and DOA can be denoted as
And, for K targets, the actual DODs and DOAs with (θk, ϕk) for k = 1,2,…,K can be achieved by using the above mentioned ambiguity elimination process.
In order to make our proposed method more comprehensive, the main steps of our scheme for angle estimation in coprime MIMO radar is summarized in Algorithm 1.
![]() | Algorithm 1
Summary of the proposed method . |
For coprime MIMO radar, the computational complexity of the proposed method mainly contains multi-SVD for extend three-order noncircular tensors
![]() | Table 1.
Comparison of the complexity. . |
In order to provide an intuitive computational complexity comparison, the complexity versus different arrays and snapshots are demonstrated in Fig.
According to Eqs. (
In order to provide a benchmark for parameter estimation performance, we derive the closed-form expression of the deterministic CRB for noncircular sources in bistatic coprime MIMO radar. Firstly, we stack all the array output of the first sparse subarray and the second sparse subarray as follows:
The estimated parameters can be written as the vector form
Define the total steering matrix as follows:
According to Ref. [35], the CRB matrix can be represented by
![](cpb_29_5_054304/cpb_29_5_054304_ieqn66.gif)
Thus, the closed-form expression about the deterministic CRB for noncircular sources in coprime MIMO radar is obtained.
In this section, several simulations are carried out to evaluate the effectiveness and superiority of the proposed method. And, the state-of-the-art RD PSS algorithm, PARAFAC algorithm, ESPRIT algorithm, and CRB in Eq. (
![](cpb_29_5_054304/cpb_29_5_054304_ieqn67.gif)
![](cpb_29_5_054304/cpb_29_5_054304_ieqn68.gif)
In the first experiment, a coprime bistatic MIMO radar system with 8 transmit antennas and 8 receive arrays is considered, where M1 = 4 and M2 = 5 both in transmitter and receiver. Assume two equi-powered uncorrelated narrowband noncircular sources located at (θ1, ϕ1) = (10°,20°), θ2, ϕ2) = (30°,40°), and noncircular phase
In the second simulation, the RMSE performance as a function of SNR and snapshots for coprime multiple-input multiple-output radar is considered. Other simulation parameters are same as the first simulation. In Fig.
![]() | Fig. 5. RMSE performance versus SNR and snapshots: (a) RMSE versus SNR with snapshots = 100 and (b) RMSE versus snapshots with SNR = 0 dB. |
In the third simulation, the RMSE performance comparison versus different numbers of transmit and receive arrays in bistatic coprime MIMO radar is investigated. Two types of coprime MIMO radar with (M1,M2) = (4,5) and (M1, M2) = (3,4) are adopted. The number of imping sources, the range of snapshots, and SNR are the same as those as the second simulation. As shown in Fig.
![]() | Fig. 6. RMSE performance versus different numbers of sensors: (a) RMSE versus SNR with snapshots = 100 and (b) RMSE versus snapshots with SNR = 0 dB. |
In the last experiment, we compare the angular super-resolution capabilities of the above mentioned algorithms. Two closely targets are considered, the DOD and DOA of the first target are (θ1, ϕ1) = (10°, 20°) and the DOD and DOA of the second target are (θ2, ϕ2) = (10° + Δa,20° + Δ), where Δ varies from 1° to 10° with a step size 1°.
In Figs.
In this paper, an efficient tensor-based estimator has been proposed to deal with the angle parameter estimation of noncircular source in bistatic coprime MIMO radar. In the proposed method, two extended three-order tensor models are constructed by using the noncircular property of the output data. Due to the utilization of the multidimensional structure, the estimated signal subspace matrix exhibits better accuracy by suppressing the effect of the noise. Thereafter, the estimated ambiguous DODs and DOAs can be deterimined via rotational invariance criterion and the angle parameters are automatically paired without extra pairing process. Moreover, the ambiguous angle parameters can be eliminated with the aid of coprime property. And, the closed-form expression of the deterministic CRB for noncircular sources in bistatic coprime MIMO radar is also derived to measure the parameter estimation performance. Theoretical analyses and numerical simulations are carried out to demonstrate that the proposed tensor-based noncircular signal model exhibits excellent estimation performance for coprime MIMO radar compared with other state-of-art methods. In the near future, we will focus on the flexible sparse array design to further improve the angular parameter estimation performance for bistatic MIMO radar system.
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