A multi-direction virtual array transformation algorithm for 2D DOA estimation
Introduction
DOA estimation is the key technology in radar, sonar, mobile communication and so on. During its development, one dimensional (1D) estimation has been widely investigated [1]. Two of its most important representatives are MUSIC [2] and ESPRIT [3], which are both categorized as eigen decomposition based methods. MUSIC algorithm is well-known for its high resolution capability. It can be applied in arbitrary array manifold to estimate multiple parameters of each source. However, MUSIC algorithm must know the analytical relation of its array response matrix to all possible combinations of source parameters. The ESPRIT algorithm, without spectral peak search and complex computation, has been considered to be a very extensive application value algorithm since it yields. Its principal advantage is that the DOA parameters are obtained with a closed form solution.
At first, ESPRIT algorithm can only be applied to uniform linear array (ULA) for 1D DOA estimation, while 2D estimation is of greater practical importance. ESPRIT algorithm cannot be directly used to uniform circular array (UCA) or arbitrary array. However, its application reduces the effective array aperture, usually it needs to pair the parameters in 2D estimation [4]. In 2D scenario, MUSIC algorithm costs a majority of run time during the procedure of space spectrum peak search.
Recently, virtual array transformation technology [5] has improved the applicability of 2D ESPRIT algorithms and can be applied in arbitrary array manifold. Usually, there are three ways to implement virtual array transformation, i.e. array interpolation [6], higher-order cumulant [7] and extrapolation algorithm [8]. However, higher-order cumulant is suffered from large amount of computation complex [9], and extrapolation algorithm can not distinguish two targets when they are relatively close each other. Array interpolation method is proposed by Bronez [10] firstly and later in different formulations by Friedlander [11]. Most of these methods can be directly applied to arbitrary array [12]. In [13] a single direction virtual interpolation algorithm termed TVIA was proposed via two virtual subarrays. TVIA uses the virtual interpolation technology twice to get the rotational invariance factor, from which the elevation angle can be estimated via rotational invariance technique. Then the estimated elevation angle is substituted into the real array manifold, hence the azimuth angle is obtained by one dimensional search. TVIA achieves higher DOA estimation accuracy than the well-known algorithm proposed by Bronez [11], and it can be applied to arbitrary array, meanwhile the estimated elevation and azimuth can be automatically paired. However, TVIA still needs 1D spectrum peak search which leads to the high computation and storage costs, at the same time it cannot distinguish signals with the same elevation but different azimuths. In [14], Yang proposed a novel 2D DOA finding algorithm for arbitrary array applications. He decomposes the 2D problem into two 1D problems by 2D DFT beam space ESPRIT algorithm, and the estimated 2D angle need to be paired.
In this paper, we propose a novel multi-direction virtual array transformation algorithm (MVATA) for 2D DOA estimation, which can effectively solve the existing problems of traditional virtual array transformation algorithms. The theoretical derivation and simulation results of the algorithm are given. Compared with the TVIA for 2D DOA estimation, the proposed MVATA has the following advantages. (1) It does not need spectrum peak search and fractal dimension processing, hence the computation and storage costs can be reduced efficiently. (2) As the azimuth and its corresponding elevation angle are obtained from the same element of a diagonal matrix, they need not the support of pairing algorithm. (3) The accuracy of the 2D DOA estimation is obviously improved especially in the scenarios of low SNR and small snapshots. (4) Two dimension angels which have the same azimuth or elevation angle can also be distinguished. The proposed algorithm can be applied not only to UCA but also to arbitrary structure arrays. Simulation results demonstrate its effectiveness.
Section snippets
Signal model
Consider a UCA composed of elements impinged by narrowband signals , where is the time variable. The distance between the neighboring elements is no greater than half the wavelength. As shown in Fig. 1, the coordinate of the mth element in the Cartesian coordinate system is . The sources are assumed as far field with azimuth and elevation , . Assume the radius of UCA is and the noise is Gaussian white noise. From Fig. 1, we know
Principle of the proposed algorithm (MVATA)
For simplicity, we illustrate our proposed algorithm based on the array model above. Assume there is a signal, whose elevation is in the sector , and its azimuth is in the sector . Let be the interpolation step, then and can be discretized as
In the sector, the real array manifold is
In other words, is a rows and columns matrix. It is the response of
Computational complexity analysis
In this section, we carefully analyze the computational complexity of TVIA and the proposed MVATA. Let MDN denote number of multiplications and divisions and represent number of snapshots. in Table 1 denotes the number of searches conducted along the DOA axis, it is set usually much bigger than in order to obtain an accurate DOA estimation, Fig. 4 shows the difference of computational complexity, where and represent the computational complexity of TVIA and proposed MVATA.
Experimental results
In this section, five experiments are presented to show the performances improvement of proposed MVATA compared with the TVIA. A UCA of radius with is employed, and the search step size of TVIA is set to be in all experiments. denotes the number of snapshots in the simulations. The signals and noise are assumed to be stationary, zero mean, and uncorrelated Gaussian random processes. Noise is both spatially and temporally white. SNR is defined as the ratio of the received
Conclusions
In order to improve the performance of virtual interpolation algorithm in finite snapshots and low SNR, a new algorithm termed MVATA for 2D DOA estimation is proposed. MVATA can derive the azimuth and elevation by using only once ESPRIT. It can be applied to arbitrary structure array and the estimated 2D angles can be paired automatically. Moreover, two dimension angles which have the same azimuth or elevation angle can also be distinguished. Compared with the TVIA, theoretical calculation and
Acknowledgments
The authors would like to thank very much the Handling Editor Jean Pierre Delmas, Journal Manager Chitra and the anonymous reviewers for their valuable comments and suggestions that have significantly improved the manuscript.
This work was supported by the National Natural Science Foundation of China under Grants 61373177, 61271293, 61272461 and 61572402; Natural Science Basic Research Plan in Shaanxi Province of China under Grant 2013JM8008.
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