International Journal of Antennas and Propagation

Volume 2018, Article ID 3568286, 10 pages

https://doi.org/10.1155/2018/3568286

## Beam-Doppler Unitary ESPRIT for Multitarget DOA Estimation

^{1}School of Information Science and Technology, Northwest University, Xi’an 710069, China^{2}School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China^{3}National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Yan Zhou; nc.ude.uwn@uohznay

Received 16 August 2017; Accepted 24 January 2018; Published 26 March 2018

Academic Editor: Lorenzo Crocco

Copyright © 2018 Cai Wen 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

High-resolution direction of arrival (DOA) estimation is a critical issue for mainbeam multitarget tracking in ground-based or airborne early warning radar system. A beam-Doppler unitary ESPRIT (BD-UESPRIT) algorithm is proposed to deal with this problem. Firstly, multiple snapshots without spatial aperture loss are obtained by using the technique of time-smoothing. Then the conjugate centrosymmetric discrete Fourier transform (DFT) matrix is used to transform the extracted data into beam-Doppler domain. Finally, the rotational invariance property of the space-time beam is exploited to estimate DOA of the target. The DOA estimation accuracy is improved greatly because the proposed algorithm takes full advantage of temporal information of the signal. Furthermore, the computational complexity of the presented algorithm is reduced dramatically, because the degree of freedom after beam transformation is very small and most of the operations are implemented in real-number domain. Numerical examples are given to verify the effectiveness of the proposed algorithm.

#### 1. Introduction

Mainbeam multitarget separation is a key problem for multitarget tracking in ground-based or airborne early warning radar system. When multiple targets fly in a formation, it is common that some targets fall into the same range cell or adjacent range cells. In this case, it is difficult to separate these targets in range domain only using pulse compression. Meanwhile, it is also difficult to distinguish them in Doppler domain, since the radial velocities of these targets are usually very close; meanwhile, the coherent accumulation time is usually very short in an early warning radar system. In this paper, we consider resolving closely spaced targets in spatial domain through high-accuracy DOA measurement.

Traditional angle measurement algorithms such as the classical monopulse technique [1] becomes invalid in the presence of multiple targets. In order to resolve multiple targets, a typical way is to improve radar resolution by narrowing antenna beam width, widening bandwidth, or extending dwell time. However, these solutions require additional hardware cost. Another possible solution is to achieve superresolution by employing array signal processing techniques. Classical spatial superresolution methods include maximum likelihood (ML) [2], multiple signal classification (MUSIC) [3], estimation of signal parameters via rotational invariance technique (ESPRIT) [4], and their variants [5–7]. In these methods, only spatial information is used, and the temporal pulses are treated as snapshots. Their performance degrades significantly in the presence of closely spaced targets, especially in the condition of low signal-to-noise ratio (SNR) and small number of snapshots. In fact, for early warning radar, the temporal information such as the difference of Doppler frequencies between two targets is extremely useful for DOA estimation. Essentially, the problem of spatial parameter estimation could be transformed into the problem of space-time parameter estimation if we make use of the temporal information of the signal. In general, the spatial distance between two targets will be amplified in space-time plane. Therefore, the accuracy of spatial parameter estimation, especially in the case of closely spaced targets, could be improved if the temporal information is used.

For the space-time signal model, the problem is to estimate multitarget DOA with a single snapshot. The ML method could be used to joint estimate of DOA and Doppler frequency [8]. However, the ML method requires a 2D parameter search process, which usually results in heavy computational load. In order to improve efficiency, the reduced-dimension ML method was proposed in [9]. In this method, the problem of DOA and Doppler frequency estimation is decoupled into two sequential 1D optimization problems. Nevertheless, this method needs a 1D parameter search. Moreover, it is easy to converge to the local optimal solution. The subspace-based superresolution methods combined with aperture smoothing could be practically extended for space-time parameter estimation, such as the 2D MUSIC method [10]. However, the 2D MUSIC needs a 2D parameter search yet. The joint angle and frequency estimation (JAFE) method based on ESPRIT is proposed to estimate DOA and carrier frequency of multicarrier signal for the communication system [11]. The JAFE method does not need parameter search, and thus the computational complexity of JAFE is much lower than that of search-based subspace method. The real value-based unitary JAFE (U-JAFE) is proposed in [12] with lower computational load than that of JAFE. However, in the JAFE and U-JAFE methods, the 2D parameters are estimated in element-pulse domain, which means the signal subspace is also estimated via eigenvalue decomposition (EVD) or singular-value decomposition (SVD) in the element-pulse domain. In the case of large temporal degree of freedom (DOF) or spatial DOF, the computational complexity of JAFE and U-JAFE is still very high. Recently, compressive sensing [13–15] (CS) is widely studied and has been applied in the array signal processing community [16–18]. Although CS could be extended to estimate the space-time parameter and a reasonable performance could be obtained in the case of small snapshots [19], it needs to solve a high-dimensional optimization problem with high computational complexity.

In this paper, the problem of estimating DOAs of multiple targets within mainbeam is considered. In order to improve the DOA estimation accuracy, the temporal information of these targets is exploited. Considering the computational complexities of existing methods are very high in original element-pulse domain, a beam-Doppler unitary ESPRIT algorithm is proposed, which aims to reduce the computational load with improved DOA estimation accuracy. Firstly, multiple space-time snapshots without array aperture loss are extracted from a single sample via time-smoothing. Then, a few space-time beams are formed pointing towards the targets according to the coarse-resolution information provided by the target detection stage. Evidently, the parameter could be estimated in the low-dimensional space-time beamspace with a reduced computation load. Further, the conjugate centrosymmetric DFT matrix is utilized for beamforming with a rotational invariance property being retained. Finally, the real value-based ESPRIT is carried out for estimating DOA with further reduced computational complexity. The proposed method is quite different from the beamspace ESPRIT method [20], in which only spatial information is utilized for DOA estimation. It is also different from the JAFE [11] and U-JAFE [12] methods, which estimate DOA in element-pulse domain, while our method is performed in a low-dimensional beam-Doppler domain.

The remainder of this paper is organized as follows. Section 2 develops the data model for an early warning radar system. Section 3 proposes a beam-Doppler unitary ESPRIT method for joint DOA and Doppler frequency estimation with some analyses of computational complexity and Cramer-Rao bound (CRB), and Section 4 gives some discussion of critical aspects for the proposed method. In Section 5, simulation results are presented to illustrate the performance improvement introduced by the proposed algorithm, while the conclusions are given in Section 6.

*Notation. *, , and denote transpose, conjugate-transpose, and inverse operations, respectively; represents the Kronecker product; diag(**v**) stands for diagonal matrix whose diagonal element is a vector **v**; is a identity matrix; and denotes the real part and imaginary part of a complex number, respectively.

#### 2. Problem Formulation

In this paper, we consider a narrowband pulse-Doppler (PD) early warning radar system operating with a rectangular planar phased array. The sketch map of target detection for early warning radar is depicted in Figure 1, where the transmitting beam is formed by the whole antenna arrays and the receiver is composed of *N* uniformly spaced column-synthesized subarrays. The pulse number for coherent accumulation is *K*, assuming that statistically independent targets are located in the same range cell. The DOA and Doppler frequency of the th () target is and , respectively. After range matched filtering, the space-time data vector of one range cell is
where is a steering matrix; denotes the space-time steering vector of the *p*th target; and are the spatial steering vector and temporal steering vector of the *p*th target, respectively, where is the normalized spatial frequency, is the normalized Doppler frequency, *d* is the spacing of column subarrays, is the operating wavelength, and is the pulse repetition period; is a vector that consists of complex amplitudes of targets, where is the complex amplitude of the *p*th target; and is the complex Gaussian white noise with zero mean and covariance matrix , where denotes the variance of the white noise. It is worth noting that both the spatial steering vector and temporal steering vector are conjugate centrosymmetric.