Time-frequency DOA estimation of chirp signals based on multi-subarray
Introduction
DOA estimation is an important problem in array signal processing and has been widely used in sonar, radar, wireless communication, and navigation [1], [2], [3], [4], [5]. The beamforming techniques represented by the conventional beamforming and the capon's beamforming are dependent upon the array aperture and clearly on the SNR. Parametric methods [6], [7] can still obtain better estimation performance at low SNR scenarios, their computational complexity is considerable. For the comprehensive consideration of DOA estimation performance and computational complexity, the subspace-based methods including multiple signal classification (MUSIC) [8] and estimation of signal parameters by rotational invariance techniques (ESPRIT) [9] are optimal, which exploits the rotation invariance of the signal subspace or the orthogonality of the signal subspace and the noise subspace to achieve DOA estimation. The above-mentioned methods which were based on the assumption that the received signals are stationary. For radar and sonar imaging system, the chirp signal is transmitted to improve image resolution [10], [11], which results in the degradation of DOA estimation performance due to the non-stationary of the chirp signals [12], [13]. The DOA estimation performance based on time-frequency methods can be improved due to spreading the noise power while localizing the source energy in the time-frequency (t-f) domain, so these methods have received considerable attention. Belouchrani and Amin proposed the concept of STFD [14] and combined it with the MUSIC algorithm (t-f MUSIC) to give good angle estimation performance for various time-frequency chirp signals [15]. Gershman and Amin applied t-f MUSIC to DOA estimation of wideband chirp signals [16]. However, the above methods [15], [16] have needed to calculate the autoterm and crossterm regions to construct the STFD matrix, which increased the computational complexity. Zhang and Amin [17] proposed a spatial average time-frequency distribution matrix method, which reduced the computational load and achieved good performance in blind source separation. However, it is not suitable for DOA estimation. Huang etc. [18] proposed a wideband signal DOA estimation method of the symmetrical array based on the time-frequency analysis. However, it required that the element spacing is less than or equal to (λ is wavelength corresponding to the highest frequency of the signal) and reduced the effective aperture of the array. To reduce the computational complexity, the time-frequency point selection algorithms [19], [20], [21] based on selecting multiple single-source time-frequency points in the time-frequency domain were proposed, and the multi-invariant (MI) characteristic [22], [23], [24] was used to further improve the DOA estimation performance of the time-frequency ESPRIT (t-f ESPRIT) algorithms [25], [26], [27], [28].
Although the aforementioned methods based on the STFD matrix improved the DOA estimation performance of chirp signals, which assumed that these signals have better time-frequency point selection performance. However, in practical scenarios, especially in the case of low SNR, the time-frequency point selection error increased, leading to a decrease of the ideal energy collection features of chirp signals, which reduced the SNR and the DOA estimation performance in the time-frequency domain. Therefore, improving the accuracy of signal time-frequency point selection at low SNR is the key to the time-frequency DOA estimation methods.
To address the above-mentioned problems, in this paper, we propose a time-frequency DOA method based on beamforming and time-frequency analysis, which improves the accuracy of time-frequency point selection. This method partitions the array into several subarrays and exploits beamforming to transform the received signals by the array from the element space to the beamspace, which realizes the spatial separation of these signals and improves the SNR of the target signal. Owing to the chirp signal possesses the ideal energy collection features in the time-frequency plane, the time-frequency analysis will enhance these signal separation and SNR, which further improves the accuracy of time-frequency point selection. Finally, the averaged STFD matrix can be obtained by selecting the time-frequency points located in the energy area of chirp signals, and the DOAs is resolved directly by combining with subspace-based methods. Theoretical analysis and simulation results indicate that the proposed method improves the DOA estimation performance at low SNR.
The organization of this paper is as the following. The signal model is described in Section 2. The influence of time-frequency point selection error on SNR is analyzed in Section 3. Then in Section 4, the time-frequency DOA estimation method based on multiple subarrays is introduced, followed by simulation results and discussions in Section 5. Finally, this paper is concluded in Section 6.
Symbols: matrices, vectors and scalars are represented by capital bold letters, lower-case bold letters and lower-case letters, respectively. (.)H, (.)T and (.)⁎ represent conjugate transpose, transpose, and conjugate, respectively. E[.] represents mathematical expectation. , and diag{.} represent zero matrix, unit matrix and diagonal matrix, respectively.
Section snippets
Signal model
Consider a uniform linear array (ULA) which is comprised of M isotropic elements distributed with equal spacing as shown in Fig. 1. Suppose that there are P narrowband far-field chirp signals impinging on the array from distinct directions . Let the element 1 be the reference, the complex envelope of each signal at the reference can be expressed as and the spacing between the adjacent elements is , where λ is the carrier wavelength.
Influence of time-frequency point selection error on SNR
The time-frequency point selection methods [19], [20], [21], [25], [26], [27], [28] have received attention due to their low computational complexity. In practice, the true frequency point f of the signal at a given time-instant is unknown and can be expressed as . Then, (10) can be rewritten as
First, inserting (1) into (14), we obtain
Under the uncorrelated signal and the noise assumed to
The proposed method
It is shown in (25) that the SNR improvement is inversely proportional to the number of signal sources contained in the STFD matrix. When each signal source is processed separately, the maximum SNR improvement can be obtained. Therefore, we propose a time-frequency DOA estimation method based on multiple subarrays, which exploits beamforming and time-frequency analysis to improve the signal separation and the time-frequency point selection accuracy, then further produce better DOA estimation
Numerical results
Consider a ULA of 10-elements separated by half of the wavelength () as shown in Fig. 2. Two narrowband chirp signals with carrier frequency of impinge on the array from the sources located at and . After downconversion, the source waveforms are modeled as [26] where the initial discrete-time frequencies of the source signal are chosen to be and while their chirp rates are assumed to be and ,
Conclusions
A time-frequency algorithm is proposed for DOA estimation of chirp signals. This proposed method is a joint beamforming and STFD matrix reconstruction that improves the effective SNR in the time-frequency domain. Thus, this results in the improvement of the time-frequency point selection accuracy and further improves the DOA estimation performance. In particular, for the cases with a low SNR, the proposed method provides a better RMSE. The convergence of the proposed method is also faster than
CRediT authorship contribution statement
Bingbing Qi: Conceptualization, Data curation, Methodology, Software, Writing – original draft. Huansheng Zhang: Data curation, Writing – review & editing. Xiaobo Zhang: Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Bingbing Qi received the B.S. degree from Yanshan University, Hebei, China, in 2010, and the M.S. degree from the Yanshan University, Hebei, China, in 2013. Currently, he is pursuing the Ph.D. degree in communication and information systems with the Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include radar, sonar and array signal processing.
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Bingbing Qi received the B.S. degree from Yanshan University, Hebei, China, in 2010, and the M.S. degree from the Yanshan University, Hebei, China, in 2013. Currently, he is pursuing the Ph.D. degree in communication and information systems with the Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include radar, sonar and array signal processing.
Huansheng Zhang received B.S. degree and M.S. degree from Xi'an University of science and technology Xi'an China in 2000 and 2003, and received PH.D. degree from Institute of Electrics, Chinese Academy of Sciences Beijing China in 2006. Currently, he is a researcher in the 3rd Research Institute of China Electronics Technology Group Corporation and his research interests include radar, sonar and array signal processing.
Xiaobo Zhang mainly focuses on signal processing, artificial intelligence, and big data applications in recent years. He has already undertaken 3 projects funded by China's government, been granted 2 patents and published more than 10 papers. He was invited to participant the 4th China-America Frontier of Engineer (CAFOE 2015) and on the 1st Indo-Chinese Young Engineering Leaders Conclave (ICON-1). Due to his achievements on signal processing, he was selected into the Young Talents Promotion Project funded by Chinese Association of Science and Technology. Now, he is the General Manager of Beijing China Electronics Intelligent Acoustics Technology Co. Ltd.