Short communicationA robust STAP method for airborne radar with array steering vector mismatch
Introduction
Space-time adaptive processing (STAP) is a two dimensional adaptive beamforming algorithm, which is able to not only suppress wideband interferences [1] but also detect moving targets in the presence of clutter [2], [3], [4]. However, a small error in either direction-of-arrival (DOA) or Doppler frequency may result in the mismatch of steering vector and subsequently the cancellation of signal-of-interest (SOI). Then the STAP filter, i.e., the beamformer in the following, misinterprets the SOI as clutter and tries to suppress it [5].
Numerous robust adaptive beamforming (RAB) techniques have been proposed to deal with the steering vector mismatch in the literature [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The eigenspace-based beamformer method is simple and efficient for strong interferences [6]. However, it would suffer from subspace swap at low signal-to-noise ratio (SNR) levels [7]. Several worst-case optimization beamformers were proposed in [8], [9], [10], where constraints on the uncertainty set are utilized to preserve a certain array gain against uncertainties in the steering vector. However, the norm bound of the uncertainty set should be carefully prescribed, otherwise, the performance may be significantly deteriorated. In [11], [12], [13], [14], the constraints on the magnitude response, which have the flexibility in controlling the beamwidth and response ripple in the desired direction, are imposed to improve the robustness. However, these methods may reduce the signal-to-interference-plus-noise ratio (SINR) owing to the fact that the noise power is also amplified in the constrained beamwidth accordingly. Contrary to the worst-case optimization and the magnitude response approaches, the robust beamformer in [15], [16] firstly correct the mismatched steering vector before the determination of beamformer weight vector. These two algorithms do not make assumptions on either the norm of the mismatch vector or the bounds of magnitude response constraints. However, the iterative quadratic convex optimization in [15] is a tedious process and the upper bound of the constraint in [16] is difficult to determine. More recently, a response vector constrained beamformer, which can yield a lower beampattern sidelobe than that of the traditional linearly constrained minimum variance (LCMV) beamformer [17], was reported in [18], [19]. However, its signal-to-clutter-plus-noise ratio (SCNR) loss performance may be considerably degraded when the Doppler frequency of the SOI is close to the clutter ridge.
In this work, the region-of-interest (ROI) in which the DOA and Doppler frequency of the SOI locates is first chosen to construct a spatial-temporal integral covariance matrix. With the so-obtained covariance matrix, the subspace of clutter-plus-noise can be determined, which can be efficiently employed for constraint imposition based on subspace orthogonality. By maximizing the output power of beamformer, the problem of steering vector estimation is formulated as a non-convex quadratically constrained quadratic program (QCQP), which can be efficiently solved using the semidefinite programming (SDP) relaxation technique [20]. Unlike the approach in [15], the proposed robust STAP method is free of an iterative procedure. Simulation results illustrate that the proposed robust STAP method can provide superior performance in terms of output SCNR over the state-of-the-art approaches.
Section snippets
Array signal model
Consider a side-looking airborne radar system with a uniform linear array (ULA) consisting of N elements, as shown in Fig. 1. Assume that each antenna element transmits M coherent pulses and the inter-element spacing is half wavelength. The spatial-temporal steering vector of echo signal can be expressed aswhere ⊗ denotes the Kronecker product, θ denotes the DOA, denotes the Doppler frequency, and denote, respectively, the spatial and temporal steering
The proposed robust STAP method
In order to overcome the shortcomings of existing approaches as mentioned earlier, a new robust STAP method for airborne radar is introduced. For ease of illustration, the ROIs of the DOA and Doppler frequency are illustrated in Fig. 2. The nominal position of DOA and Doppler frequency is considered as the center which is not coincident with the actual one. denotes the spatial region covering the possible target DOA θ and denotes the frequency region including the possible target Doppler
Simulation results
The main simulation parameters are listed in Table 1 unless otherwise specified. The additive noise is modeled as a spatially and temporally independent complex Gaussian process with zero mean and unit variance. For each scenario, 100 Monte-Carlo runs are performed. The number of dominant eigenvalues of the matrix is chose as P=4 since the sum of these four dominant eigenvalues gives over 99% of the sum of all eigenvalues [15]. Assume that , , , the
Conclusion
The problem of robust STAP for airborne radar with array steering vector mismatch is considered. A spatial-temporal integral covariance matrix including the actual steering vector component is firstly formulated. The corresponding subspace is then used for robust STAP design. By estimating the actual steering vector with an SDP technique, the weight vector for STAP can be achieved by traditional beamforming methods. Simulation results demonstrate that the superiority of the proposed method in
Acknowledgment
The work described in this paper was supported by the National Natural Science Foundation of China under Grants U1501253, 61271420, 61401284, and 61471365, the Natural Science foundation of Guangdong Province, P.R. China (No. 2015A030311030), the Foundation of Shenzhen under Grants JCYJ20140418091413566 and 827-000071, the China Postdoctoral Science Foundation Grant 2015M582414.
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