International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 9170403, 11 pages

http://dx.doi.org/10.1155/2016/9170403

## Separate DOD and DOA Estimation for Bistatic MIMO Radar

^{1}School of Electrical and Information Engineering, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China^{2}National Mobile Communications Research Laboratory, Southeast University, 2 Sipailou Road, Nanjing 210096, China

Received 26 February 2016; Revised 6 June 2016; Accepted 20 June 2016

Academic Editor: Qinghua Guo

Copyright © 2016 Lin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A novel MUSIC-type algorithm is derived in this paper for the direction of departure (DOD) and direction of arrival (DOA) estimation in a bistatic MIMO radar. Through rearranging the received signal matrix, we illustrate that the DOD and the DOA can be separately estimated. Compared with conventional MUSIC-type algorithms, the proposed separate MUSIC algorithm can avoid the interference between DOD and DOA estimations effectively. Therefore, it is expected to give a better angle estimation performance and have a much lower computational complexity. Meanwhile, we demonstrate that our method is also effective for coherent targets in MIMO radar. Simulation results verify the efficiency of the proposed method, particularly when the signal-to-noise ratio (SNR) is low and/or the number of snapshots is small.

#### 1. Introduction

Multiple-input multiple-output (MIMO) radar, which utilizes multiple antennas to simultaneously transmit diverse waveforms and receive reflected signals, has many potential advantages over the conventional phased-array radar [1–4]. Direction of departure (DOD) and direction of arrival (DOA) estimation [5–8] is a key issue in MIMO radar signal processing, and it has attracted a lot of attention. Many algorithms for DOD and DOA estimation have been established in the literatures [9–11]. By exploiting the invariance property of both the transmit array and the receive array, [12] developed a subspace method based on the classical rotational invariance techniques (ESPRIT) algorithm, but additional pair matching is required. To avoid pair matching, an improved ESPRIT algorithm was presented in [13], whose complexity is lower than the algorithm in [12]. A real-valued ESPRIT algorithm was proposed in [14], where all the complex computations are transformed into real-valued ones. As a consequence, it can further reduce the computational complexity and ameliorate the performance for DOD and DOA estimation. Moreover, a propagator method (PM) algorithm for DOD and DOA estimation for MIMO radar was investigated in [15], which can construct the signal subspace without the eigenvalue decomposition of covariance matrix. So, the PM algorithm has lower complexity than ESPRIT-type methods [12–14, 16, 17], but it has a low performance of DOD and DOA estimation.

It is well known that multiple signal classification (MUSIC) algorithms have better performance than ESPRIT-type and PM-type algorithms. It has been proved that two-dimension MUSIC (2D-MUSIC) algorithm can be used for DOD and DOA estimation in MIMO radar and has a good angle estimation accuracy; however, it requires high computation complexity. The method in [18] combines ESPRIT and root-MUSIC to achieve the compromise between the complexity and estimation performance. In [19], a reduced-dimension MUSIC (RD-MUSIC) algorithm was proposed for DOD and DOA estimation, which can reduce the computational cost by replacing the two-dimensional searching with one-dimensional searching. All these MUSIC-type algorithms [19–22] can pair DOD and DOA estimation automatically. However, none of them can avoid the interference between DOD and DOA estimations. For example, if DOA is first estimated in these methods, then the estimation of DOD will be influenced by the estimation error of DOA. Therefore, the performance of these MUSIC-type algorithms might seriously degrade.

To solve the aforementioned problem, in this paper, a separate MUSIC algorithm for DOD and DOA estimation is presented. The algorithm first addresses the DOD estimation and then rearranges the received signal matrix to estimate DOA. Compared with conventional MUSIC-type algorithms, the proposed algorithm, due to the utilization of separate DOD and DOA estimation, avoids the interference between DOD and DOA estimations effectively; therefore, the algorithm gives a better angle estimation accuracy and has a much lower computational complexity, when signal-to-noise ratio (SNR) is low and/or the number of snapshots is small. Meanwhile, this paper guarantees that the algorithm is also effective for coherent targets in MIMO radar.

This paper is organized as follows. Section 2 addresses the data model for bistatic MIMO radar. Section 3 reviews the existing MUSIC-type algorithms for DOD and DOA estimation. Section 4 proposes our separate MUSIC algorithm for DOD and DOA estimation. Finally, simulation results and conclusions are given in Sections 5 and 6, respectively.

#### 2. Data Model

Consider a bistatic MIMO radar system with transmit antennas and receive antennas, both of which are half-wavelength spaced uniform linear arrays (ULAs). At the transmit site, different narrowband waveforms are emitted simultaneously, which have identical bandwidth and central frequency but are temporally orthogonal. In each receiver, the echoes are processed for all of the transmitted waveforms. Assume that there are uncorrelated targets located at the same range, and the DOD and DOA of the th target relative to the transmitter and the receiver are denoted by and , respectively.

The output of the matched filters at the receiver can be expressed aswhere is an array manifold matrix withbeing the receive steering vector and the transmit steering vector, respectively. is the Kronecker product and is the transpose operation. is a column vector consisting of the amplitudes and phases of the targets at time . represents an complex Gaussian white noise vector of zeros mean and covariance matrix .

#### 3. Review of MUSIC-Type Algorithms

Let be denoted as the data matrix composed of snapshots of ; then the matrix can be expressed aswhere

The covariance matrix of is defined as where is the source covariance matrix and is the Hermitian transpose. Let the eigenvalue decomposition of bewhere denotes a diagonal matrix formed by largest eigenvalues and denotes a diagonal matrix formed by the rest of the smaller eigenvalues. and represent the signal subspace and noise subspace, respectively, of which contains the eigenvectors corresponding to the largest eigenvalues and consists of the rest of the eigenvectors. As the noise subspace is orthogonal to the actual target directional vector, we can construct the 2D-MUSIC spatial spectrum function as follows:Here, we have the largest peaks of corresponding to the estimates of the DODs and DOAs for the targets.

Since 2D-MUSIC requires exhaustive two-dimensional searching, it is normally inefficient due to high computational cost, and therefore Zhang et al. [19] proposed a RD-MUSIC algorithm which has a much lower computational complexity. Firstly, letThen, the RD-MUSIC algorithm rewrites asThe DOAs can be found from the quadratic optimization problem . In order to avoid a trivial zero solution, some normalized constraint (e.g., , where ) should be added to the optimization problem; that is,According to the Karush-Kuhn-Tucker (KKT) conditions, we haveInserting (11) into the objective function , DOAs can be estimated viaSearching the objective function of along , we can find the largest peaks that correspond to DOAs (). Substituting (11) for the estimated DOAs , we may obtain vectors . Finally, the DODs can be found after some algebraic calculation with .

#### 4. Separate MUSIC Algorithm for DOD and DOA Estimation

Obviously, the DOD estimation of RD-MUSIC algorithm is based on the estimate of the DOA. It is likely to degrade the DOD estimation performance due to the interference caused by the DOA estimation. In this section, we will propose a separate MUSIC algorithm to handle the problem. To this end, we first address the DOD estimation and then rearrange the received signal matrix to estimate DOA. Finally, we will address the pair matching.

##### 4.1. DOD Estimation

In order to simplify the notation, we rewrite with . Then, the array manifold matrix becomesWe partition the matrix , defined in (3), into submatrices:whereand denotes a diagonal matrix constructed by the vector .

In order to independently estimate DOD, we construct a new matrix containing the information of DOD; that is,According to the structure of in (14), can be rewritten byNote that and can be seen as the signal matrix and noise matrix corresponding to the new measurement matrix , respectively.

The covariance matrix of is defined aswhere is the source covariance matrix corresponding to . Performing the eigenvalue decomposition of , we can get the noise subspace corresponding to . Then, we construct the following MUSIC spatial spectrum function for DOD estimation:Searching , we can obtain largest peaks of . The corresponding () are taken as the estimates of the DODs.

Actually, our method is effective for coherent DOD estimation. Considering a completely coherent environment [23–25], we havewhere () represents the complex attenuation of the th signal with respect to the first signal . Using (20) with , it is easy to see thatwhere . So the source covariance matrix takes the form [26]where

Clearly, the rank of is equal to the rank of . Since and the diagonal matrix is of full rank, the rank of is the same as that of . Now, the rank of the Vandermonde matrix is , and, hence, if . We can find that the rank of is the same as the number of the targets. Therefore, the proposed algorithm can be applied to solve multiple coherent targets for DOD estimation.

##### 4.2. DOA Estimation

Define a new matrix , where . In order to simplify the notation, we rewrite with ; then, we have

Obviously, there exists an transformation matrix corresponding to the finite number of row interchanged operations such that , where

Using the structure of the matrix , we introduce a virtual data matrix , or, equivalently,where . Divide the matrix into submatrices:where

In order to estimate DOA independently, we construct a new matrix :which contains the information of DOA. According to the structure of in (27), can be expressed asNote that and can be seen as the virtual signal matrix and noise matrix corresponding to the measurement , respectively. The covariance matrix of can be calculated bywhere is the source covariance matrix corresponding to . Performing the eigenvalue decomposition of , we obtain the noise subspace matrix of . Thus, the MUSIC spatial spectrum function for DOA estimation can be expressed asSearching , we obtain the largest peaks of corresponding to the estimated DOAs (). Moreover, since has similar structure characteristics to , the rank of is also the same as the number of the targets. Therefore, the proposed algorithm can also be applied to solve multiple coherent targets for DOA estimation.

##### 4.3. Pair Matching for DOD and DOA

Now, the estimates of DODs and DOAs have been acquired from the functions and . Then, we form easily vectors and vectors , respectively. The paired DOD and DOA, denoted by , where and , maximize the spatial spectrum function (7):

Summarily, we show the major steps of our proposed algorithm as follows.

*Step 1. *Form and according to (16) and (29).

*Step 2. *Calculate the sample covariance matrix of and , respectively, and obtain the noise subspace matrices and correspondingly.

*Step 3. *Acquire the estimates of DODs and DOAs by searching through (19) and through (32), respectively.

*Step 4. *Pair the estimates of DODs and DOAs with (33).

#### 5. Simulation Results

In this section, we introduce the simulation results of the proposed algorithm in comparison with the standard ESPRIT method [13], the Unitary ESPRIT method [14], the RD-MUSIC method [19], and the PM method [15]. The root mean squared error (RMSE) of the DOD or DOA estimation is defined aswhere is the estimate of DOD or DOA corresponding to true value of DOD or DOA of the Monte Carlo trial. In the following simulations, we adopt the 200 Monte Carlo trials for a bistatic MIMO radar system with uniform linear arrays with half-wavelength spacing.

Simulation 1 presents the angle paired results of the proposed algorithm. In this simulation, we assume a ULA composed of transmit antennas and receive antennas, and we consider uncorrelated targets located at angles , , , , and , respectively. The number of snapshots is , and the SNR is dB. As can be seen from Figure 1, the proposed algorithm can correctly estimate and pair the DODs and DOAs.