Abstract
The beamspace unitary ESPRIT (B-UESPRIT) algorithm for estimating the joint direction of arrival (DOA) and the direction of departure (DOD) in bistatic multiple-input multiple-output (MIMO) radar is proposed. The conjugate centrosymmetrized DFT matrix is utilized to retain the rotational invariance structure in the beamspace transformation for both the receiving array and the transmitting array. Then the real-valued unitary ESPRIT algorithm is used to estimate DODs and DOAs which have been paired automatically. The proposed algorithm does not require peak searching, presents low complexity, and provides a significant better performance compared to some existing methods, such as the element-space ESPRIT (E-ESPRIT) algorithm and the beamspace ESPRIT (B-ESPRIT) algorithm for bistatic MIMO radar. Simulation results are conducted to show these conclusions.
1. Introduction
Multiple-input multiple-output (MIMO) radar [1–6] is developed on the basis of MIMO communication theory, which has gained increasing attention and wide investigation in recent years. MIMO radar can emit orthogonal waveforms simultaneously through multiple antennas and also extract the orthogonal waveforms by using a bank of matched filters. Compared with the traditional phased array radars, a lot of potential advantages of MIMO radars, such as more degrees of freedom (DOFs) [2], better parameter identifiability [2], and higher angular estimation accuracy [3], increasingly appear. According to the configuration of transmitting/receiving arrays, MIMO radars can be grouped into two classes: the former is called statistical MIMO radar, where the transmitting and receiving antennas are widely spaced [1, 3, 4]. It aims at overcoming the radar cross section (RCS) scintillation effect which was encountered in radar systems by capitalizing on the spatial diversity [5]. The latter is known as monostatic MIMO radar or bistatic MIMO radar [2, 6–10], where the transmitting and receiving antennas are closely spaced. Monostatic or bistatic MIMO radar, which can form receiving beam and virtual transmitting beam jointly at the receiver [2], has many advantages, such as narrower beamwidth, lower sidelobes, higher angular resolution, and higher angular estimation accuracy [6, 11]. And this paper focuses on the bistatic MIMO radar.
For the bistatic MIMO radar, one of the most important issues is to estimate the directions of departure and arrival of multiple targets from the received signals corrupted by noise. So far various approaches have been put forward. In [7], the two-dimensional (2D) Capon estimator is introduced to estimate the DOAs and the DODs of the targets in bistatic MIMO radar. To reduce the computational cost, a polynomial rooting estimator is introduced in [8]. The 2D multiple signal classification (MUSIC) algorithm and reduced-dimension MUSIC (RD-MUSIC) algorithm are discussed in [12, 13]. To alleviate the computational burden, the ESPRIT algorithm [14] is used for target direction estimation by utilizing the invariance property of both the transmitting array and the receiving array. Nevertheless, an additional matched pair is required in the ESPRIT algorithm. In order to solve this problem, a unitary ESPRIT algorithm for bistatic MIMO radar is proposed in [15], which fully exploits the real-valued rotational invariance equations of signal subspace to estimate DODs and DOAs that are paired automatically. When some a priori knowledge of the angle information of sources is known, the beamspace ESPRIT (B-ESPRIT) [16] algorithm is raised to reduce computational complexity; however, an additional pair matching is also needed and a degradation of the algorithm’s performance can be observed as well.
To overcome the aforementioned problems, a beamspace unitary ESPRIT algorithm is developed to estimate DODs and DOAs of the targets in bistatic MIMO radar in this paper. First, the conjugate centrosymmetrized DFT matrix is employed to retain the rotational invariance structure in the beamspace transformation for both the receiving array and the transmitting array. Then the DODs and DOAs can be estimated in accordance with a new version of ESPRIT for the bistatic MIMO radar. The ESPRIT algorithm works in the beamspace and involves only real-valued computation from start to finish. Over the beamspace ESPRIT algorithm, the proposed algorithm has better performance that can be paired automatically for DODs and DOAs estimation. In some situation, the proposed algorithm also has a better performance over the element ESPRIT.
And the structure of the rest of the paper will be organized as follows. The bistatic MIMO radar signal model is presented in Section 2. The proposed beamspace unitary ESPRIT for DODs and DOAs estimation is proposed in Section 3. Section 4 gives the computational complexity analysis of the proposed algorithm, unitary ESPRIT, and beamspace ESPRIT; advantages and disadvantages of the proposed algorithm also have been discussed in Section 4. And simulation results are provided to verify the performance of the proposed algorithm in Section 5; the Cramer-Rao Bound (CRB) also has been derived in this section. Finally, Section 6 concludes this paper.
2. Signal Model of the Bistatic MIMO Radar
Considering a narrowband bistatic MIMO radar system with an -element transmitting array and an -element receiving array, we found that both are half-wave length spaced uniform linear arrays (Figure 1).
At the transmitting site, different orthogonal narrowband waveforms are emitted simultaneously. In each receiver, the echoes are processed for all of the transmitted waveforms. Assume that there are uncorrelated targets in the same range bin, located in the far field of the array. The DOD and DOA of the th target with respect to the transmitting array and the receiving array are denoted by and , respectively. Thus, the output of all the matched filters in receivers can be written aswhere is an matrix, the column vectors , are steering vectors, which can be written as , is the combined steering vector of th target, where and are the receiving and transmitting steering vector of the th target, and and , respectively; denotes the Kronecker product; is a column vector consisted of the amplitudes and phases of the sources at time ; is usually in the form of with being the complex amplitude and the Doppler frequency of the th target; is the additive noise, which is modeled as a zero-mean, spatially white Gaussian process with covariance matrix , where denotes an identity matrix.
Note that the dimension of will be , and is the number of time samples. When and are large, the computational load and time will be huge. To overcome this shortcoming, the beamspace unitary ESPRIT is proposed as follows.
3. Proposed Method for DOD and DOA Estimation
In a radar application, the operation of reducing dimension could be facilitated in beamspace when a priori information on the general angular locations of the signal arrivals presents. In this case, by utilizing the beamspace transform matrix, beams, which involve the sector of interest, would be formed, thereby reducing computational complexity. And if there is no a priori information, one may apply angle estimation algorithm via parallel processing to each of the number of sets of successive overlapped sectors, which will also reduce computational complexity.
To retain the rotational invariance structure in the beamspace transformation, in this paper, the conjugate centrosymmetrized DFT matrix is applied as the beamspace transformed matrix. Let be the receiving beamspace transformed matrix and let be the transmitting beamspace transformed matrix. The th row of and is formulated as [17]Both and are conjugate centrosymmetrized matrixes and their th row vector represents a DFT beam steered at the spatial frequency and , respectively. An important property of the Kronecker operator that will prove useful throughout the transformation iswhere is an matrix, which is not the matrix mentioned above in (1). is a matrix, which is also not the matrix mentioned below in (5). is an matrix and is a matrix.
The receiving beamspace manifold is defined as and the transmitting beamspace manifold is defined as , respectively. Then the final beamspace manifold can be written asAccording to property (3), the equation holds. Thus, the final beamspace transformed matrix is defined as is an matrix, where and are the number of transmitting beam and receiving beam, respectively. A new beamspace received signal is defined aswhere is the noise of beamspace.
Next the rotational invariance structure in the beamspace will be examined. Considering the receiving beamspace manifold , the th component of is . And the th elements of areComparing (7) with , it is observed that the numerator of is the negative of that of . Then the two adjacent components of are related as [18]Trigonometric equations lead toDue to the fact that the beams with indices and are steered at the spatial frequencies and , where the two beams are physically adjacent to each other. First, we observe thatThen, the first and the last elements of are related by setting in (9), and there isAll equations lead to an invariance relationship for the as follows:where and are the two selection matrices defined as When some a priori knowledge of the angle information of sources is known, a reduced dimension processing can be achieved by applying a subset of row vectors defined in (4) to the data matrix . Hence, only those subblocks of the selection matrices and which are correlated with the corresponding components of will be used.
With targets, (12) translates into the receiving beamspace matrix relation:where is a real-valued diagonal matrix whose diagonal elements contain the desired DOA information. Similar to the relation of receiving beamspace matrix, the relation of transmit beamspace matrix iswhere and are defined similar to (13) with replaced by and is a real-valued diagonal matrix whose diagonal elements contain the desired DOD information.
According to the use of the property of the Kronecker operator in (3), we find that the whole beamspace manifold satisfieswhere and are the receiving beamspace selection matrices and and are the transmitting beamspace selection matrices, respectively.
Looking back on the beamspace received signal defined in (5), the proper matrix of signal subspace for the proposed algorithm can be formed by “largest” left singular vectors of the real-valued matrix . As we know, the signal subspace may be spanned by , which is expressed aswhere is an unknown real-valued matrix. Substituting into (16) yields where and . Equation (18) can be solved by the least squares (LS) or the total least squares (TLS) algorithm. Note the fact that all of the quantities of and are real-valued. Automatic pairing of the spatial frequency and estimation can be obtained by decomposing as follows:Hence, the real and imaginary parts of the eigenvalues are the estimation of , . Then the DODs and DOAs of targets can be derived as follows:The beamspace unitary ESPRIT algorithm based on this development is summarized below.To form the beamspace transformed matrix , where .To transform the receiving data into beamspace , where , is the number of time samples.To compute the SVD of . The dominant left singular vectors will be called . And if is not known as the a priori information, the number of sources should be estimated [19].To compute by solving the overdetermined system of equation via the least squares or the total least squares algorithm. The selection matrices and , where and are defined in (13). When one has the a priori information on the general angular locations of the signal arrivals, only the appropriate subblocks of and are employed.To compute by solving the overdetermined system of equation by the least squares or the total least squares algorithm. The selection matrices are and .To compute the eigendecomposition of matrix . Then largest eigenvalues could be obtained as , where and are the real and imaginary parts of , respectively.To compute the DODs and DOAs as the solution to , and , which are paired automatically.
4. Computational Complexity Analysis
In Table 1, the beamspace unitary ESPRIT is compared against the unitary ESPRIT and the beamspace ESPRIT algorithm in terms of computational complexity. For the element ESPRIT and the unitary ESPRIT algorithm, the computational complexity of eigendecomposition is , which is very heavy. Unlike the element ESPRIT and the unitary ESPRIT algorithm, the beamspace ESPRIT and the beamspace unitary ESPRIT algorithm transform the original data vector into several lower-dimensional beamspaces. Since the data processing of each beamspace is independent, it can be parallel processed. If only transmitting beams and receiving beams are selected, the computational complexity could be reduced from to . It is noted that the computational saving is quite significant. Comparing the beamspace unitary ESPRIT with the beamspace ESPRIT algorithm, the proposed algorithm involves only real-valued computation from start to finish. So the computational complexity of beamspace unitary ESPRIT is slightly lower than the beamspace ESPRIT algorithm. And the beamspace unitary ESPRIT algorithm can estimate DOAs and DODs that are paired automatically, which need to be paired additionally in the beamspace ESPRIT algorithm, to save computational complexity.
4.1. Advantages and Disadvantages of the Proposed Algorithm
The proposed algorithm has the following advantages.(1)The proposed algorithm involves only real-valued computation after the initial transformation of beamspace.(2)The proposed algorithm has a low complexity for the fact that peak searching is not required. And when a priori information is known, only several beams encompassing the sector of interest need to be formed, thereby yielding further reduced computational complexity.(3)The proposed algorithm can obtain automatically paired DOD and DOA angle estimations.(4)The proposed algorithm has a better angle estimation performance than B-ESPRIT algorithm which has been explained in the following section.The proposed algorithm also has the following disadvantages.(1)The proposed algorithm shows that, due to the dramatic reduction in computational complexity, the performance degradation would appear.(2)The proposed algorithm is sensitive to array errors. The rotational invariance structure will be damaged in the presence of array errors.
5. Simulation Results
In this section, some numerical examples are presented to assess the effectiveness of the proposed method. In all simulations, 2000 Monte Carlo trials are performed. The RMSE is defined as which are employed as the performance metric, where and denote the DOD and DOA of the th source and and are the estimation of and in the th Monte Carlo trail, respectively. denotes the number of snapshots. According to [16], the CRB for the angle estimation in bistatic MIMO radar is derived as follows:where , , , with being the th column of , , , , stands for the noise variance, and is the number of time samples.
Assuming that there are sources, whose angles are , , and . For the B-ESPRIT and the proposed algorithm, there are five beams are formed, which involve the sector of interest.
Figures 2 and 3 depict the DOD and DOA estimation results of the proposed algorithm for the three sources in bistatic MIMO radar with , , , , and 20 dB, respectively. It shows that the DODs and DOAs of sources can be clearly observed, and the performance will be improved as the SNR increases.
The proposed algorithm is compared against the beamspace ESPRIT algorithm, the element ESPRIT algorithm, the unitary ESPRIT, and CRB. And in this Simulation, a new source located at is added. Figures 4 and 5 present angle estimation performance comparison with , , and . It can be found that the angle estimation performance of proposed algorithm is better than the B-ESPRIT algorithm because unitary ESPRIT doubles the number of data samples effectively [20]. It is observed that the proposed algorithm performs slightly worse than the U-ESPRIT algorithm does in the DOD estimation in despite of the dramatic reduction in computational complexity (where the number of transmit antenna is and the number of transmit beam is ). And the proposed algorithm and U-ESPRIT algorithm both have similar estimation performance of DOA due to the fact that the number of receiving beam is nearly equal to the number of transmitting antenna (where and ). Compared with the E-ESPRIT algorithm, although the beamspace unitary ESPRIT algorithm is a unitary algorithm, its performance will also be degraded because of the dramatic reduction in computational complexity. When the degradation is not large enough, the proposed algorithm also has a better performance over the element ESPRIT.
Figures 6 and 7 present the angle estimation performance of the proposed algorithm with different , where , . The proposed algorithm is compared with the unitary ESPRIT algorithm. It illustrates that the angle estimation performance of our algorithm is improved as the number of snapshots increases. Meanwhile, it shows that performance of our algorithm is slightly worse than the unitary ESPRIT algorithm when the computational complexity is reduced all at once.
In Figure 8, values of transmitting array antennas and 10 are examined for the SNR support. From Figure 8, it can be seen that the angle estimation performance of proposed algorithm will gradually increase with the increasing transmitting antenna number.
Figure 9 gives the resolution performance of the proposed algorithm, the beamspace ESPRIT algorithm, the element ESPRIT algorithm, and the unitary ESPRIT algorithm. Considering a scenario with two uncorrelated sources of equal power, the angle parameters are and , and , . Then the angular source separation is parameterized by [21]The two sources are considered as resolved if the estimation of angle is close to the true angle parameter; that is,It can be seen that the probability of resolution of proposed algorithm is better than both the beamspace ESPRIT algorithm and the element ESPRIT algorithm but slightly worse than the unitary ESPRIT algorithm.
Figure 10 depicts an evaluation of the computational complexity using TIC and TOC instructions that can serve for calculating the runtime of an algorithm in MATLAB. It can be seen that when all the beams are formed, the runtime of proposed algorithm is smaller than the U-ESPRIT algorithm in the case of the number of sensors is larger than 10. And when only five beams are formed, the runtime of proposed algorithm is much smaller than the U-ESPRIT algorithm. Also, the runtime of the B-UESPRIT is smaller than the B-ESPRIT algorithm, which is still in the case of only five beams are formed. It is observed that as the number of sensors increases, the runtime of the proposed algorithm increases slowly. The reason is that the computational complexities and runtime of proposed algorithm are influenced by the number of beams formed, which is still five.
6. Conclusion
In this paper, a beamspace unitary ESPRIT is developed to estimate angles of the targets in bistatic MIMO radar. The conjugate centrosymmetrized DFT matrix is applied to transform the received data into beamspace. Then the invariance property of the transmitting beam and the receiving beam is exploited, respectively, to calculate the DODs and the DOAs of targets. Unlike the beamspace ESPRIT, the B-UESPRIT involves only real-valued computation from beginning to end. Therefore, a reduction of the computational complexity is obtained, which is demonstrated by the analysis of computational complexity and the runtime of algorithm. Furthermore, the simulation results prove that the B-UESPRIT requires no matched pair but possesses better angle estimation performance than the E-ESPRIT and the B-ESPRIT. Additionally, the CRB has been derived to analyze the performance.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.