Research on the Signal Reconstruction of the Phased Array Structural Health Monitoring Based Using the Basis Pursuit Algorithm

The signal processing problem has become increasingly complex and demand high acquisition system, this paper proposes a new method to reconstruct the structure phased array structural health monitoring signal. The method is derived from the compressive sensing theory and the signal is reconstructed by using the basis pursuit algorithm to process the ultrasonic phased array signals. According to the principles of the compressive sensing and signal processing method, non-sparse ultrasonic signals are converted to sparse signals by using sparse transform. The sparse coefficients are obtained by sparse decomposition of the original signal, and then the observation matrix is constructed according to the corresponding sparse coefficients. Finally, the original signal is reconstructed by using basis pursuit algorithm, and error analysis is carried on. Experimental research analysis shows that the signal reconstruction method can reduce the signal complexity and required the space efficiently.


Introduction
Structural Health Monitoring (SHM) technology is an important application of the intelligent material structure practical enginee+96ring [Cao, Thaker, Oseng et al. (2015); Andrea, Pietro, Marco et al. (2016)]. The data acquisition and processing system is an important part of the structural health monitoring. Many scholars internationally have implemented different signal processing methods in structural health monitoring, such as model analysis [Bassoli, Forghieri and Vincenzi (2017)], system identification [Nagarajaiah (2107)], wavelet transform, optimization calculation [Li (2015)] and so on. However, the 410 Copyright © 2019 Tech Science Press CMC, vol.58, no.2, pp.409-420, 2019 majority of research findings are based on the simplified structure and carried out in a controllable laboratory environment. Research has been very successful at the real collected sensor signals which contains various signals. In general, due to the environment noise and the complex properties of composites, the extraction of the damaged characteristic signal is very difficult. In a real-time and on-line monitoring operation a large amount of data can be generated, which increases the complexity of data processing. Thus choosing a suitable method of signal processing is particularly important. Signal processing based on compressive sensing technology has been applied in many disciplines and engineering fields [Jin, Yang, Chen et al. (2014); Liu (2012)]. For example, in the analysis of signal channel bandwidth in wireless communication, the data is sampled in a compressed sampling manner [Haupt, Bajwa, Raz et al. (2010)]. In image processing, Rice University used compressive sensing technology to develop a new "single pixel camera" [Duarte, Davenport, Takhar et al. (2008)]. In the late 1990s, A. Abbate [Abbate, Frenkel and Das (1995)] proposed a new type of signal processing device, and the ultrasonic signal processing is not limited to the analysis of amplitude, phase and frequency domain. It can provide a powerful guarantee for the qualitative and quantitative analysis of the echo signal. This article proposes a method to realize the structural health monitoring of composite materials using ultrasonic phased array technology. The ultrasonic phased array detection technology uses different shapes of multi array elements to generate and receive ultrasonic beam [Sun, Zhang, Qian et al. (2013); Sun and Ji (2015)]. Where, the phased array transducer contains a plurality of piezoelectric elements in a certain way, where each piezoelectric element can independently transmit and receive ultrasonic signals [Liu (2012)]. This method consists of 3 steps: Firstly it proposes the sparse signal using discrete cosine transform; secondly designs a phased array signal acconding to the observation matrix; finally reconstructs the signal based on basis pursuit. The reconstruction uses the compressive sensing theory to process the data acquisition and signal compression at the same time, which greatly reduces the computational complexity and reduces the complexity of the data signal processing and saves the storage space. The paper is organized as follows: Section 1 introduces the research status of structural health monitoring based on compressive sensing; Section 2 explains the principles of compressive sensing and sparse signal processing methods; Section 3 uses experimental verification signal reconstruction methods to reduce signal complexity and save space complexity degree of effect.

Compressive sensing fundamentals
Compressive Sensing (CS), which is derived from the traditional signal sampling theory, is a new sampling and recovery theory for sparse signal [Friedland, Li and Schonfeld (2014)]. The sparse representation of signal, the design of observation matrix and the reconstruction of signal are introduced in Yu et al. [Yu, Li, Wang et al. (2012)]. If the original signal is sparse in the time domain or a certain transformation base, the signal can be collected at low sampling rate, and then achieve high probability reconstruction with compressive sensing technology. Compared to other sampling methods, this method can effectively reduce the complexity of the data, and has a wider range of superiority and universality [Donoho (2006)]. The whole signal processing is shown in Fig. 1

Sparse representation
In the structural health monitoring signal acquisition process, the signal obtained through the piezoelectric array has continuous amplitude. The number of non-zero values in the sparse coefficient vector can be reduced by selecting the appropriate sparse transform base, and a sparse representation of the phased array signal can be obtained. In the process of sparse representation, the signal is projected onto an orthogonal base. If the obtained transformation vector satisfies the characteristics of sparsity, it can be used as a sparse representation of the original signal, which is the prior condition of the compressive sensing. That is to say if a signal satisfies certain conditions, the sparse representation of the signal can be obtained by selecting one transformation base. In general, the commonly used transform bases are: Fourier Transform, Discrete Cosine Transform, Wavelet Basis [Song, Guo and Zhu (2014)], Curvelet Basis [Shahidi, Tang and Ma (2013)], and so on. At present, in addition to the traditional Fourier Analysis, the advanced signal processing methods are widely used in the study, such as Wavelet Analysis [Soni, Jain and Roshan (2013)], HHT analysis method [Petrov (2016)], and so on. Discrete Cosine Transform (DCT) is an orthogonal transformation method, which has the best correlation, and is often considered as a quasi-optimal transformation, moreover, it is also often used to lossy data compression of signals or image [Bayer and Cintra (2017)]. As the widely use of the special integrated circuit, the discrete cosine transform is constantly strengthened. Discrete Fourier Transform (DFT) is a discrete form of Fourier transform in time domain and frequency domain. The time domain sampling of signal is transformed into its DFT frequency domain sampling [West, Harwell and McCall (2017)]. Principal Component Analysis (PCA) can be used to deal with the data of high dimension, noise and high correlation by projecting the data into the low dimensional space and the most possible features of the original data [Singh, Sharma and Dandapat (2016)]. In compressive sensing, many complex signals in real life can expressed in a more concise way, or can be more succinctly expressed under some orthogonal basis transforms. This indicates that these complex signals are sparse under some orthogonal bases. In many cases, there is only a small amount of useful information in the highdimensional signal, and therefore, the key to obtain a sparse representation is to remove the redundant degree of signal [Zhu (2014); Gleichman and Eldar (2011) CMC, vol.58, no.2, pp.409-420, 2019 be seen as based on i  column, that is 12 [ , , , ] N    = . Therefore any one of a limited length of one-dimensional discrete signal N xR  can be expressed as: where  is the sparse coefficient vector. If most of the elements of  are zero, the one dimensional discrete time signal N xR  is sparse. At this time, the sparse coefficients are satisfied: where the signal x is the expression in time domain, <· > is the inner product, T is transpose of a vector, the sparse coefficient  represents the form of x by the sparse transform  .
Set the signal For any real number, when Jiang (2015)].

Projection observation
The sparse coefficients can be obtained after sparse decomposition of the original signal, according to the sparse coefficients [Feng, Zhang and Liu (2016)]. Thus we can construct the observation matrix which is not related to the transform of the signal.
In Eq. (5), the solution of  is an undetermined problem. However, because of the sparsity M k  , the problem can be solved, and the original signal can be restored with high probability. The matrix  for projection transformation needs to be irrelevant to the sparse transformations matrix  and to satisfy the finite equidistant principle. It is found that the Gauss random observation matrix has the characteristics that it is not related to the majority of the fixed orthogonal bases, and meet the compressive sensing requirement. Therefore, the Gauss matrix can be worked as the projection observation matrix.

Signal reconstruction Signal reconstruction process uses the known
where s.t. means the constraint. Eq. (6) is the solution of a Linear Programming (LP) problem, which is a convex optimization problem [Li (2015)]. If the reconstruction error is taken into account, the Eq. (7) can be converted to the minimum 1 l -norm problem: In the process of signal reconstruction, the most important task is to find the sparse solution to meet Eq. (7). In order to solve this problem, there are two approaches can be considered, which are convex optimization and greedy algorithm [Xia, Wang, Sun et al. (2015)]. The convex optimization algorithm obtains the sparsest mainly by increasing the constraint. Its norm bound form is expressed as: Literature shows that the (0 1) p l p   has a good application in the sparse representation [Zhang, Xu, Li et al. (2016)]. Commonly used algorithms are basis pursuit algorithm based on the linear programming and the gradient projection sparse reconstruction algorithm. The greedy algorithm is to use matching pursuit algorithm, combined with the local optimization approach to find the non-zero coefficient. On this basis, the orthogonal matching pursuit method is developed [Needell and Vershynin (2010)]. Basis Pursuit (BP) algorithm [Xiao, Zhao and Li (2013)], which is represented in the form of norm sparsity signal, namely by means of minimizing the number of signal norm sparse representation problem into a constrained extremum problems, thereby the problem is transformed into a linear programming problem.
In the reconstruction algorithm, the minimum 1 l norm and minimum 0 l norm have equivalence, and can be interchangeable under certain conditions. Then the Eq. (8) is converted to an optimization problem under the minimum 1 l -norm: Because the number of observations is far less than the length of the signal, it is difficult to solve a set of undetermined equations in the signal reconstruction process. The undetermined equation has infinitely many solutions, the solution process is difficult to achieve. However, since the signal passes through the sparse transformation, the issue CMC, vol.58, no.2, pp.409-420, 2019 can be resolved. Restricted Isometry Property (RIP) is also a theoretical guarantee for the solution of the observation matrix. The basis pursuit algorithm is based on the above principle to reconstruct the original signal.

Experimental verification
In this study, a sensor linear array which consists of 8 piezoelectric sensors is arranged in a 800 mm×800 mm×3 mm plate. The diameter and the thickness of the piezoelectric sensors are 8 mm and 0.48 mm, respectively. The distance between two adjacent piezoelectric sensors is 12 mm. We use the phased array to scan the aluminum plate structure and obtain 8×7×181 groups data, here we select one group to process the data, which is emitted by the No. 0 array element and received by the No. 1 array element in the 90 degree direction; other angles take the same approach. The time domain waveform of the data obtained from the group is shown in Fig. 2.

415
According to the Eq. (10), ) 0 ( Y is the maximum constant after one-dimensional DCT transform. According to the Eq. (11), after removing the constant, the shape of the signal does not change, only a certain phase shift occurs, which can prove that the DC component ) 0 ( Y is not suitable for a signal characteristics. Then, in the process of selecting the AC component, the AC component with larger amplitude is chosen as the characteristic of the signal. DCT is used to process the waveforms received by the 0-array-element to receive No. 1 array element in 90 degree direction. Fig. 3 shown in the sparse representation of the original signal is obtained, In Fig. 3, the value of most sparse coefficients of the signal after DCT is equal to zero or close to zero, which is consistent with the nature of sparse signal.

Design the observation matrix for the phased array signals
For the processing of ultrasonic signals, the Gauss random matrix is multiplied with the sparse coefficient of the signal, and the observation vector of the signal can be obtained. Let  is MN  matrix and its general term is: Gaussian distribution. Studies show that the matrix cannot associated with most sparse orthogonal matrix [Wen, Zhang, Wong et al. (2014)], and the number of measurements value is relatively small. For the original data with length N and sparse degree K , only the log( / ) M cK N K  measurements have a high possibility to restore the original signal. Where c is a very small constant value. In our example, the ultrasonic data length N is 1024, the projection observation of the signal uses the number of observed values 400. The resulting pattern is shown in Fig. 4.

Signal reconstruction based on the basis pursuit
We use BP algorithm to minimize the number of signal norm sparse representation problem into a constrained extremum problem, thereby the problem is transformed into a linear programming problem.  According to the basis pursuit to deal with the phased array signal, you can get the results as shown in Fig. 5. Fig. 5(a) shows the reconstruction of the original signal and Fig. 5(b) shows the reconstruction of BP algorithm. Fig. 5 shows the corresponding reconstructed signal obtained by using the basis pursuit algorithm. Tab. 1 shows the effectiveness of the reconstruction algorithm. In Tab. 1, the absolute error of V  is expressed as:

Signal reconstruction error analysis
Where, 0 V is the amplitude of the maximum point of the reconstructed signal; 1 V is the amplitude of the original phased array signal. The relative error of  is present in Eq. (15).   Tab. 2 shows the error comparison of some common transform base. It can be seen from the experimental error analysis that the BP algorithm used in this paper has low error.

Conclusion
This paper studies phased array signal reconstruction for structural health monitoring using the basis pursuit algorithm. The proposed method consists of three steps: Signal sparse representation, observation matrix design, signal reconstruction. Finally making error analysis of reconstructed signal. The signal from a sensor liner array on an aluminum plat is used as a study case to demonstrate how this approach effectively saves storage space and reduce the data complexity. The experimental also process its superiority in signal processing method by comparison to other commonly used algorithm such as PCA, DCT and DFT.