1-Bit direction of arrival estimation via improved complex-valued binary iterative hard thresholding☆
Introduction
Direction-of-arrival (DOA) estimation using sensor arrays has been a hot research topic in many fields such as communication, radar, and speech signal processing, etc [1], [2], [3], [4]. Accurate DOA estimation always requires high-resolution quantization, which is a strict requirement for analog-to-digital converters (ADCs). Decreasing the quantization bits can apparently reduce the power consumption and error caused by nonlinear components [5]. As a special case, the 1-bit ADC, which only retains the sign of input, has been widely used in multiple-input multiple-output (MIMO) systems [6], [7], [8], [9], [10], [11].
In recent decades, many 1-bit DOA estimation approaches have been proposed. According to the arcsine law [12], Bar-Shalom reconstructed the unquantized covariance matrix and utilized the minimum variance distortionless response (MVDR) algorithm to estimate DOA [13]. In [14], Liu estimated the normalized covariance of the unquantized data by arcsine law and designed a sparse array to achieve more accurate DOA estimation. Huang proved that the covariance matrix of 1-bit array observation was an approach of the unquantized covariance matrix with ignorable errors so that multiple signal classification (MUSIC) algorithm could be utilized to 1-bit DOA estimation directly [15]. In [16], Li extended the relationship between the covariance matrices of 1-bit observation and that of unquantized observation to the tensor domain and proposed a 1-bit parallel factor analysis (PARAFAC) estimator to solve both DOA and frequency at the same time. Yoffe derived a maximum likelihood (ML) estimator for 1-bit DOA estimation and proposed a suboptimal approach [17].
Sparse representation can reconstruct the signals with fewer measurements by utilizing the structured information, which can also be applied to reduce the quantization bits. In [18], Stockle proposed the complex-valued binary iterative hard thresholding (CBIHT) algorithm, which extended the binary iterative hard thresholding (BIHT) algorithm [19] into complex-valued and multi-snapshot cases. Meng applied the generalized sparse Bayesian learning (Gr-SBL) algorithm [20] to solve the 1-bit DOA estimation problem [21]. Meng's method improved the recovery performance by leveraging the joint sparsity of the real and imaginary parts of signals. In [22], Gao solved the problem via the support vector machine (SVM) framework. Wei proposed a robust and gridless approach via atomic norm denoising in [23].
Based on sparse representation theory, the CBIHT algorithm [18] performs better than the traditional approaches in DOA estimation. However, because of the column correlation of the array manifold, CBIHT sometimes retains the wrong bases, which reduces the accuracy of DOA estimation as well as slows the convergence, especially when the real DOA is off-grid. Inspired by CBIHT, an improved complex-valued binary iterative hard thresholding (iCBIHT) algorithm is proposed in this work. Firstly, an error function of signal reconstruction is defined and the signals are updated by gradient descending. By complex-valued gradient descending, the joint sparsity of the real and imaginary parts of the reconstructed signals can be automatically utilized. Secondly, a refined hard threshold function is designed by the spatial spectrum, and a stopping criterion is defined to judge the convergence. Thirdly, a bases-updating strategy is introduced to solve the off-grid DOAs. By updating the searching grids and the manifold matrix iteratively, the accuracy of DOA estimation is improved. Fourthly, aiming to avoid the repeated selection and abandonment of bases, a backtracking strategy is utilized to refine the estimated signals processed by hard thresholding, which also promotes convergence. Simulations and analyses show that the proposed method can estimate DOA more accurately than 1-bit MUSIC and CBIHT with lower signal-to-noise ratio (SNR) and fewer snapshots.
This paper is arranged as follows: Section 2 reviews the formulations of the signal model and sparse representation. Section 3 introduces the specific steps of the proposed iCBIHT algorithm. In section 4, the performance of iCBIHT is presented by simulations and analyses. Section 5 concludes the whole paper.
Section snippets
Signal model
Considering D far-field narrowband signals impinging on an array made of M sensors from direction , the observation of the mth sensor at the nth snapshot is where denotes the phase delay of the dth source between the mth element and the reference element, is the dth signal received by the reference element at the nth snapshot, and is the additive complex white Gaussian noise. is the 1-bit complex-valued quantizer, which keeps
Proposed method
In this section, we introduce an improved complex-valued binary iterative hard thresholding (iCBIHT) algorithm to realize 1-bit DOA estimation. The details are shown as follows.
Simulation settings
The performance of iCBIHT is presented by simulation. 1-bit MUSIC [15] and CBIHT [18] are compared with iCBIHT in this section. Moreover, the performance of unclipped MUSIC (infinite-bit quantization) [24] is shown in Fig. 1, Fig. 6, and Fig. 9, and the Cramer-Rao bound (CRB) [25] is shown in Fig. 1 and Fig. 6.
The proposed method is available on sensor arrays with arbitrary shapes. For simplicity, a uniform linear array (ULA) with half-wavelength spacing is used in the simulations. The initial
Conclusion
In this paper, we aim to estimate DOA with 1-bit observation of a sensor array. After the formulation of the signal model and sparse representation, an improved complex-valued binary iterative hard thresholding (iCBIHT) algorithm is proposed. Firstly, we define an error function of signal reconstruction and update the signals by gradient descending, which automatically utilizes the joint sparsity of the real and imaginary parts of the reconstructed signals. Secondly, we design a refined hard
CRediT authorship contribution statement
Pengyu Wang: Conceptualization, Formal analysis, Investigation, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. Huichao Yang: Formal analysis, Investigation, Writing – review & editing. Zhongfu Ye: Supervision, Visualization, 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.
Pengyu Wang received the B.E. degree in electronic and information engineering from Xidian University of, Xian, China, in 2019. He is currently pursuing the M.S. degree at the University of Science and Technology of China, Hefei, China. His research interests include speech signal processing and array signal processing.
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Pengyu Wang received the B.E. degree in electronic and information engineering from Xidian University of, Xian, China, in 2019. He is currently pursuing the M.S. degree at the University of Science and Technology of China, Hefei, China. His research interests include speech signal processing and array signal processing.
Huichao Yang received the B.E. degree in information engineering from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2019. He is currently pursuing the M.E. degree at the University of Science and Technology of China, Hefei, China. His research interests include speech signal processing and array signal processing.
Zhongfu Ye received the B.E. and M.S. degrees in electronic and information engineering from the Hefei University of Technology, Hefei, China, in 1982 and 1986, respectively, and the Ph.D. degree from the University of Science and Technology of China, Hefei, China, in 1995, where he is currently a Professor. His current research interests are in statistical and array signal processing, speech processing, and image processing.