Computationally Efficient DOA Tracking Algorithm in Monostatic MIMO Radar with Automatic Association

Abstract

We consider the problem of tracking the direction of arrivals (DOA) of multiple moving targets in monostatic multiple-input multiple-output (MIMO) radar. A low-complexity DOA tracking algorithm in monostatic MIMO radar is proposed. The proposed algorithm obtains DOA estimation via the difference between previous and current covariance matrix of the reduced-dimension transformation signal, and it reduces the computational complexity and realizes automatic association in DOA tracking. Error analysis and Cramér-Rao lower bound (CRLB) of DOA tracking are derived in the paper. The proposed algorithm not only can be regarded as an extension of array-signal-processing DOA tracking algorithm in (Zhang et al. (2008)), but also is an improved version of the DOA tracking algorithm in (Zhang et al. (2008)). Furthermore, the proposed algorithm has better DOA tracking performance than the DOA tracking algorithm in (Zhang et al. (2008)). The simulation results demonstrate effectiveness of the proposed algorithm. Our work provides the technical support for the practical application of MIMO radar.

1. Introduction

Multiple-input multiple-output (MIMO) radar employs multiple antennas to simultaneously transmit diverse waveforms and utilizes multiple antennas to receive the reflected signals [1–5]. Direction of arrival (DOA) algorithms in MIMO radar have been recently investigated in [6–24], which contain estimation of signal parameters via rotational invariance technique (ESPRIT) algorithms [6–8], Capon algorithms [10, 11], multiple signal classification (MUSIC) algorithms [9, 12–15], parallel factor (PARAFAC) analysis algorithms [16–18], propagator method [19, 20], quaternion method [21, 22], compressive sampling methods [23, 24], and so on. However, the algorithms mentioned above are, generally, used in offline situation, and they are not applicable for tracking moving targets. The online algorithms can be used for real-time application of MIMO radar.

DOA tracking for array antenna has been investigated for a long time, which contains projection approximation and subspace tracking (PAST) algorithm [25], projection approximation and subspace tracking of deflation (PASTd) algorithm [26], Bi-iterative least-square method [27], Bi-iteration single value decomposition [28], and others [29, 30]. The DOA tracking algorithms in [25–30] have a high computational complexity. A low-complexity method was proposed in [31, 32] to track DOA of moving sources using the elements of the covariance matrix of the received signal in array signal processing, and the DOA tracking algorithm can implement automatic association, which is a key technique in DOA tracking [33].

DOA tracking for MIMO radar is to track the DOA of the moving targets. PARAFAC adaptive algorithms [17, 18] and PASTd [34] were used for DOA tracking for MIMO radar, but an extra data association is required. Kalman-PASTd was proposed in [35] for DOA tracking for monostatic MIMO Radar with automatic association. PARAFAC adaptive algorithm, PASTd algorithm, and Kalman-PASTd algorithm have high computational complexity. In the paper, we propose a computationally efficient DOA tracking algorithm in monostatic MIMO radar with automatic association.

In this paper, we reference the array-signal-processing DOA tracking idea in [31] to propose a DOA tracking algorithm which is suitable for MIMO radar. Using the reduced-dimension transformation for the received signal of MIMO radar, we obtain the covariance matrix of reduced-dimension transformation signal, and then we adopt an improved version for DOA tracking. The proposed algorithm can realize automatic association in DOA tracking. Error analysis and Cramér-Rao lower bound (CRLB) are also derived in this paper. Finally, the simulation results demonstrate the effectiveness and robustness of the proposed algorithm.

There are some differences between the work in [31] and the proposed algorithm. (1) Reference [31] proposed an effective DOA tracking algorithm in array signal processing, while we address DOA tracking problem for MIMO radar in the paper. (2) The DOA tracking algorithm in array signal processing in [31] requires the direction matrix of Vandermonde form. The direction matrix in MIMO radar is not a Vandermonde matrix, and the DOA tracking algorithm in [31] cannot be used directly for DOA tracking in MIMO radar. We employ the reduced-dimension transformation for the received signal to obtain the reduced-dimension direction matrix of Vandermonde form. (3) Our work improves the DOA tracking algorithm in [31] to enhance the DOA tracking performance in MIMO radar, since it fully uses the Toeplitz matrix property to eliminate the noise. Therefore, the proposed algorithm not only can be regarded as an extension of the work in [31], but also is an improved algorithm. Simulation results show that the proposed algorithm has much better DOA tracking performance than the DOA tracking algorithm in [31].

The reminder of this paper is structured as follows. Section 2 develops the data model for monostatic MIMO radar. Section 3 establishes our DOA tracking algorithm based on the elements of the covariance matrix of the reduced-dimension transformation signal. In Section 4, error analysis and Cramer-Rao lower bound (CRLB) are derived. In Section 5, simulation results are presented to verify the effectiveness of the proposed algorithm, while the conclusions are made in Section 6.

Notation. (⋅)^𝑇,(⋅)^𝐻,(⋅)⁻¹, and (⋅)⁺ denote transpose, conjugate-transpose, inverse, and pseudoinverse operations, respectively. diag(𝐯) stands for diagonal matrix whose diagonal is a vector 𝐯;𝐈_𝐾 is a 𝐾×𝐾 identity matrix; ⊗, ∘, and ⊕ are the Kronecker product, Khatri-Rao product, and Hadamard product, respectively. 𝐸(⋅) denotes the expectation operator. 𝐗 denotes the estimated value of 𝐗.

2. Data Model

We consider a monostatic MIMO radar system equipped with both uniform linear arrays (ULAs) for the transmit/receive array, in which 𝑀 elements and 𝑁 elements are arranged with half-wavelength spacing between adjacent antennas, respectively. The array structure of monostatic MIMO radar is shown in Figure 1. We assume that there are 𝐾 uncorrelated targets. At time 𝑡, the output of the matched filters at the receiver can be expressed as [9] 𝐫(𝑡)=𝐚_𝑟𝜃_𝑡,1⊗𝐚_𝑡𝜃_𝑡,1,…,𝐚_𝑟𝜃_𝑡,𝐾⊗𝐚_𝑡𝜃_𝑡,𝐾𝐬(𝑡)+𝐧(𝑡)=𝐀_𝑡𝐬(𝑡)+𝐧(𝑡), (1) where 𝜃_𝑡,𝑘 is the DOA of the 𝑘th target at time 𝑡. 𝐚_𝑡(𝜃_𝑡,𝑘)=[1,exp(−𝑗𝜋sin𝜃_𝑡,𝑘),…,exp(−𝑗(𝑀−1)𝜋sin𝜃_𝑡,𝑘)]^𝑇,𝐚_𝑟(𝜃_𝑡,𝑘)=[1,exp(−𝑗𝜋sin𝜃_𝑡,𝑘),…,exp(−𝑗(𝑁−1)𝜋sin𝜃_𝑡,𝑘)]^𝑇. 𝐬(𝑡)=[𝑠₁(𝑡),𝑠₂(𝑡),…,𝑠_𝐾(𝑡)]^𝑇, and 𝑠_𝑘(𝑡)=𝜌_𝑘𝑒^{𝑗2𝜋𝑓𝑘𝑡} with 𝑓_𝑘 being Doppler frequency and 𝜌_𝑘 being the amplitude. We assume that 𝑠_𝑖(𝑡) and 𝑠_𝑗(𝑡) are uncorrelated for the different targets. 𝐧(𝑡) is the received additive white Gaussian noise with noise vector of zeros mean and covariance matrix 𝜎²𝐈_𝑀𝑁 an 𝑀𝑁×1 noise vector. ⊗ stands for Kronecker product. The matrix 𝐀_𝑡 is 𝐀_𝑡=𝐚_𝑟𝜃_𝑡,1⊗𝐚_𝑡𝜃_𝑡,1,…,𝐚_𝑟𝜃_𝑡,𝐾⊗𝐚_𝑡𝜃_𝑡,𝐾. (2) During the interval ((𝑡−1)𝑇_𝑠,𝑡𝑇_𝑠], 𝜃_𝑡,𝑘 is a constant and 𝐽 snapshots of sensor data are available for the signal processing.

Figure 1 

The array structure of monostatic MIMO radar.

3. DOA Tracking Algorithm

The DOA tracking algorithm of array signal processing in [31] requires the direction matrix of Vandermonde form. The direction matrix 𝐀_𝑡 in (2) is not a Vandermonde matrix, and the DOA tracking algorithm of array signal processing in [31] cannot be used directly for DOA tracking in MIMO radar. We employ the reduced-dimension transformation for the received signal to obtain the reduced-dimension direction matrix of Vandermonde form. In this paper, we reference the array-signal-processing DOA tracking idea in [31] to propose a low complexity DOA tracking algorithm which is suitable for MIMO radar.

3.1. Reduced-Dimension Transformation

𝐚_𝑟(𝜃_𝑡,𝑘)⊗𝐚_𝑡(𝜃_𝑡,𝑘) can be expressed by [9] 𝐚_𝑟𝜃_𝑡,𝑘⊗𝐚_𝑡𝜃_𝑡,𝑘=𝐆𝐛𝜃_𝑡,𝑘, (3) where 𝐛(𝜃_𝑡,𝑘)=[1,exp(−𝑗𝜋sin𝜃_𝑡,𝑘),…,exp(−𝑗𝜋(𝑀+𝑁−2)sin𝜃_𝑡,𝑘)]^𝑇, and𝑀𝑀⋮𝑀𝐆=10⋯00⋯001⋯00⋯0⋮⋮⋱⋮⋮⋱000⋯10⋯001⋯00⋯000100⋯0⋮⋮⋮⋱⋮⋱000⋯01000⋯010⋯00⋯001⋯0⋮⋱⋮⋮⋮⋱⋮0⋯000⋯1∈ℂ^{𝑀𝑁×(𝑀+𝑁−1)}. (4)

Then we define 𝐖≜𝐆^𝐻𝐆 as follows: 𝐖=diag1,2,…,min(𝑀,𝑁),…,min(𝑀,𝑁_{||𝑀−𝑁+1}),…,2,1. (5) Using the reduced-dimension transformation 𝐖⁻¹𝐆^𝐻 for the receiver signal 𝐫(𝑡), we obtain 𝐲(𝑡)=𝐖⁻¹𝐆^𝐻𝐫(𝑡)=𝐖⁻¹𝐖𝐛𝜃_𝑡,1,…,𝐛𝜃_𝑡,𝐾𝐬(𝑡)+𝐖⁻¹𝐆^𝐻𝐧(𝑡)=𝐁_𝑡𝐬(𝑡)+𝐖⁻¹𝐆^𝐻𝐧(𝑡), (6) where 𝐁_𝑡=𝐛𝜃_𝑡,1,…,𝐛𝜃_𝑡,𝐾∈ℂ^{(𝑀+𝑁−1)×𝐾}. (7)

𝐁_𝑡∈ℂ^{(𝑀+𝑁−1)×𝐾} is the direction matrix of Vandermonde form. Since the reduced-dimension matrix is sparse, its transformation adds less computational load.

The covariance matrix of 𝐲(𝑡) in (6) is 𝐑_𝑡∈ℂ^{(𝑀+𝑁−1)×(𝑀+𝑁−1)}. Consider 𝐑_𝑡=𝐸𝐲(𝑡)𝐲^𝐻(𝑡)=𝐁_𝑡𝐑_𝑠𝐁_𝑡^𝐻+𝐑^′_𝑛(𝑡), (8) where 𝐑_𝑠 = 𝐸[𝐬(𝑡)𝐬^𝐻(𝑡)]=diag|𝜌₁|²|𝜌₂|²⋯|𝜌_𝐾|² and 𝐑^′_𝑛(𝑡)=𝐖⁻¹𝐆^𝐻𝐑_𝑛𝐆𝐖⁻¹ with 𝐑_𝑛=𝐸[𝐧(𝑡)𝐧(𝑡)^𝐻].

3.2. DOA Tracking

The direction matrix in the reduced-dimension signal in (6) is a Vandermonde matrix, and we use the improved version of the DOA tracking in [31] for DOA tracking of MIMO radar.

We assume that 𝜃_𝑡,𝑘 is slowly varying. The DOA of the 𝐾 targets at time 𝑡 is 𝜃_𝑡=[𝜃_𝑡,1,𝜃_𝑡,2,…,𝜃_𝑡,𝐾]. Similarly, the DOA at time 𝑡+1 is 𝜃_𝑡+1=[𝜃_𝑡+1,1,𝜃_𝑡+1,2,…,𝜃_𝑡+1,𝐾], with 𝜃_𝑡+1,𝑘 being the DOA of the 𝑘th target at time 𝑡+1. We define Δ𝜃_𝑡=𝜃_𝑡+1−𝜃_𝑡, (9) where Δ𝜃_𝑡=[Δ𝜃_𝑡,1,Δ𝜃_𝑡,2,…,Δ𝜃_𝑡,𝐾] with Δ𝜃_𝑡,𝑘=𝜃_𝑡+1,𝑘−𝜃_𝑡,𝑘.

We define 𝐑_𝑡+1 as the covariance matrices of the signal 𝐲(𝑡) at time 𝑡+1. The covariance matrix is 𝐑_𝑡+1=𝐁_𝑡+1𝐑_𝑠𝐁^𝐻_𝑡+1+𝐑^′_𝑛(𝑡+1), (10) where 𝐁_𝑡+1=[𝐛(𝜃_𝑡+1,1),…,𝐛(𝜃_𝑡+1,𝐾)] is the direction matrix at time 𝑡+1 and 𝐛(𝜃_𝑡+1,𝑘)=[1,exp(−𝑗𝜋sin𝜃_𝑡+1,𝑘),…,exp(−𝑗𝜋(𝑀+𝑁−2)sin𝜃_𝑡+1,𝑘)]^𝑇. 𝐑^′_𝑛(𝑡+1) is the covariance matrix of the noise.

Then we can obtain [31, 36] Δ𝐑_𝑡=𝐑_𝑡+1−𝐑_𝑡. (11)Δ𝐑_𝑡 is denoted by Δ𝐑_𝑡=𝐁_𝑡+1𝐑_𝑠𝐁^𝐻_𝑡+1−𝐁_𝑡𝐑_𝑠𝐁^𝐻_𝑡+𝐑^′_𝑛(𝑡+1)−𝐑^′_𝑛(𝑡). (12)

We assume that the noise covariance matrix at time 𝑡+1 is approximately equal to that at time 𝑡, and then we have Δ𝐑_𝑦≃𝐁_𝑡+1𝐑_𝑠𝐁^𝐻_𝑡+1−𝐁_𝑡𝐑_𝑠𝐁^𝐻_𝑡. (13)

Using the Vandermonde characteristic of the matrices 𝐁_𝑡 and 𝐁_𝑡+1, the noiseless Δ𝐑_𝑡 in (13) can be denoted by [31, 32, 36] Δ𝐑_𝑡=0𝑏₁𝑏₂⋯𝑏_{𝑀+𝑁−2}𝑏^∗₁0𝑏₁⋯𝑏_{𝑀+𝑁−3}𝑏^∗₂𝑏^∗₁0⋯𝑏_{𝑀+𝑁−4}𝑏⋮⋮⋮⋱⋮_{∗𝑀+𝑁−2}𝑏^∗_{𝑀+𝑁−3}𝑏^∗_{𝑀+𝑁−4},⋯0 (14)

where 𝑏_𝑛=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋sin(𝜃𝑡,𝑖+Δ𝜃𝑡,𝑖)}−𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖},𝑛=1,2,…,𝑀+𝑁−2, (15) where 𝑠_𝑖 is the (𝑖,𝑖) element of the matrix 𝐑_𝑠 and 𝑠_𝑖=|𝜌_𝑖|². 𝑏_𝑛 in (15) can be expanded as 𝑏_𝑛=∑^𝐾_𝑖=1𝑠_𝑖(𝑒^{𝑗𝑛𝜋[sin𝜃𝑡,𝑖cosΔ𝜃𝑡,𝑖+cos𝜃𝑡,𝑖sinΔ𝜃𝑡,𝑖]}−𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}). Considering that Δ𝜃_𝑡,𝑖 is very small, sin𝜃_𝑡,𝑖+Δ𝜃_𝑡,𝑖≃sin𝜃_𝑡,𝑖+Δ𝜃_𝑡,𝑖cos𝜃_𝑡,𝑖. (16)

Then 𝑏_𝑛 in (15) can be denoted by 𝑏_𝑛≃^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋(sin𝜃𝑡,𝑖+Δ𝜃𝑡,𝑖cos𝜃𝑡,𝑖)}−𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝜋𝑛sin𝜃𝑡,𝑖}𝑒^{𝑗𝑛𝜋Δ𝜃𝑡,𝑖cos𝜃𝑡,𝑖}−1. (17)

Considering that 𝑥 is very small, 𝑒^𝑥−1≃𝑥. And then 𝑏_𝑛 in (17) can be rewritten as 𝑏_𝑛≈^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑛𝑗𝜋sin𝜃𝑡,𝑖}𝑗𝑛𝜋Δ𝜃_𝑡,𝑖cos𝜃_𝑡,𝑖. (18)

According to (18), we can construct the following matrix:^{𝑏1⋮𝑏𝑀+𝑁−2}_{𝑏∗1⋮𝑏∗𝑀+𝑁−2𝐛}=^{𝑠1𝑒𝑗𝜋sin𝜃𝑡,1𝑗𝜋cos𝜃𝑡,1⋯𝑠𝐾𝑒𝑗𝜋sin𝜃𝑡,𝐾𝑗𝜋cos𝜃𝑡,𝐾𝑠⋮⋱⋮1𝑒𝑗(𝑀+𝑁−2)𝜋sin𝜃𝑡,1𝑗(𝑀+𝑁−2)𝜋cos𝜃𝑡,1⋯𝑠𝐾𝑒𝑗(𝑀+𝑁−2)𝜋sin𝜃𝑡,𝐾𝑗(𝑀+𝑁−2)𝜋cos𝜃𝑡,𝐾}_{−𝑠1𝑒−𝑗𝜋sin𝜃𝑡,1𝑗𝜋cos𝜃𝑡,1⋯−𝑠𝐾𝑒−𝑗𝜋sin𝜃𝑡,𝐾𝑗𝜋cos𝜃𝑡,𝐾⋮⋱⋮−𝑠1𝑒−𝑗(𝑀+𝑁−2)𝜋sin𝜃𝑡,1𝑗(𝑀+𝑁−2)𝜋cos𝜃𝑡,1⋯−𝑠𝐾𝑒−𝑗(𝑀+𝑁−2)𝜋sin𝜃𝑡,𝐾𝑗(𝑀+𝑁−2)𝜋cos𝜃𝑡,𝐾𝐕𝑡}^{Δ𝜃𝑡,1Δ𝜃𝑡,2}_{⋮Δ𝜃𝑡,𝐾Δ𝜽𝑡}. (19)

Equation (19) can be rewritten with matrix form 𝐕_𝑡Δ𝜽_𝑡=𝐛. (20)

From (14), we find that Δ𝐑_𝑡 is a Toeplitz matrix, whose elements in a straight line paralleled to the principal diagonal are equal. We take the following processing to estimate 𝑏_𝑛 and 𝑏^∗_𝑛 (𝑛=1,…,𝑀+𝑁−2) to eliminate the noise. Consider 𝑏̂_𝑛=1𝑀+𝑁−1−𝑛^{𝑀+𝑁−1−𝑛}_𝑖=1Δ𝐑_𝑡()𝑏̂𝑖,𝑛+𝑖_∗𝑛=1𝑀+𝑁−1−𝑛^{𝑀+𝑁−1−𝑛}_𝑖=1Δ𝐑_𝑡(𝑛+𝑖,𝑖), (21) where Δ𝐑_𝑡(𝑖,𝑗) is the (𝑖,𝑗) element of the matrix Δ𝐑_𝑡. In the DOA tracking algorithm in [31], 𝑏̂_𝑛 and 𝑏̂^∗_𝑛 are obtained through 𝑏̂_𝑛=Δ𝐑_𝑡(1,𝑛+1) and 𝑏̂^∗_𝑛=Δ𝐑_𝑡(𝑛+1,1), respectively. The proposed algorithm fully uses the Toeplitz matrix property of Δ𝐑_𝑡 to eliminate the noise and improves the parameter estimation performance.

Using the method of least square, we get Δ𝜽_𝑡=(𝐕_𝑡^𝐻𝐕_𝑡)⁻¹𝐕^𝐻_𝑡𝐛, (22) where 𝐛=𝑏̂₁⋯𝑏̂_{𝑀+𝑁−2}𝑏̂^∗₁⋯𝑏̂^∗_{𝑀+𝑁−2}^𝑇.

Through the above analysis, the angles at time of 𝑡 and 𝑡+1 are automatically associated, and the proposed algorithm avoids an extra data association.

For the finite samples, the covariance matrix in (8) is estimated by 𝐑_𝑡=∑^𝐽_𝑙=1𝐲(𝑡_𝑙)𝐲^𝐻(𝑡_𝑙), and we also get 𝐑_𝑡+1, which is the estimate of the covariance matrix at time 𝑡+1. We estimate 𝑏_𝑛 and 𝑏̂^∗_𝑛 as follows:𝑏̂_𝑛=1𝑀+𝑁−1−𝑛^{𝑀+𝑁−1−𝑛}_𝑖=1Δ𝐑_𝑡(𝑖,𝑛+𝑖)𝑏̂^∗_𝑛=1𝑀+𝑁−1−𝑛^{𝑀+𝑁−1−𝑛}_𝑖=1Δ𝐑_𝑡(𝑛+𝑖,𝑖), (23a)(23b)where Δ𝐑_𝑡=𝐑_𝑡+1−𝐑_𝑡. We use (22) to estimate Δ𝜽_𝑡, where 𝐛=𝑏̂₁⋯𝑏̂_{𝑀+𝑁−2}𝑏̂^∗₁⋯𝑏̂^∗_{𝑀+𝑁−2}^𝑇, whose element is estimated via (23a) and (23b).

Till now, we show the major steps of DOA tracking algorithm for monostatic radar as follows.

Step 1. Use reduced-dimension matrix 𝐖⁻¹𝐆^𝐻 for the received signal 𝐫(𝑡) and estimate the covariance matrix 𝐑_𝑡.

Step 2. Get covariance matrix 𝐑_𝑡+1 at time 𝑡+1, and we calculate Δ𝐑_𝑡=𝐑_𝑡+1−𝐑_𝑡.

Step 3. We compute 𝑏̂_𝑛 and 𝑏̂^∗_𝑛 via (23a) and (23b) and construct 𝐛=𝑏̂₁⋯𝑏̂_{𝑀+𝑁−2}𝑏̂^∗₁⋯𝑏̂^∗_{𝑀+𝑁−2}^𝑇.

Step 4. We estimate Δ𝜽_𝑡 via (22), and the DOA at time 𝑡+1 is 𝜽_𝑡+1=𝜽_𝑡+Δ𝜽_𝑡.

Step 5. Repeat Steps 1–4 to estimate DOA at next time.

Remark 1. In this paper, the number of targets in MIMO radar is assumed to be preknown. If we have no knowledge about it, we may use the existing source-number estimation technique in [37] to obtain an estimate of the number of targets.

Remark 2. The initial angles in DOA tracking are obtained by ESPRIT algorithm or other DOA algorithms. The initial DOA is 𝜽₁, and we get angles 𝜽_𝑚=𝜽₁+∑^𝑚−1_𝑖=1Δ𝜽_𝑖.

Remark 3. We assume that the noise covariance matrices of adjacent time are approximately equal. The noise component in (12) will be eliminated no matter which type of noise. The proposed algorithm still works well for the case of the colored noise.

Remark 4. The proposed algorithm works well in condition of small value of Δ𝜃_𝑡,𝑘. When Δ𝜃_𝑡,𝑘 becomes large, the proposed algorithm may fail to work.

3.3. Complexity Analysis and Advantages of the Proposed Algorithm

Since the reduced-dimension matrix is sparse, its transformation adds less computational load. The proposed algorithm does not require eigenvalue decomposition of the covariance matrix and avoids an extra data association. For the proposed algorithm, the calculation of the covariance matrix requires 𝑂((𝑀+𝑁−1)²𝐽), and the computation of (𝐕^𝐻_𝑡𝐕_𝑡)⁻¹𝐕^𝐻_𝑡𝐛 needs (2𝐾²(𝑀+𝑁−2)+𝐾³+2𝐾(𝑀+𝑁−2)+𝐾²). The major computational complexity of the proposed algorithm is 𝑂((𝑀+𝑁−1)²𝐽+2𝐾²(𝑀+𝑁−2)+𝐾³+2𝐾(𝑀+𝑁−2)+𝐾²). Table 1 and Figure 2 show the computational complexity comparison among the proposed algorithm and other DOA tracking algorithms. We find that the proposed algorithm has much lower computational load than PAST in [25] and PASTd in [26].

Table 1 

Complexity comparison.

Figure 2 

Complexity comparison with 𝑀=8, 𝑁=6, 𝐾=3, and different 𝐽.

The advantages of the proposed algorithm can be presented as follows.(1)The proposed algorithm does not require eigenvalue decomposition of the covariance matrix and has lower complexity than the conventional DOA tracking algorithms including PAST and PASTd.(2)The proposed algorithm can implement automatic association of DOA for monostatic MIMO radar.(3)The proposed algorithm has much better DOA tracking performance than the DOA tracking algorithm in [31], which will be shown in Section 5.

4. Error Analysis and CRLB

In this section, we derive the variance of DOA tracking and CRLB. We assume that the observation noise variances are almost the same at adjacent time. When computing Δ𝜽, we use some approximate calculations such as 𝑒^𝑥−1≈𝑥 and sin𝑥≈𝑥 when 𝑥 is smaller. This leads to a little difference compared with the perfect value. Consider 𝑥sin𝑥=𝑥−³3!+Λ𝑥cos𝑥=1−²2!+Λ^′𝑒^𝑥−1=𝑥+Λ^′′, (24a)(24b)(24c)where Λ,Λ^′,Λ^′′ are the high-order expansion terms.

According to (16), (17), (18), (24a), (24b), and (24c), we have 𝑏_𝑛=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋[sin𝜃𝑡,𝑖cosΔ𝜃𝑡,𝑖+cos𝜃𝑡,𝑖sinΔ𝜃𝑡,𝑖]}−𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋[sin𝜃𝑡,𝑖(1−(Δ𝜃2𝑡,𝑖/2!)+Λ′)+cos𝜃𝑡,𝑖(Δ𝜃𝑡,𝑖−(Δ𝜃3𝑡,𝑖/3!)+Λ)]}−𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋[sin𝜃𝑡,𝑖+cos𝜃𝑡,𝑖Δ𝜃𝑡,𝑖]}×𝑒^{𝑗𝑛𝜋[sin𝜃𝑡,𝑖(−(Δ𝜃2𝑡,𝑖/2!)+Λ′)+cos𝜃𝑡,𝑖(−(Δ𝜃3𝑡,𝑖/3!)+Λ)]}−𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}×𝑒^{𝑗𝑛𝜋cos𝜃𝑡,𝑖Δ𝜃𝑡,𝑖}×𝑒^{𝑗𝑛𝜋[sin𝜃𝑡,𝑖(−(Δ𝜃2𝑡,𝑖/2!)+Λ′)+cos𝜃𝑡,𝑖(−(Δ𝜃3𝑡,𝑖/3!)+Λ)]}=−1_𝐾𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}×𝑗𝑛𝜋cos𝜃_𝑡,𝑖Δ𝜃_𝑡,𝑖+𝑗𝑛𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+𝑗Λ^′′+𝑗Λ^′′=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}𝑗𝑛𝜋cos𝜃_𝑡,𝑖Δ𝜃_𝑡,𝑖+^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}𝑗𝑛𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+𝑗Λ^′′. (25)

𝜕𝑏_𝑛 is estimation error of 𝑏_𝑛, and 𝜕𝑏_𝑛 can be shown as follows: 𝜕𝑏_𝑛=^𝐾_𝑖=1𝑠_𝑖𝑒^{𝑗𝑛𝜋sin𝜃𝑡,𝑖}𝑗𝑛𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+𝑗Λ^′′. (26)

According to (26) and (22), the variance of Δ𝑏_𝑛 is denoted by𝐸𝜕𝑏_𝑛²=1×𝑀+𝑁−1−𝑛_𝐾𝑖=1𝑠²_𝑖𝑛𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+𝑛𝜋cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+Λ^′′.² (27a)

And we have 𝐸𝜕𝑏_𝑖𝜕𝑏^∗_𝑗=0,∀𝑖≠𝑗𝐸𝜕𝑏²_𝑛=0. (27b)(27c)

We define that 𝜕𝐛 is the estimation error of 𝐛. 𝜕𝐛=[𝜕𝑏₁,…,𝜕𝑏_{𝑀+𝑁−2},𝜕𝑏^∗₁,…,𝜕𝑏^∗_{𝑀+𝑁−2}]^𝑇. Then the variance of 𝜕𝐛 is𝐸|𝜕𝐛|²=𝐸𝜕𝑏₁²⋮𝐸𝜕𝑏_{𝑀+𝑁−2}²𝐸𝜕𝑏₁²⋮𝐸𝜕𝑏_{𝑀+𝑁−2}²=1𝑀+𝑁−2^𝐾_𝑖=1𝑠²_𝑖𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+𝜋cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+Λ^′′2_𝐾𝑖=1⋮𝑠²_𝑖(𝑀+𝑁−2)𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+(𝑀+𝑁−2)𝜋cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+Λ^′′21𝑀+𝑁−2^𝐾_𝑖=1𝑠²_𝑖𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+𝜋cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+Λ^′′2_𝐾𝑖=1⋮𝑠²_𝑖(𝑀+𝑁−2)𝜋sin𝜃_𝑡,𝑖−Δ𝜃²_𝑡,𝑖2!+Λ^′+(𝑀+𝑁−2)𝜋cos𝜃_𝑡,𝑖−Δ𝜃³_𝑡,𝑖3!+Λ+Λ^′′2. (28)

According to [38], we get the variance of Δ𝜃_𝑡,𝑖varΔ𝜃_𝑡,𝑖=𝐸𝐕⁺_𝑡,𝑖𝜕𝐛²+Re𝐸𝐕⁺_𝑡,𝑖𝜕𝐛²2, (29) where 𝐕⁺_𝑡,𝑖 denotes the 𝑖th row of 𝐕⁺_𝑡 and 𝐕⁺_𝑡 is the pseudoinverse of 𝐕_𝑡. According to (27a), (27b), and (27c), we get varΔ𝜃_𝑡,𝑖=𝐕⁺_𝑡,𝑖diag𝐸|𝜕𝐛|²𝐕^+𝐻_𝑡,𝑖+Re𝐕⁺_𝑡,𝑖𝐸𝜕𝐛𝜕𝐛^𝑇𝐕^+𝑇_𝑡,𝑖2=𝐕⁺_𝑡,𝑖diag𝐸|𝜕𝐛|²𝐕^+𝐻_𝑡,𝑖2, (30) where 𝐸[|𝜕𝐛|²] is shown in (28).

According to [39], we derive the CRLB for DOA estimation in monostatic MIMO-radar at time 𝑡. Consider (𝑡)=𝜎CRLB²𝐃2𝐽Re^𝐻Π^⟂_𝐀𝑡𝐃⊕𝐏^𝑇−1, (31) where Π^⟂_𝐀𝑡=𝐈_𝑀𝑁−𝐀_𝑡(𝐀^𝐻_𝑡𝐀_𝑡)⁻¹𝐀^𝐻_𝑡, 𝐃=[𝐝₁,𝐝₂,…,𝐝_𝐾], 𝐝_𝑘=𝜕(𝐚_𝑟(𝜃_𝑡,𝑘)⊗𝐚_𝑡(𝜃_𝑡,𝑘)/𝜕𝜃_𝑡,𝑘, and

𝚸=(1/𝐽)∑^𝐽_𝑛=1𝐬(𝑡_𝑛)𝐬^𝐻(𝑡_𝑛). ⊕ stands for Hadamard product.

Then we can define the average CRLB as follows: 1CRLB=𝑇^𝑇_𝑡=1CRLB(𝑡), (32) where 𝑇 is the total tracking time.

5. Simulation Results

The Monte Carlo simulations are adopted to assess DOA tracking performance of the proposed algorithm. We suppose that there are three moving targets. We define root-mean square error (RMSE) as 1RMSE=𝐾^𝐾_𝑘=11𝐹^𝐹_𝑓=11𝑇^𝑇_𝑡=1𝜃_{𝑘,𝑓,𝑡}−𝜃_{𝑘,𝑓,𝑡}², (33) where 𝜃_{𝑘,𝑓,𝑡} is the estimate of DOA 𝜃_𝑡,𝑘 of the 𝑓th Monte Carlo trial. 𝐹 is the times of Monte Carlo trial and 𝐹=1000. We note that 𝑀, 𝑁, and 𝐾 are the number of transmit antennas, receive antennas, and targets, respectively. During the tracking procedure, the targets are tracked over an interval of 100 s with 𝑇_𝑠 = 1 s. During each 1 s interval, 𝐽 snapshots of sensor data are generated and used to estimate DOA.

Figure 3 depicts the DOA tracking result of the proposed algorithm for 𝑀=8, 𝑁=6, 𝐾=3, 𝐽=100, and SNR=15 dB. It is shown that the proposed algorithm can track the DOA in monostatic MIMO radar effectively.

Figure 3 

DOA tracking results of the proposed algorithm at SNR = 15 dB (Scene 1).

Figures 4-5 show the DOA tracking result of the proposed algorithm for other moving trajectories of targets with 𝑀=8, 𝑁=6, 𝐾=3, 𝐽=100, and SNR=15 dB. From Figures 4-5 we find that the proposed algorithm can work well, which proves the robustness of the proposed algorithm.

Figure 4 

DOA tracking results of the proposed algorithm at SNR = 15 dB (Scene 2).

Figure 5 

DOA tracking results of the proposed algorithm at SNR = 15 dB (Scene 3).

Figure 6 shows the DOA tracking performance comparison with 𝑀=8, 𝑁=6, and 𝐾=3, where we compare the proposed algorithm against the DOA tracking algorithm in [31], Kalman-PASTd algorithm in [35], and CRLB. It is shown in Figure 6 that the proposed algorithm has better DOA tracking performance than the DOA tracking algorithm in [31], since the proposed algorithm fully uses the Toeplitz matrix property of Δ𝐑_𝑡 to eliminate the noise and improves the estimation performance. The proposed algorithm has less DOA tracking performance than Kalman-PASTd algorithm, which has a heavy computational complexity.

Figure 6 

DOA tracking performance comparison with 𝑀=8 and 𝑁=6.

Figure 7 investigates the DOA tracking performance of the proposed algorithm with different values of 𝐾. From Figure 7 we find that DOA tracking performance of the proposed algorithm improves with the decreasing of 𝐾. When 𝐾 increases, the interference will enhance.

Figure 7 

DOA tracking with different values of 𝐾.

Figures 8-9 show DOA tracking performance with different 𝑀/𝑁. It is clear that the DOA tracking performance of the proposed algorithm is improved as the number of transmit/receive antennas increases. Multiple transmit/receive antennas improve DOA tracking performance because of diversity gain.

Figure 8 

DOA tracking with different values of 𝑀.

Figure 9 

DOA tracking with different values of 𝑁.

Figure 10 depicts the DOA tracking performance of the proposed algorithm with 𝑀=8, 𝑁=6, 𝐾=3, 𝐽=100, and SNR=15 dB, and the different DOA spacing between two consecutive observations is considered. It is shown in Figure 10 that the angle tracking performance of the proposed algorithm degrades with increasing of DOA spacing between two consecutive observations.

Figure 10 

DOA tracking performance with different angle spacing.

6. Conclusions

In this paper we have presented the DOA tracking of multiple moving targets for monostatic MIMO radar. The proposed algorithm obtains DOA estimation of the target via the difference in previous and current covariance matrix of the reduced-dimension transformation signals, and the proposed algorithm reduces the computational complexity and realizes automatic association of DOA. Error analysis and CRLB of DOA tracking are derived in the paper. The simulation results demonstrate effectiveness and robustness of the proposed algorithm. Our research provides technical support for the practical application of MIMO radar.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by China NSF Grants (61371169, 61301108, and 61071164), Jiangsu Planned Projects for Postdoctoral Research Funds (1201039C), China Postdoctoral Science Foundation (2012M521099 and 2013M541661), Open Project of Key Laboratory of Modern Acoustic of Ministry of Education (Nanjing University), Aeronautical Science Foundation of China (20120152001), Qing Lan Project, Priority Academic Program Development of Jiangsu High Education Institutions, and Fundamental Research Funds for the Central Universities (NS2013024, kfjj130114, and kfjj130115).