A blind source separation method for time-delayed mixtures in underdetermined case and its application in modal identification

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Abstract

A novel blind source separation (BSS) method for time-delayed mixtures in underdetermined case is studied in this paper. The proposed method not only addresses the problem of source separation with limited sensors but also avoids the influence of propagation delay. Firstly, the sparse domain is converted by utilizing the spectrum of observed signals to perform modulus operation in time-frequency (TF) domain, which appears several clustering lines in the scatter plot. Secondly, based on the linear clustering features of observed signals in the sparse domain, the angular probability distribution of preprocessing scatter is calculated to estimate the source number. Thirdly, the frequency bin corresponding to the peak of distance between scatter and original point is selected to construct the binary TF mask according to the estimated source number, and then the spectrum of recovered source is obtained via mask. Finally, the estimated sources considering padding line are calculated to eliminate the boundary effect in time domain. Experimental results demonstrate that the proposed method can effectively recover the simulated vibration sources with time-delayed mixtures in underdetermined case. In addition, two experimental validations manifest that compared with state-of-the-art algorithms, the proposed method improves signal separation performance and identifies the natural frequency of monomodal response successfully.

Introduction

Blind source separation (BSS), as a signal processing technique that recovers the source signals based only on the observed signals, provides an effective solution for signal analysis [1], [2], [3]. However, traditional BSS methods are limited by the number of sensors, i.e., the number of observed signals not less than that of source signals. This framework is similar to the multiple-input and multiple-output (MIMO) equalization, i.e., the signal classification problem is addressed under the condition of the underdetermined or overdetermined [4]. In practice, the number of sensors is limited on account of the restrictions of cost and equipment. Hence, the number of observed signals is always less than that of the sources, which leads to the problem of underdetermined blind source separation (UBSS) [5], [6], [7]. Meanwhile, it is difficult for each source signal to reach all sensors at the same time, the time delay depends on the relative position between the sensor and the source as well as the propagation speed of the signal. It is a challenging task to install enough sensors in a limited space and the propagation delay of sources between all sensors is not avoided in the process of signal acquisition [8], [9], [10]. Nevertheless, the instantaneous mixtures do not consider the delay in propagation channel, i.e., only consider the attenuated information of source, so the mixing matrix only contains the attenuation coefficient [11], [12], [13]. Dealing with time-delayed mixtures is more challenging than instantaneous mixtures, it is of great practical significance to study the UBSS method in presence of time-delayed mixtures with modal identification.

Some methods have been developed to address the problem of UBSS caused by attenuated and time-delayed mixtures. Bofill [14] proposed a three-step approach to deal with attenuation matrix, delay matrix and source estimation, respectively. The angular clustering of the input magnitude is used to infer attenuation matrix; and then the delay matrix is deduced via maximizing clustering over phase; according to the attenuation and delay matrices, sources are recovered by second-order cone programming. In Ref. [15], two types of time-frequency (TF) BSS approaches inspired by time-frequency ratios of mixtures (TIFROM), are presented with attenuated and delayed mixtures. The detection of time delay is resorted to regression lines associated to the TF ratios of mixtures. In addition, either in constant-time or constant-frequency analysis zones the variance of TF ratios is used to detect attenuation coefficients. Zhang [16] also proposed an UBSS method for time-delayed mixtures based on the prior information exploitation. The prior information is extracted from the complex-valued mixing matrix to determine the single source points, then the agglomerative hierarchical clustering (AHC) and subspace methods are utilized to estimate mixing matrix and source signals, respectively. To solve the time-delayed mixing model of the UBSS, a mixing matrix estimation algorithm was proposed in [17]. The method creates a transformation matrix to construct a real spectrum matrix in TF domain, then the mixing matrix is derived based on the clustering centers of single source points. Meanwhile, the degenerate unmixing estimation technique (DUET) [18] can separate sources from speech mixtures in the time-delayed mixing model. Nonetheless, the DUET method must meet very rigorous sparsity. Similarly, sparse component analysis (SCA) [19] as the most useful approach for the UBSS can yield better source separation performance by using sparsity property of signals in sparse domain, especially in the application of modal identification [20], [21], [22].

In structural dynamic analysis, the mixing matrix and monotone modal sources can be accurately obtained via SCA in sparse domain, then the monotone modal identification method is used to estimate the modal parameters [20], [23]. However, traditional SCA technique works well in the case of instantaneous mixture, which indicates that the modal response signals should reach the sensors simultaneously [2], [5], [24] and this requirement is also too strict to meet. Therefore, it is imperative to study an UBSS method for modal identification in presence of time-delayed mixtures. We provide the detailed theorem to explain the sparse domain first. Then a novel approach is proposed to address the source separation problem for time-delayed mixtures in underdetermined case, which consists of three main parts, i.e., source number estimation, binary mask prediction and elimination of boundary effect. Finally, relevant simulation and experiments testify the advantage of the proposed approach aiming at the delay mixtures in the UBSS problem.

The rest of this paper is structured as follows. The theoretical background is listed in Section 2, which includes the introduction of time-delayed mixture model and detailed analysis on the theorem of delay influence. Section 3 describes the main components of the UBSS method for time-delayed mixtures, i.e., source number estimation, binary mask prediction and elimination of boundary effect. In Section 4, simulation analysis validates the effectiveness and applicability of the proposed method. Meanwhile, two experimental validations illustrate the superiority of the proposed method, compared with state-of-the-art algorithms in Section 5. Lastly, the conclusions are drawn in Section 6.

Section snippets

Time-delayed mixture model

This paper tackles UBSS problem with the time-delayed mixtures as observed signals, i.e., considering the influence of delay involved in mixing system, which depends mainly on distance between sensor and source. The mathematical expression of time-delayed mixing model without considering noise can be described asxi(k)=j=1naijsj(kσij),i=1,2,,m where k denotes the discrete time; xi(k) is the ith mixing signal; sj(k) is the jth source signal; m and n are the total number of observed signals and

Source number estimation

In practical applications, the number of sources determines the accuracy of source signals recovery. Regarding the problem of source number estimation in signal processing, most methods are only for overdetermined BSS model and the specific estimation process is quite complex [28], [29]. In the problem of UBSS, source number estimation is an indispensable part of sources recovery [30] especially considering the time delay in the mixing process. Therefore, the source number estimation is carried

Simulation analysis

In this section, four simulated vibration signals are used to verify the effectiveness of the proposed UBSS method in presence of time-delayed mixtures. The vibration signal is mainly generated by mechanical rotating parts, e.g., the periodic signal is a common form of vibration signal, the mutual meshing of gears and bearings is often accompanied the modulation signal and the mechanical vibration shock also produces impact signal. Therefore, four source signals are adopted as shown in Eq. (16)

Case 1

In this section, the setup of the experiment and data are directly from [36]. Here the sampling frequency is 2560 Hz, sampling point is 1280. Both STFT and padding line choose the same Hamming window, window size is a quarter of the number of frequency bins, and parameters of all padding line are set to 0.625. The waveform and the spectrum of measured signals are collected via sensors, as shown in Fig. 10. It can be seen from the spectrum that the signals collected by three sensors all contain

Conclusion

The acquisition process of recorded signals is strictly limited by the number of sensors, which indicates that the number of source signals often greater than the sensors number, i.e., the underdetermined case. In addition, considering the complexity of the transmission channel, we cannot avoid the propagation delay of sources between all sensors. Therefore, it is of great practical significance to study the BSS method for time-delayed mixtures in underdetermined case. In this paper, an UBSS

CRediT authorship contribution statement

Baoze Ma: Conceptualization, Methodology, Software, Writing – original draft. Tianqi Zhang: Formal analysis, Investigation, Methodology, Supervision. Zeliang An: Data curation, Resources, Software, Writing – review & editing. Tiecheng Song: Formal analysis, Methodology, Visualization, Writing – review & editing. Hui Zhao: Conceptualization, Data curation, Supervision, Visualization.

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.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 61671095, 61371164).

Baoze Ma received the B.S. degree in electrical and information engineering from Inner Mongolia Normal University, China, in 2014, and the M.S. degree in information and communication engineering from the Chongqing University of Posts and Telecommunications, China, in 2017, where he is currently pursuing the Ph.D. degree with the School of Communication and Information Engineering. His research interests include blind source separation, signal feature extraction, and deep learning.

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    Baoze Ma received the B.S. degree in electrical and information engineering from Inner Mongolia Normal University, China, in 2014, and the M.S. degree in information and communication engineering from the Chongqing University of Posts and Telecommunications, China, in 2017, where he is currently pursuing the Ph.D. degree with the School of Communication and Information Engineering. His research interests include blind source separation, signal feature extraction, and deep learning.

    Tianqi Zhang received the B.S. degree in physics from Southwest University, China, in 1994, and the M.S. degree in communication and electronic system and the Ph.D. degree in circuits and systems from the University of Electronic Science and Technology of China, in 1997 and 2003, respectively. From 2003 to 2005, he was a Postdoctoral Fellow in communication and information system with Tsinghua University. Since August 2005, he has been a Professor with the School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications. His research interests include the areas of communication, and image and speech signal processing.

    Zeliang An received the B.S. degree of Electronic Information Engineering from Huainan Normal University, China, in 2015, and the M.S. degree for Communication and Information Systems from Nanchang Hangkong University, China, in 2019. He is currently pursuing the Ph.D. degree at Chongqing University of Post and Telecommunications. His research interests include multicarrier waveform recognition, deep learning, signal feature extraction and modulation recognition.

    Tiecheng Song received the Ph.D. degree in signal and information processing from the University of Electronic Science and Technology of China, in 2015. From October 2015 to April 2016, he was a Visiting Student with the Multimedia Laboratory, Nanyang Technological University, Singapore. He is currently an Associate Professor with the School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications (CQUPT), China. His research interests include image processing and computer vision.

    Hui Zhao was born in 1980. She received the M.S. degree in fundamental mathematics and the Ph.D. degree in electronic science and technology from the Harbin Institute of Technology, China, in 2006 and 2010, respectively. She is currently a Professor with the School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications. Her research interests include signal and information processing, digital communications, and wireless networks.

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