Frequency Aliasing-Based Spatial-Wavenumber Filter for Online Damage Monitoring

&e spatial-wavenumber filter method can extract the specific mode of the Lamb wave, thereby distinguishing the incident wave and the damage reflection wave.&is method has been widely studied for damage imaging. However, the diameter of piezoelectric transducer (PZT) sensor limits the spatial sampling wavenumber of the linear PZT sensor array, which limits the application of this method because of the Nyquist–Shannon sampling theorem. &erefore, the wavenumber filtering range of spatial-wavenumber filter should be less than half of the spatial sampling wavenumber. In this paper, a frequency aliasing based spatialwavenumber filter for online damage monitoring is proposed. In this method, the wavenumber filtering range is extended to the spatial sampling wavenumber, and two wavenumber results will be calculated as for the frequency aliasing. Subsequently, the wavenumber of the received Lamb wave signal can be obtained according to the average arrival time difference between the two adjacent sensors in the linear PZTsensor array. Finally, the damage is localized using the spatial-wavenumber filter and cruciform PZT sensor array. &is method was validated on an epoxy laminate plate. &e maximum damage localization errors are less than 2 cm. It is indicated that this method can extend the spatial-wavenumber filtering range to the spatial sampling wavenumber and the application of spatial-wavenumber filter-based online damage monitoring.


Introduction
According to the concept of smart material structure, structural health monitoring (SHM) technology involves the application of embedded sensor networks to obtain information related to structural health online. e characteristic parameters of the signal related to structural health are extracted by an advanced signal processing algorithm. us, we can determine whether the structure is damaged, localize the damage, analyze the degree of damage, and predict the failure form and remaining life of the damaged structure. erefore, SHM technology can be used to prevent the occurrence of major accidents, improve safety of the structure, and reduce economic losses [1,2]. As a type of elastic stress wave, the Lamb wave is widely applied in damage identification of composite structures because of its long propagation distance, small energy attenuation, and sensitive response to damage. erefore, SHM technology based on the Lamb wave has been widely studied and is one of the most promising SHM technologies. In such applications, the Lamb wave is excited and collected by a low-cost piezoelectric transducer (PZT) sensor [3][4][5].
In previous spatial-wavenumber filter damage imaging research, the Lamb wave was collected using linear PZT sensor array. According to the Nyquist-Shannon sampling theorem, the frequency of the Lamb wave must be less than half of the sampling frequency. Similarly, the wavenumber of the Lamb wave also must be less than half of the spatial sampling wavenumber. However, the diameter of PZT sensor which is difficult to increase limits the spatial sampling wavenumber.
us, it will limit the application of spatial-wavenumber filter based online damage monitoring. In this study, a frequency aliasing based spatial-wavenumber filter for online damage monitoring is proposed, which extends the spatial-wavenumber filtering range to the spatial sampling wavenumber. e basic principle of the spatialwavenumber filter is introduced in Section 2.
en, the damage is localized using the spatial-wavenumber filter and cruciform PZT sensor array that is described in Section 3. e proposed spatial-wavenumber filter is validated on an epoxy laminate plate in Section 4. Finally, the conclusions are stated in Section 5.

Spatial-Wavenumber Filter
eoretic Foundation. Figure 1 shows a linear PZT sensor array placed on a structure. ere are M PZT sensors in the linear PZT sensor array and are numbered m � 1, . . ., M.
e spatial sampling interval is Δx, which is also equivalent to the distance between the centers of two adjacent PZT sensors in the linear PZT sensor array. A Cartesian coordinate was built on the structure. e original point was set at the center point of the linear PZT sensor array, and the X-axis was set along the linear PZT sensor array.
As illustrated in Figure 1, the acoustic source located at (θ a , L a ) excites the Lamb wave signal in the structure. e acoustic source is in the far-field area of the linear PZT sensor array. e propagation of the Lamb wave in the structure can be expressed using [36] f( where x and t represent the propagation distance and time of the Lamb wave, respectively; A denotes the amplitude term of the Lamb wave; ω a and k a are the central frequency and wavenumber of the Lamb wave; φ 0 is the initial phase of the Lamb wave. e wavenumber response can be obtained by Fourier Transform of the spatial response shown in (1) and (2). en, the wavenumber domain of the Lamb wave can be obtained as follows: where δ is the Dirac function and Using the linear PZT sensor array, the discrete spatial sampling signal f ′ (x, t) can be obtained by spatial sampling of Lamb wave with a spatial sampling interval of Δx, as shown in (4). In other words, the spatial sampling signal f ′ (x, t) is the product of the Lamb wave propagating continuously and the periodic impact signal p(x): where n is an integer and Using the Fourier Transform, the wavenumber response P(x) of the periodic impact signal p(x) can be obtained, as shown in where k s � 2π/Δx is the spatial sampling wavenumber of the linear PZT sensor array.

Shock and Vibration
According to (2) and (6), the wavenumber response of the discrete spatial sampling signal f′(x, t) is shown in Equation (7) shows that in the range of wavenumber domain (−k s , k s ) we have the following: ① k a ≥ 0: (a) If k s ≥ 2k a , the wavenumber results of the discrete spatial sampling signal f′(x, t) are k a and (k a − k s ), and (k a − k s )∈(−k s , −k a ], which is negative (b) If k a ≤ k s < 2k a , the wavenumber results of the discrete spatial sampling signal f′(x, t) are k a and (k a − k s ), and (k a − k s )∈(−k s , 0], which is negative (c) If 0.5k a ≤ k s < k a , the wavenumber results of the discrete spatial sampling signal f′( (a) If k s ≥ 2|k a |, the wavenumber results of the discrete spatial sampling signal f′(x, t) are k a and (k a + k s ), and (k a + k s )∈[k a , k s ), which is positive (b) If |k a| ≤k s < 2|k a |, the wavenumber results of the discrete spatial sampling signal f′(x, t) are ka and (k a + k s ), and (k a + k s )∈[0, k s ), which is positive (c) If 0.5|k a | ≤ k s < |k a |, the wavenumber results of the discrete spatial sampling signal f′(x, t) are (k a + k s ) and (k a + 2k s ); (k a + k s )∈(−k s , 0], which is negative, and (k a + 2k s )∈(0, k s ], which is positive As discussed above, the spatial sampling wavenumber k s should be greater than twice that of the Lamb wave according to the Nyquist-Shannon sampling theorem, as shown in (8). Furthermore, there is only one calculated wavenumber result k a in the range of (−0.5k s , 0.5k s ) which is the wavenumber k a of the Lamb wave in the previous research: erefore, if k a ≠ 0, there will be two calculated wavenumber results k a and ((k a − k s ) or (k a + k s )) in the range of (−k s , k s ) and k s ≥ 2|k a | or |k a | ≤ k s < 2|k a |. e signs of the two calculated wavenumber results are opposite. us, the wavenumber k a of the Lamb wave can be obtained when the sign of the wavenumber k a can be determined. Figure 1, the spatial sampling wavenumber of the linear PZT sensor array exceeds that of the Lamb wave, k s > |k a |. In addition, the received Lamb wave signals collected by the linear PZT sensor array can be expressed as shown in

Principle of the Method. As shown in
where L → a is the vector of the distance L a from the position of the acoustic source to the origin point; X → m is the vector of the X-axis coordinate x m of the No.m PZT sensor: According to (7), the received Lamb wave signal shown in (9) is transformed to the wavenumber response, as shown in A spatial-wavenumber filter is designed for the received Lamb wave signal, as shown in (12). Using Fourier Transform, the wavenumber response of the spatial-wavenumber filter can be obtained, as shown in where k′ is the pass-through wavenumber of the spatialwavenumber filter. Equation (13) shows that the spatial-wavenumber filter can selectively pass through the signal with the wavenumber that is k � k′ and reject the signal with the other wavenumbers k ≠ k′.
Next, the designed spatial-wavenumber filter is applied to the received Lamb wave signal when the wavenumber filtering range is (−k s , k s ), as shown in (14). e filtered wavenumber response of the received Lamb wave signal can be expressed as Finally, the spatial-wavenumber filtered synthesis signal of the linear PZT sensor array can be obtained using

Shock and Vibration
According to (15), the amplitude value of the spatialwavenumber filtered synthesis signal is small when k ′≠ (k a ·cos θ a − n·k s ). When k′ � (k a ·cos θ a − n·k s ), the amplitude value will be maximum. erefore, by applying the designed spatial-wavenumber filter to the Lamb wave received signal with the wavenumber filtering range from −k s to +k s , the (k a ·cos θ a − n·k s ) value corresponding to the maximum value of spatial-wavenumber filtered synthesis signal can be obtained.
According to the analysis in the previous section, there are two positive and negative wavenumber results in the range of (−k s , k s ) when k a ·cos θ a ≠ 0. If k a ·cos θ a > 0, (k a ·cos θ a − k s ) will be negative. If k a ·cos θ a < 0, (k a ·cos θ a + k s ) will be negative.
ere will be only one value of 0 rad/m which can be obtained when k a ·cos θ a � 0, which is the wavenumber of the received Lamb wave signal collected by the linear PZT sensor array.
In addition, Figure 1 shows that if the damage is at the right side of the Y-axis, that is, the positive half axis of the X-axis, the arrival time of the received Lamb wave signal collected by the No.M PZT sensor will be earlier than that of the signal collected by the No.1 PZT sensor and k a ·cos θ a > 0. Otherwise, the arrival time of the Lamb wave received signal collected by the No.M PZT sensor will be later than that of signal collected by the No.1 PZT sensor and k a ·cos θ a < 0, when the damage is at the left side of the Y-axis. erefore, when the spatial-wavenumber filtering result has two values, the wavenumber of the received Lamb wave signal can be finally determined by comparing the arrival times of the received Lamb wave signals collected by the No.M and No.1 PZT sensors in the linear PZT sensor array.
In practical application, the spatial-wavenumber filtering result and the calculated arrival time considerably fluctuate because of various factors which can easily cause misjudgment. erefore, the average arrival time difference t a between two adjacent sensors can be calculated using where t m is the arrival time of the received Lamb wave signal collected by the No.m PZT sensor and t (m+1) is the arrival time of the received Lamb wave signal collected by the No.(m + 1) PZT sensor. Equation (17) shows that the arrival time of the received Lamb wave signals collected by the No.M PZT sensor will be later than that of the signal collected by the No.1 PZT sensor if t a > 0, which means k a ·cos θ a < 0. Otherwise, if t a < 0, the arrival time of the received Lamb wave signals collected by the No.M PZT sensor will be earlier than that of signal collected by the No.1 PZT sensor and k a ·cos θ a > 0.
Finally, the wavenumber of the received Lamb wave signal can be obtained.
Using the linear PZT sensor array, the received Lamb wave signals can be collected for a certain length of time.
en, a wavenumber-time image can be obtained by spatialwavenumber filtering of the received Lamb wave signals at each time, as shown in Figure 2. In Figure 2, the wavenumber and time corresponding to the image point of the highest pixel value can be judged to be the spatial-wavenumber filtering result (k a ·cos θ a − n·k s ) and the arrival time t R of the received Lamb wave signal. erefore, the wavenumber k a ·cos θ a and the arrival time t R of the Lamb wave received signal can be obtained simultaneously by the spatial-wavenumber filter.

Damage Localization
ere is a cruciform PZT sensor array in the structure which is constructed by two linear PZT sensor arrays, as shown in Figure 3. e two linear PZT sensor arrays of the cruciform PZT sensor array are labeled as No.I and No.II. A Cartesian coordinate is built on the structure. e original point is set at the cross point of the cruciform PZT sensor array, and the X-and Y-axis are set along the No.I and No.II PZT sensor arrays. e Lamb wave is excited from the center point and propagation in the structure. If there is damage in the structure, it will scatter the incident Lamb wave [37]. e damage scattering signal can be collected by the cruciform PZT sensor array for a certain length of time.
e values of k a-I � k a cos θ a and t R-I can be obtained by spatial-wavenumber filtering of the damage scattering signal collected by No.I linear PZT sensor array, as shown in Figure 3. In addition, k a-II � k a cos(90°− θ a ) and t R-II can be obtained by spatial-wavenumber filtering of the damage scattering signal collected by No.II linear PZT sensor array.
us, the X-axis and Y-axis projection wavenumbers of the damage scattering signals all can be calculated using the spatial-wavenumber filter and cruciform PZT sensor array. en, the angle θ a of damage can be calculated using (18). Furthermore, the distance L a of damage can be calculated using the following equation. Finally, the damage position (θ a , L a ) is localized: where c g is the Lamb wave group velocity and t e is the Lamb wave start time.

Experimental
Setup. e validation experimental system comprises an integrated SHM system, a cruciform PZT sensor array, and an epoxy laminate plate, as shown in No.II PZTsensor array, respectively. e Lamb wave velocity c g is measured by a PZT sensor pasted at the position of 90°a nd 30 cm, which is labeled as PZT 8. e integrated SHM system is developed by Professor Yuan research group [38].
In this experimental verification, the excitation signal was a modulated 5-peak narrowband signal [39]. e frequency and amplitude of the excitation signal are 50 kHz and ±70 V. e sampling frequency and length of the Lamb wave are 10 MHz and 8000 samples with 1000 presamples. e experimental process is as follows: first, the Lamb wave velocity c g is measured using the Shannon wavelet transform [40]. e Lamb wave is excited by the excitation PZT sensor and propagates in the epoxy laminate plate. e corresponding Lamb wave signal is collected by PZT 8. e excitation time and arrival time are calculated through the continuous complex Shannon wavelet transform. en, the Lamb wave group velocity can be calculated as c g � 1370 m/s and applied to the following damage localization.
Second, the epoxy laminate plate is in the healthy status. e Lamb wave is excited by the excitation PZT sensor and propagates in the epoxy laminate plate. e corresponding Lamb wave signals collected by the cruciform PZT sensor array are the health reference signals f HR .
ird, six damages labeled A to F are applied to the epoxy laminate plate. Next, the corresponding Lamb wave signals collected by the cruciform PZT sensor array are the online monitoring signals f OM . e positions of these damages are shown in Figure 4(b) and Table 1.

Damage Localization.
e damage F is chosen as an example to validate in detail the proposed method and is located at 180°and 20 cm. First, the health reference signals f HR are collected by the cruciform PZT sensor array, as shown in Figure 5.
After the damage F is applied to the epoxy laminate plate, the online monitoring signals f OM collected by the cruciform PZT sensor array are shown in Figure 6. Shock and Vibration 5 e damage scattering signals of damage F can be extracted by subtracting the health reference signals f HR from the online monitoring signals f OM , as shown in Figure 7.
According to the spatial sampling interval Δx � 0.9 cm, the wavenumber filtering range was set to be from −680 rad/ m to 680 rad/m with the wavenumber filtering interval Δk � 0.1 rad/m. en, the wavenumber-time images of   According to (18), the damage direction (θ a � 179.4°) can be obtained using the wavenumbers k a-I and k a-II , and the damage direction error is −0.6°. e excitation time (t e � 0.1031 ms) of the Lamb wave is calculated by the continuous complex Shannon wavelet transform. en, the distance L a � 20.4 cm of the damage F can be calculated by (19). Finally, the damage position (179.4°and 20.4 cm) is localized, and the damage localization error becomes Δl � 0.5 cm.
According to the signal processing flow of damage F discussed above, the six damage localization results and errors are listed in Table 1, and the damage localization image is shown in Figure 9. It can be seen from Table 1 that the X-axis projection wavenumbers of damages A and F  Shock and Vibration exceed half of the spatial sampling wavenumber; the wavenumber-time images obtained by the conventional spatial-wavenumber filter method [29,30], with the wavenumber filtering range from −340 rad/m to 340 rad/m, are shown in Figure 10. In Figure 10, the X-axis projection wavenumber of damage A is k a-I � -315.8 rad/m, and the damage direction is θ a � 169.5°with the damage direction error 159.5°. Similarly, the X-axis projection wavenumber of damage F is k a-I � 310.3 rad/m, and the damage direction is θ a � 0.7°with the damage direction error 179.3°. It means that if the wavenumber of collected signal exceeds half of the spatial sampling wavenumber, the damage direction cannot be acquired correctly.
In the proposed method, the maximum filtering wavenumber is set to the spatial sampling wavenumber, and the two wavenumber filtering results are distinguished according to the average arrival time difference. e maximum damage localization errors are less than 2 cm in this experiment. e results indicate that the proposed method can improve the limitation of Nyquist-Shannon sampling theorem to the conventional spatial-wavenumber filter method, expand the filtering range of spatial-wavenumber filter to the spatial sampling wavenumber of the linear PZT sensor array, and thus expand the application of the spatialwavenumber filter based online damage monitoring.

Conclusion
In this paper, a frequency aliasing based spatial-wavenumber filter for online damage monitoring is proposed. In this method, the wavenumber filtering range of the spatialwavenumber filter is expanded to the spatial sampling wavenumber of the Lamb wave. en, the wavenumber of the received Lamb wave signal is determined according to the average arrival time difference between two adjacent sensors in a linear PZT sensor array. e damage can be localized using this method and a cruciform PZT sensor array. We validated the results using an epoxy laminate plate, and the results show that the damage localization errors are less than 2 cm. is method extends the wavenumber processing ability of the linear PZT sensor array using a software algorithm, without adding any hardware equipment. It is easily expanding the application of the spatial-wavenumber filter based online damage monitoring. However, depending on the group velocity of damage localization, the application of the proposed method may be limited; hence, further study is required. In addition, the influence of various factors on this method also needs to be studied further.

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

Conflicts of Interest
e authors declare no conflicts of interest.