Theoretical and Experimental Study on Homologous Acoustic Emission Signal Recognition Based on Synchrosqueezed Wavelet Transform Coherence

Te acoustic emission (AE) technique has been widely investigated for its ability to locate damage in structures. However, the selection of the arrival point of AE signals and the existence of nonhomologous AE signals can signifcantly afect the location accuracy of damages. Te synchrosqueezed wavelet transform (SWT) was used in our previous research to pick the accurate arrival point, but the existence of the nonhomologous signals was neglected in the picking process. To address this limitation, the synchrosqueezed wavelet transform coherence (SWTC) method is proposed to improve the accuracy by recognizing homologous signals and suppressing the spectral leakage in this paper. Compared with the wavelet transform coherence (WTC) method previously used, the SWTC method using the squeezing wavelet coefcients obtained by the SWT can constitute a more explicit coherence graph of AE signals. Tis clear coherence graph can help reduce the efect of subjective factors in observing the coherence and improve the recognition accuracy of homologous signals. Te efectiveness of the proposed method is experimentally verifed on a steel pipe and a concrete beam. Te results demonstrate that the SWTC accurately identifes homologous AE signals and efectively improves the localization accuracy across diferent signal densities, localization distances, and materials.


Introduction
Structural health monitoring (SHM) has been advancing rapidly in recent years to address the impacts of common adverse factors, such as corrosion [1], vibration and fatigue [2], and excessive loads [3], on structures across diferent engineering felds. Various SHM methods, including visionbased [4], ultrasonic wave-based [5], modal-based [6], and vibration-based [7] methods, have been developed with the goal of rapidly and accurately detecting damage. Among these nondestructive testing approaches, the acoustic emission (AE) technique [8,9], which is a passive monitoring technique that measures the elastic wave released by deformation or fracture of the materials under stress [10], is more sensitive to damage and suitable for continuous health monitoring in real time [11]. Tese advantages make the AE technique widely applied in mechanical engineering [12,13], civil engineering [14,15], mining engineering [16], and other felds [17,18]. Since the AE signals are generated when the materials become damaged, the AE signals can not only contain much information about the damage but can also be used to locate the damage [19,20]. However, the AE-based localization method faces challenges in accurately picking the arrival point of AE signals and recognizing the homologous AE signals. Our previous research [21] proposed the synchrosqueezed wavelet transform (SWT) to accurately pick the arrival point, but the recognition of homologous AE signals is a yet unsolved problem. Te homologous AE signals are emitted by the same source of damage. Terefore, by analyzing the homologous signals collected by AE sensors, it is possible to accurately determine the location of the damage source. However, the collected AE signals usually contain the homologous and nonhomologous signals. Tese nonhomologous signals infuence the localization accuracy by interfering with the time diference of arrival (TDOA) of signals [22]. Hence, the accurate identifcation of the homologous signals is benefcial to improve localization accuracy.
Many methods have been proposed for identifying homologous signals in the AE, speech, and volcanic tremor. Chen et al. [22] proposed the assumption that homologous signals should have similar characteristics (such as waveform), and they used concrete thermal-cracking experiments to demonstrate the validity of the assumption. Multiple algorithms, such as Pearson's correlation coefcient [23], the longest common subsequence algorithm [24], and edit distance on the real sequence [25], are available for investigating the similarity of a time series. Among such algorithms, dynamic time wrapping (DTW) [26,27], which fnds the minimum space distance between two time series by providing nonlinear alignments based on the standard of dynamic time warping [28], is one of the most popular algorithms used to calculate the Euclidean distance (ED) between time series. Te ED between two time series is commonly used to evaluate their similarity degree, with smaller ED denoting higher similarity [29,30]. Te wrapping process allows the DTW to evaluate the similarity of nonequal-length time series [31]. Sakoe and Chiba [32] leveraged this wrapping process to identify continuous speech. Sharma et al. [33] proposed a new road surface monitoring system based on the DTW to identify road irregularities. Besides the ED, the area between time series is also an often-used metric to evaluate similarity. Te normalized cross-correlation (NCC) [34], which calculates the inner product between two diferent signals, is a widely used area-based similarity recognition algorithm [35]. It has been widely used in electrocardiogram (ECG) signal processing [36], audio/ speech signal processing [37][38][39], and big data analysis [40]. Klausen and Robbersmyr [41] applied the NCC to calculate the cross-correlation between a whitened vibration signal and its envelope to analyze bearing faults. Nguyen et al. [42] used the cross-correlation image to identify the defects in pipelines. Te convolution operation in the NCC helps determine the time-based shift between two signals [40], which can be used to calculate the time delay of signals used for localization [43,44]. Permana et al. [45] used the characteristic of the NCC to locate a volcanic tremor source. However, the convolution causes the NCC to become overly sensitive towards minor distortions in the time axis [46] and also causes the recognition accuracy to become vulnerable to noise and frequency dispersion. However, the above similarity recognition algorithms are operated in the time domain, which precludes their usage in the time-frequency domain. In general, time-frequency analysis more comprehensively refects the intrinsic characteristics of signals than with time domain analysis [47]. Terefore, the time-frequency analysis has played a crucial role in SHM [48], and many time-frequency analysis algorithms, including fast Fourier transform (FFT), wavelet transform (WT), and Hilbert-Huang transform (HHT), have served as the foundation for developing methods to analyze damage signals. Katunin and Sun et al. combined the vibration-based method and wavelet transform (WT) to detect the damage of sandwich composites [49,50]. Chakraborty et al. utilized wavelet transform to identify the corresponding mode shapes from the transient response of the system under ambient vibration conditions [51] and proposed an online time-varying stifness monitoring system [52]. Furthermore, they combined time-frequencybased signal processing methods with clustering algorithms to efciently identify modal parameters [53]. For analyzing the similarity in the time-frequency domain, Torrence and Compo [54] proposed the wavelet transform coherence (WTC), which is a time-frequency similarity recognition algorithm. Tis algorithm converts the time domain signals to time-frequency matrices through wavelet transform and evaluates the similarity by calculating the coherence between time-frequency matrices. As a result, the coherence graph shows the similarity between signals in the time and frequency axes. Gao et al. [55] applied the WTC and AE analysis to study the relationship between the pure metal burn signals and grinding burn signals. Grinsted et al. [56] applied WTC to study the Arctic Oscillation index and the Baltic maximum sea ice extent record. Kramer et al. [57] used the WTC to distinguish tremor types between organic and functional types efectively.
Despite a large volume of research for WTC, there are still some crucial problems yet to be solved. First, the pointto-point coherency process [58] shows that the WTC is very sensitive to the energy distribution change in the timefrequency domain. Terefore, precise and clear energy distribution in the time-frequency domain is crucial for accurate coherency analysis. However, the spectrum leakage using continuous wavelet transform (CWT) causes the distribution of instantaneous amplitudes in the timefrequency matrix to be blurred [59], suggesting that the WTC based on the CWT cannot accurately refect the similarity between AE signals. Second, the accuracy in recognizing homologous signals through the coherence graph is susceptible to subjective factors, and thus, an objective quantitative coefcient for analyzing a coherence graph is necessary. Finally, it will take much computing time to sequentially obtain the coherence graphs of every two signals in a larger amount of data. Te defects mentioned previously can cause the low recognition accuracy of existing methods of recognizing homologous signals.
In this paper, the synchrosqueezed wavelet transform coherence (SWTC) recognition method is proposed to improve the recognition accuracy of homologous AE signals. Te proposed method uses the SWT to replace the CWT, thereby improving the clarity degree of coherence graphs by suppressing the spectrum leakage. Meanwhile, the SWTC coefcient is used to gauge the similarity degree of the coherence graph, reducing the infuence of subjective factors in observing the coherence graph. In addition, the proposed method also improves computational efciency by combining the time-order approach. Te pencil lead fracture (PLF) experiments on a steel pipe and a concrete beam were conducted to verify the efectiveness of the proposed method.

Synchrosqueezed Wavelet Transform
Coherence-Based Recognition of Homologous AE Signals 2.1. Flowchart of the Novel Method. Tis paper proposes the SWTC method to improve the accuracy in recognizing homologous signals, and a detailed fowchart of this method is presented in Figure 1. Te collected AE signals are frst divided into diferent localization groups by the time-order method, which can preliminarily identify the homologous signals by the arrival time of AE signals. Each localization group is composed of the localization pair and velocity measurement pair. Te localization pair is used to locate the damage source, and the velocity measurement pair at a known distance is used to determine the wave velocity of AE signals. Ten, the SWTC coefcients of these pairs are computed to refect the similarity degree of AE signals. When the SWTC coefcients of the localization pair and velocity measurement pair in the localization groups are higher than the mean value of all SWTC coefcients, the localization groups are used to compute the accurate localization values. Te detailed process is described in the following section.

Detailed Process of the Novel Method.
A large amount of AE signals can be collected in an experiment (as shown in Figure 2). Te traditional similarity recognition methods, which sequentially compute the coherence value between the selected signals and other signals, require much computing time. To solve this problem, the time-order method [21] is used to flter out the interference signals and divide the potential homologous AE signals into diferent localization groups ( g 1 , g 2 , · · · , g n ) with less computation time. However, the localization groups are composed of the homologous AE signals and the nonhomologous AE signals, and thus the localization groups need to be further fltered. For a random localization group g i from g 1 , g 2, · · · , g n , it contains four AE signals x 1 , x 2 , x 3 , x 4 . Te AE signals x 2 , x 3 collected from sensors s 2 and s 3 (as shown in Figure 2) are the localization pair p l , and the AE signals x 1 , x 2 (or x 3 , x 4 ) collected from sensors s 1 and s 2 (or s 3 and s 4 ) are the velocity measurement pair p v . For the AE signal x 1 from the velocity measurement pair p v , the wavelet coefcient can be expressed as follows: where a is the scale variable, which controls the dilation of the mother wavelet function ψ a,b , and b is the time variable, which controls the translation of the mother wavelet function [60]. Considering that the analyzed signals are AE signals, the complex Morlet wavelet is chosen as the wavelet basis [61]. According to reference [62], the candidate's instantaneous frequency ω 1 (a, b) of the AE signal x 1 is described as follows: Te wavelet coefcients (the red circle in Figure 3) with frequency for which the distance away from ω 1 (a, b) is ξ in the frequency domain will be squeezed and reassigned around the candidate's instantaneous frequency ω 1 (a, b): where ξ is the frequency variable, δ is the Dirac delta function, and T 1 (ξ, b) is the quantity transformed from CWT W 1 (a, b) on the time-frequency plane [60]. Te squeezing and reassigning process is defned as the synchrosqueezed wavelet transform, and T 1 (ξ, b) is the synchrosqueezed wavelet coefcient. Trough squeezing and reassigning the scale variable of the CWT into a candidate instantaneous frequency variable during the wavelet transform [63], the spectral leakage caused by CWT can be suppressed. Xue et al. [64] compared the performances of the SWT and CWT in monitoring wind turbine blades and found that the SWT can efectively suppress the spectral leakage caused by WT. Hence, the SWT is introduced to replace the CWT in the wavelet transform coherence Perform time-order method for {s 1 , s 2 , … , s n } to divide the AE signals into different localization groups {g 1 , g 2 , … g i , … , g n } Calculate the SWTC coefficients g l = {c l,1 , c l,2 , … , c l,i , … , c l,n } of localization pairs and c v,n } of velocity measurement pairs in these localization groups.
Calculate the mean value mean l of g l and the mean value mean v of g v If any g i from {g 1 , g 2 , … g i , … , g n } satisfies C l,i ≥ mean l and C v,i ≥ mean v , the g i is regarded as the homologous signals group  recognition method, and the detailed process is described as follows.
Based on the SWT, the autospectrum of the synchrosqueezed wavelet coefcient of the AE signal x 1 can be expressed as follows: where * is the complex conjugate, and the synchrosqueezed wavelet power of the AE signal x 1 is defned as Te cross-wavelet transform accurately describes the coherency of two signals in the time-frequency domain [65]. Specifcally, for the velocity measurement pair x 1 , x 2 , the cross synchrosqueezed wavelet transform (XSWT) is defned as follows [55]: Te synchrosqueezed wavelet power of the XSWT is |T 1,2 (η, b)|. Ten, the SWTC matrix R v of the velocity measurement pair can be obtained by normalizing and smoothing |T 1,2 (η, b)| [58]: where S is the smoothing function that prevents R v � 1 at all scales and times and η − 1 is the factor to convert the wavelet power to energy density [65]. Te smoothing function S is expressed as follows: where S scale and S time are smoothing operations along the wavelet scale axis and time axis, respectively. Te smoothing functions S scale and S time should have a similar footprint as used by the wavelet; therefore, S scale and S time used in the Morlet wavelet are defned as follows [55]: where t 1 denotes the smoothing range. Considering the length of AE signals (1024 signal points), the smoothing range is set to [−30, 30], which can keep a good balance between frequency resolution and signifcance [55]; c is the scale decorrelation length for the Morlet wavelet, and it is set to 0.6 [54]; Π is the rectangle function. Equations (4)-(9) are similar to the equations of the WTC recognition method [55,58], but the coefcients T, T 1 , and T 2 in the above equations are obtained by the SWT. By suppressing the spectral leakage, the coherence matrix R v obtained by the SWTC can more accurately characterize the similarity of the velocity measurement pair than the WTC. Te improved ability to characterize similarity will be verifed by an AE localization experiment described in the Section "Recognition Accuracy of the SWTC Recognition Method." In order to reduce the infuence of subjective factors in observing the coherence graphs, the SWTC coefcient of the coherence matrix R v is proposed. Te element R i,j v indicates the similarity of the AE signals x 1 and x 2 at the i th line and j th column in the coherence matrix R v . Te line and column of the coherence matrix R v correspond to the frequency and time, and thus R i,j v is a gauge for the similarity in the timefrequency domain. Hence, the average value of the coherence matrix R SWTC v can also refect the similarity of the velocity measurement pair x 1 , x 2 in the time-frequency domain: where M and N are the total numbers of lines and columns of the coherence matrix R v , respectively. c v is denoted as the SWTC coefcient of the velocity measurement pair. Similarly, the SWTC coefcient c l of the localization pair can be obtained in the same way. According to the Schwartz inequality [66], the range of the SWTC matrix is between [0, 1]; thus, the range of c v and c l are also between [0, 1]. In this range, 0 means low coherency and 1 means high coherency, with higher coherency of signals denoting higher similarity. Terefore, the SWTC coefcients of g 1 , g 2 , · · · , g n can be expressed as follows: Te mean values of g l and g v are defned as follows: Structural Control and Health Monitoring mean l � mean g l � mean c l,1 , c l,2 , · · · , c l,n , If a localization group satisfes the following: the localization group will be regarded as the homologous signal group. Ten, the time diference ∆t among the homologous signal group is calculated by the SWT picker method [21], and the damage location can be obtained by the following equation: where x, y, and z are each the damage location; x i , y i , and z i are each the sensor location; and v is the wave velocity of the AE signal. Terefore, localization accuracy can be improved by recognizing the localization groups composed of the homologous signals. Furthermore, since the noise is a nonhomologous signal, this recognition process can also flter out the noise.
To sum up, the SWTC recognition method can improve the accuracy in recognizing homologous signals and localization accuracy by suppressing the spectral leakage and quantifying the coherency matrix. In addition, by combining the time-order method, the SWTC recognition method can also improve computational efciency. An AE localization experiment is used to test the proposed method in the following section.

Experimental Setup.
A steel pipe and a concrete beam served as test specimens. As shown in Figures 4 and 5(a), the inside and the outer diameter of the steel pipe are 78.9 and 88.9 mm, respectively, and the length is 3000 mm. Te possible locations of AE sources (the red dots in Figure 4) are assumed to be distributed near the center of the pipe and are separated from each other by 20 mm. Te AE signals generated from the AE sources propagate along the steel pipe and are received by the AE sensors (the red rectangles in Figure 4). Five localization distances, including 330 mm, 550 mm, 770 mm, 1210 mm, and 2130 mm, are selected for testing in the experiment.
Te size of the concrete beam is 900 × 115 × 80mm, as shown in Figures 5(b) and 6. Te strength of the concrete is graded at C60 with its detailed composition listed in Table 1. Te locations of manual AE sources (the red dots in Figure 6) are also distributed near the center of the concrete beam and are separated from each other by 20 mm. Te possible locations of AE sources (the red dots in Figure 6) are assumed to be distributed along the concrete beam and are separated from each other by 20 mm. Due to the size limitation of the concrete beam, only two localization distances, including 330 mm and 550 mm, are tested in the experiment.
A PCI-2 8-channel AE system (Physical Acoustic Corporation) interrogated four R6a AE sensors (Physical Acoustic Corporation) is shown in Figure 5. Te AE sensors have a 35-100 kHz frequency band and a resonance frequency of 55 kHz. Since the approximate range of damage sources can be preliminarily determined, the distributions of sensors are shown in Figure 5, and the detailed information is shown in Figures 4 and 6. According to the authors of [21,67], the preamplifer is type 2/4/6 (Physical Acoustic Corporation), and its gain and threshold are set to 40 dB and 45 dB, respectively.
As the damage in the specimen grows, the density of collected AE signals gradually increases. However, pencil lead fracture (PLF) [68], a standard method of simulating AE signals, has long interval times and thus only acts on lowdensity AE signals. To solve this problem, the authors of [21,69] proposed ball dropping and wrench knocking to simulate the high-density AE signals. In this experiment, PLF is used to simulate low-density AE signals, and wrench knocking (for the steel pipe) and ball dropping (for the concrete beam) are used to simulate high-density AE signals.
Te main valuable components of signals are preserved to maintain higher recognition accuracy, while noise components are fltered out. Figure 7 shows the waveform and spectrogram of an AE signal, and it can be found that the energy is concentrated in the 7-70 kHz. Hence, in the frequency domain, the SWTC coefcient and the WTC coefcient are calculated in the range of 7-70 kHz (horizontal red diamond-dashed lines in Figure 7(b)) and thereby avoid noise. In the time domain, the starting point (vertical black triangle-dashed line in Figure 7) is identifed by the double-AIC method. Te endpoint (vertical orange x-dashed line in Figure 7) corresponds to 0.000556 s. Te selection of the area within these points can help minimize the interference of refected and refracted waves. As illustrated in Figure 7, the computing intervals of the NCC and DTW are between the vertical black triangle-dashed line and orange x-dashed line in Figure 7(a); the computing intervals of the WTC recognition and SWTC recognition methods are the rectangles enclosed by the above four lines in Figure 7(b). For this signal, four methods use the same computing interval of 0.00031 s. In practical engineering, the parameters of computing intervals can be easily obtained from the preexperiment in the laboratory. Furthermore, the central frequency of the complex Morlet wavelet is 0.955 Hz which can provide a good balance between the time and frequency localization [61].
In the following section, the experimental results are sequentially used to test the recognition accuracy and the localization accuracy of the SWTC recognition method, respectively. Te localization error gauges the recognition accuracy of the SWTC recognition method, where a lower localization error denotes higher recognition accuracy. Te results of the proposed method are compared with those of three other similarity recognition methods (i.e., the NCC recognition method, the DTW recognition method, and the WTC recognition method). A part of collected AE signals is used to compare the computing time between the traditional recognition methods and the SWTC recognition method.

Structural Control and Health Monitoring
where s i,1 and s i,2 are a pair of homologous AE signals, which indicate high similarity, and m � 20. Ten, by sequentially computing the similarity values sim j between the AE signals s j,1 from group 1 and all AE signals from group 2 , the results can be expressed as follows: where j � 1, 2 · · · m { } and k � 1, 2 · · · m { }. Tis similarity computing process is called the j th match. If the maximum value r max k,j of sim j satisfes k � j in the j th match, the match will be regarded as a successful match. However, when using DTW to calculate the similarity values, it should choose the minimal value r min k,j of sim j to check whether the value satisfes k � j in the j th match. Te rate of successful matches is where n is the number of successful matches and N is the total number of matches. In the following sections, the results of the SWTC recognition method are compared with those of other similarity recognition methods, including the NCC, DTW, and WTC recognition methods. Te results are shown in Figure 8, and the detailed information is listed in Table 2. Te NCC and DTW recognition methods have similar successful match rates, which are around 50%. Te low successful match rates suggest that the two time domain similarity recognition methods are unsuitable for identifying homologous AE signals. Compared with the NCC and DTW recognition methods, the WTC recognition method has better successful match rates (around 70%) because the WTC recognition method extends the analysis domain from the time domain to the time-frequency domain. Te SWTC recognition method further improves the successful math rate to be around 80%.
Te observed diference in successful match rates among the WTC and SWTC recognition methods can be attributed to the spectrum leakage, which blurs the timefrequency graph in the WTC recognition method. As shown in Figure 9(a), two AE signals are selected from the above experiment, and the computing intervals of SWTC and WTC coefcients are identical (as shown in Figure 7). Te two AE signals are nonhomologous, which means the coherence value between the two signals should be low. Figure 9(b) shows the time-frequency graph transformed by WTC. Te fgure shows that the Structural Control and Health Monitoring spectrum leakage causes the wavelet energy to be vaguely distributed across 10-100 kHz. Te vague time-frequency graph further causes the coherency coefcient to difuse in the coherency graph (as shown in Figure 9(c)), which in turn causes the WTC coefcient of the two AE signals to be high (0.58). Figure 10 shows the results obtained by the SWTC recognition method. Compared with the results of WTC (as shown in Figure 9), the SWTC efectively improves the clarity of the time-frequency graphs by concentrating the wavelet energy to central frequency. Meanwhile, the clear time-frequency graph further enhances the accuracy of the coherence graph (as shown in Figure 10(b)). Te SWTC coefcient of the two AE signals is 0.19, which indicates low similarity. Terefore, the SWTC recognition method is demonstrated to have higher accuracy than the WTC recognition method in determining the similarity of AE signals.
Te above results verify that the SWTC recognition method yields higher accuracy than other methods in recognizing homologous signals. Hence, the SWTC recognition method should have better localization accuracy than other recognition methods (i.e., the NCC recognition method, DTW recognition method, and WTC recognition method).
Te localization accuracy will be demonstrated in the following section.

Diferent Signal Densities.
In this section, the above four recognition methods (SWTC recognition method, NCC recognition method, DTW recognition method, and WTC recognition method) are applied to obtain the homologous signals set from the collected signals. Ten, the SWT localization method is used to obtain the fltered location values from the homologous signals set. Meanwhile, the unfltered location values are obtained from the collected signals.
Te mean absolute errors (MAE) of the localization values in the low signal density experiments are shown in Figure 11, and the improvement rates (IR) of the localization values are listed in Table 3. Te MAE and IR are defned as follows: where l mea,i is the calculated localization value; l true,i is the accurate localization value; N is the number of localization values; MAE unfiltered is the MAE of unfltered location values, and MAE filtered is the MAE of fltered location values. Trough observing equations (22) and (23), it can be found that lower MAE indicates higher localization accuracy and higher IR indicates higher recognition accuracy. Te MAE (31.74 mm) of the DTW recognition method is close to the original MAE (32.21 mm), which causes the IR of the DTW recognition method to be only 1.48%. Te NCC recognition method, which operates in the time domain, demonstrated a better performance by improving the MAE and IR to 29.59 mm and 8.13%, respectively. However, the time-frequency similarity recognition methods have higher IR and lower MAE than the time domain similarity recognition methods. Te MAE and IR of the WTC recognition method are 17.17 mm and 46.69%, respectively. Te SWTC recognition method has the lowest MAE (10.67 mm) and the highest IR (66.86%) out of the four recognition methods, demonstrating that the SWTC has superior recognition accuracy.
Te same conclusion can also be obtained from the high signal density experiment (as shown in Figure 11 and Table 3). Te SWTC recognition method has the best recognition accuracy by providing the lowest MAE (13.77 mm) and highest IR (56.28%). However, in the SWTC recognition method, the IR (56.28%) of the high signal density experiment is lower than that of the one of the low signal density experiment (66.86%). To assist in understanding this problem, the SWTC recognition method results obtained from the steel pipe experiment and concrete beam experiment are listed in Tables 4 and 5, respectively. Te mean amplitudes for low and high signal    densities in the steel pipe experiment and concrete experiment are also listed in Tables 4 and 5, respectively. Te tables show that the mean amplitudes of the steel pipe experiment are similar (around 84 dB) regardless of signal density, and the IRs are also similar (around 67%). In contrast, in the concrete beam experiment, the IR (41.63%) and mean amplitude (87.71 dB) of the low signal density experiment are larger than those of the high signal density experiment (26.2% and 77.17 dB). Te lower amplitude of signals means more severe attenuation in transmission, which is not benefcial to the identifcation of homologous signals. Hence, the problem can be attributed to the low mean amplitudes of the high signal density in the concrete beam experiment. In summary, the SWTC recognition method is better than other recognition methods in improving the localization accuracy across diferent signal densities by efectively fltering out nonhomologous signals. Figure 12 shows the MAEs of four recognition methods in diferent localization distances, and the detailed information is listed in Table 6. Since the concrete beam experiment only has two localization distances (0.33 and 0.55 m), the data in the frst two localization distances (0.33 and 0.55 m) contain the results from the steel pipe experiment and the concrete beam experiment. Furthermore, the data in the    With the increase in localization distances, the MAEs of the SWTC recognition method are always the lowest among the four recognition methods. Te SWTC IR curve, as shown in Figure 12(b), is also almost above 50%. Particularly, in the steel pipe experiment, as shown in Figure 13 and Table 7, the SWTC MAEs reduced signifcantly in the frst two localization distances compared to the results of Table 6, and the IR curve, as shown in Figure 13(b), is almost above 60% in fve localization distances. Te reason for such performance is discussed in the following section. Compared with other recognition methods, the DTW recognition method has the highest MAEs in the later four localization distances, and the IR curve is below 10% in all localization distances. Particularly at 2.13 m, the IR of the DTW recognition method is negative, which means that the DTW enlarges the localization errors. Te same situation also occurred for the NCC recognition method at 0.33 m. Tese results indicate that the DTW recognition method and NCC recognition method cannot efectively recognize the homologous signals. Te above discussion shows that the time-frequency similarity recognition methods outperform the time domain similarity recognition methods in the fve localization distances. For the diferent localization distances, the SWTC recognition        Figure 14, and the detailed information is listed in Table 8. Compared with other recognition methods, the SWTC recognition method has demonstrated its superior recognition accuracy by providing the lowest MAEs and highest IRs in the two tested material specimens. However, the IR of the SWTC recognition method in the steel pipe experiment is larger than that in the concrete beam experiment, and the same situation also occurred for the NCC and WTC recognition methods. Te reason can be attributed to the inhomogeneity of concrete. Inhomogeneity in the material can cause severe frequency dispersion and severe attenuation when the stress wave propagates through the material. Te dispersion and attenuation can unpredictably modulate the energy distribution in the time-frequency matrix and cause the coherent matrix to be inaccurate. Te IRs of the DTW recognition method are below 10% in both the steel pipe and concrete beam, which means that this method cannot accurately identify the homologous signals. In summary, the SWTC recognition method efectively improves the localization accuracy by fltering out the nonhomologous signals for both tested materials, including one that is inhomogeneous.

Discussion
Te analysis of the above results shows that the DTW and NCC recognition methods have high MAEs and low IRs, and they fail to identify the homologous signals accurately and cannot improve the localization accuracy. Te WTC recognition method ofered improved recognition than the above two recognition methods. However, the proposed SWTC recognition method has the lowest MAEs and highest IRs regardless of signal density, location distance, and material. Tus, this method has the best recognition accuracy of homologous signals and localization accuracy among the four tested recognition methods. Te conclusion of the section is consistent with the Section "Recognition Accuracy of the SWTC Recognition Method." To verify the computational efciency of the proposed SWTC recognition method, 40 AE signals are randomly selected from the above AE localization experiment. Te computer CPU is an Intel Core i5-6402P, and the computing software is Matlab R2018a (MathWorks). Te computing time taken by the traditional algorithm is 515.74 s, and that of the SWTC recognition method is only 10.19 s. Terefore, the SWTC recognition method efectively reduces the computing time.

Conclusion
In this study, a novel SWTC recognition method is proposed to improve the recognition accuracy of homologous AE signals. Te spectral analysis of AE signals shows that the proposed recognition method can obtain the explicit coherence graph of AE signals by efectively suppressing the spectral leakage in the time-frequency graph. Ten, the SWTC coefcient obtained from the explicit coherence graph can reduce the infuence of subjective factors. Two AE localization experiments were implemented on a steel pipe and a concrete beam to test the proposed recognition method. Te results demonstrate that the SWTC recognition method has the best recognition accuracy among the four tested recognition methods and further improves the AE localization accuracy across diferent materials, signal densities, and localization distances by fltering out nonhomologous signals. In particular, for the steel pipe experiment, the improving rate ofered by the SWTC recognition method is above 60%, which is much higher than that of the other recognition methods. Hence, the SWTC recognition method can improve localization accuracy by accurately fltering out the nonhomologous AE signals.
However, the proposed method also has some limitations. Te selection of the frequency bands for fltering and calculation range requires prior information about the structure and its boundary conditions; the damage size and shape could not be identifed. In future research, there are plans to combine the clustering technique and active detection technique with the SWTC recognition method. Tis integration aims to automatically search for the appropriate calculation range in the scalogram and assess the shape of the damage. To validate the efectiveness of the proposed method, extensive experiments will be conducted on largescale specimens. Furthermore, the accurate damage source and AE parameters will be used to convert the timefrequency domain to the wavenumber-frequency domain, which can show more information about the damage. Te types and degree of damage will be easily identifed in the wavenumber-frequency domain.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that there are no conficts of interest.