Time-frequency DOA estimation of chirp signals based on multi-subarray

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Abstract

Spatial time-frequency distribution (STFD) utilizes the spatial time-frequency characteristics of chirp signals and effectively improves the direction of arrival (DOA) estimation performance. However, the existing methods based on the STFD matrix assumed that each source has good time-frequency point selection performance. In practice, the time-frequency point selection accuracy of chirp signals suffers from the signal-to-noise ratio (SNR). Especially in the case of low SNR, the time-frequency point selection error increased, leading to the reduction of the SNR and degradation of the DOA estimation performance in the time-frequency domain. To solve the above problems, we propose a time-frequency DOA estimation method based on multiple subarrays. The array is firstly divided into several overlapping subarrays, and the data received by the array is transformed from the element space into beamspace by using beamforming, which improves the source separation and the SNR. Then, chirp signals possess the ideal energy collection features in the time-frequency domain so that the time-frequency analysis tools are used for separated sources in beamspace to further improve the source separation and the SNR in the time-frequency domain. This results in better time-frequency point selection accuracy of chirp signals at a low SNR regime. Finally, the averaged STFD matrix can be obtained through averaging over multiple single-source time-frequency points in the time-frequency domain, and the DOAs can be obtained by combining with the subspace-based method. The theoretical analysis and simulation results indicate that compared with the existing STFD-based methods, the proposed method in this paper provides good performance on estimation and resolution in cases with low input SNRs due to beamspace processing. Furthermore, in cases where the DOAs between the coherent sources are closely spaced and the snapshot number is low, our proposed method significantly improves the performance of the DOA estimation.

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/4 (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. 0m×n, Im and diag{.} represent m×n zero matrix, m×m 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 θi,i=1,2,,P. Let the element 1 be the reference, the complex envelope of each signal at the reference can be expressed as si(t),i=1,2,,P and the spacing between the adjacent elements is d=λ/2, where λ is the carrier wavelength.y(t)=[y1(t),,yM(t)]T=x(t)+n(t)

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 fˆ. Then, (10) can be rewritten asDˆyy(t,fˆ)=τ=(L1)/2(L1)/2y(t+τ)yH(tτ)ej4πfˆτ

First, inserting (1) into (14), we obtainDˆyy(t,fˆ)=Dˆxx(t,fˆ)+Dˆxn(t,fˆ)+Dˆnx(t,fˆ)+Dˆnn(t,fˆ)

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 (d=λ/2) as shown in Fig. 2. Two narrowband chirp signals with carrier frequency of f=150kHz impinge on the array from the sources located at θ1=5 and θ2=5. After downconversion, the source waveforms are modeled as [26]s1(t)=ej2π(f1+β1t)ts2(t)=ej2π(f2+β2t)t where the initial discrete-time frequencies of the source signal are chosen to be f1=0 and f2=0.5 while their chirp rates are assumed to be β1=0.001 and β2=0.001,

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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      At the same, the beamspace eigenvalue ratio of the signal subspace to the noise subspace is a linear function of the input SNR, which enhances the beamspace signal noise suppression ability at low SNR. Theoretical analysis and simulation results indicate that compared with the existing Toeplitz matrix reconstruction methods, the proposed method improves the DOA estimation performance at low SNR, and this ideal has also been used to provides good performance on estimation and resolution in the time-frequency domain [31]. The main contributions of the proposed method are as follows:

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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.

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