# Mode Separation for Multimodal Ultrasonic Lamb Waves Using Dispersion Compensation and Independent Component Analysis of Forth-Order Cumulant

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Lamb Wave Dispersion Compensation

#### 2.2. Mode Separation for Multimodal Overlapped Lamb Waves

**R(t)**.

**R(t)**= E[

**Y**(t)

**Y**

^{T}(t)]. The fourth-order cumulant ICA algorithm is used to get the signals of the modes from the signal matrix

**Y**(t).

**Y**(t) [20,21,22]. The matrix is whitened and the fourth-order cumulant matrix of the whitened matrix is calculated. Then the diagonalization of the cumulant matrix is performed to find the best estimation matrix.

**Y**(t) is defined as

**Z**(t),

**V**is the whitening matrix. It can be obtained by eigenvalue decomposition of the covariance matrix

**E**is the orthogonal matrix corresponding to the eigenvectors of

**R**.

**D**is the diagonal matrix of

**Y**(t).

**G**is the rotation matrix. Let

**D**,

**R**is the covariance matrix of $Z(t)$.

_{Z}## 3. Results

_{1}mode signal at the distance of 1.6 cm is retrieved, as shown in Figure 4. The multimodal Lamb wave signal at the distance of 2.7 cm after compensation using the A

_{1}mode dispersion characteristic is shown in Figure 4a. The separated A

_{1}mode signal using the proposed method at the distance of 1.6 cm is shown in Figure 4b. Its amplitude spectrum is shown in Figure 4d, and its time-frequency distribution is shown in Figure 4f. The separated A

_{1}mode signal using the time-frequency blind signal separation method at the distance of 1.6 cm is shown in Figure 4c. Its amplitude spectrum is shown in Figure 4e, and its time-frequency distribution is shown in Figure 4g.

_{1}mode signal at the distance of 1.6 cm is also retrieved, as shown in Figure 5. The multimodal Lamb wave signal at the distance of 2.7 cm after compensation using the S

_{1}mode dispersion characteristic is show in Figure 5a. The separated S

_{1}mode signal using the proposed method at the distance of 1.6 cm is shown in Figure 5b. Its amplitude spectrum is shown in Figure 5d and its time-frequency distribution is shown in Figure 5f. The separated S

_{1}mode signal using the time-frequency blind signal separation method at the distance of 1.6 cm is shown in Figure 5c. Its amplitude spectrum is shown in Figure 5e, and its time-frequency distribution is shown in Figure 5g.

_{1}and S

_{1}mode signals are added and shown in Figure 6a,b. The original measured mixed Lamb wave signal at the distance of 1.6 cm is also shown in the figure. In Figure 6a, the reconstructed signal is obtained using the time-frequency blind signal separation method. In Figure 6b, the reconstructed signal is obtained using the proposed method. The time-frequency distributions of the original signal and the reconstructed signal are shown in Figure 6b,c, respectively.

## 4. Discussion

_{1}and S

_{1}modes will continue to disperse. The amplitude of the signals will decrease, and the duration will extend.

_{1}mode signal at the distance of 1.6 cm is retrieved, as shown in Figure 4. The multimodal Lamb wave signal at the distance of 2.7 cm after compensation using the A

_{1}mode dispersion characteristic is shown in Figure 4a. The signal start time after compensation has been compensated back to around 2.8 μs. The separated A

_{1}mode signal using the proposed method at the distance of 1.6 cm is shown in Figure 4b. The separated A

_{1}mode signal using the time-frequency blind signal separation method at the distance of 1.6 cm is shown in Figure 4c. Their amplitude spectra are shown in Figure 4d,e, respectively. Their time-frequency distributions are shown in Figure 4f,g, respectively. It can be seen from the time signals that the extracted signal using the time-frequency blind signal separation method has large errors due to the influence of Lamb wave dispersion and the source signal cross-term interference or local correlation under the calculation of global time-frequency distribution. It can be seen from the amplitude spectra and the time-frequency distributions that the separated A

_{1}mode signal using the time-frequency blind signal separation method still contains some part of the S

_{1}mode signal. There is still some part of the S

_{1}mode amplitude on the right side of the amplitude spectrum, and the tail energy remains in the time-frequency distribution. Therefore, the separation precision is low. However, the accuracy of separation is improved by using the proposed method. The separation result has better smoothness, which is consistent with the dispersion of the A

_{1}mode Lamb wave. From Figure 4f, the time-frequency of the extracted A

_{1}mode signal is non-stationary. From Figure 4d, the central frequency of the extracted A

_{1}mode signal is 2.98 MHz, and the frequency range is from 2.5 to 3.35 MHz.

_{1}mode signal at the distance of 1.6 cm is retrieved, as shown in Figure 5. The multimodal Lamb wave signal at the distance of 2.7 cm after compensation using the S

_{1}mode dispersion characteristic is shown in Figure 5a. The separated S

_{1}mode signal using the proposed method at the distance of 1.6 cm is shown in Figure 5b. The separated S

_{1}mode signal using the time-frequency blind signal separation method at the distance of 1.6 cm is shown in Figure 5c. Their amplitude spectra are shown in Figure 5d,e, respectively. Their time-frequency distributions are shown in Figure 5f,g, respectively. It can be seen from the time signals that the extracted signal using the time-frequency blind signal separation method has large errors due to the influence of Lamb wave dispersion and the source signal cross-term interference or local correlation under the calculation of global time-frequency distribution. There are even some irregular signals in the time signal. It can be seen from the amplitude spectra and the time-frequency distributions that the separated S

_{1}mode signal using the time-frequency blind signal separation method still contains some part of the A

_{1}mode signal and lacks some part of its mode signal. Therefore, the separation precision is low. However, the accuracy of separation is improved by using the proposed method. The separation result has better smoothness, which is consistent with the dispersion of the S

_{1}mode Lamb wave. The right part of the amplitude spectrum is slightly irregular, which means it has some minor errors. From Figure 5f, the time-frequency of the extracted S

_{1}mode signal is non-stationary. From Figure 5d, the central frequency of the extracted S

_{1}mode signal is 3.14 MHz, and the frequency range is from 2.7 to 3.55 MHz.

_{1}mode is slow, delaying on the time domain, and the time waveform changes less along the distance. On the contrary, the phase speed of the S

_{1}mode changes quickly in the low frequency band; the time signal changes significantly along the distance, and the amplitude becomes smaller and the duration longer with the increase of the distance. Figure 4d and Figure 5d show the amplitude spectra of the separated A

_{1}mode and S

_{1}mode. Except for a slight irregularity at both ends, it is in line with the theoretical analysis of the Lamb wave dispersion. The A

_{1}mode center frequency is 2.98 MHz, and the S

_{1}mode is 3.14 MHz.

_{1}and S

_{1}modes are added and compared with the experimental original measured signal at the distance of 1.6 cm in Figure 6. From Figure 6a, to the time-frequency blind signal separation method, the reconstructed signal is very different from the original signal. The result is obvious because the extracted signal mode signals are inaccurate and incomplete. In this method, the dispersion of Lamb waves are not considered resulting in the inaccuracy and incompleteness of single mode separation. On the contrary, to the proposed method, from Figure 6b, it can be seen that the mixed signals after separation and addition is the same as the original signal. The start and end time points are also the same. The front part of the reconstructed signal is more consistent with that of the original signal, and only a few peaks are slightly larger than the original signal. The tail of the signal is slightly less than that of the original signal. Considering that the tail of the signal is almost the main component of the S

_{1}mode and its dispersion degree is relatively fast, there may have been incomplete compensation in the dispersion compensation, which might lead to errors in the signal separation. Figure 6c,d are the time-frequency distributions of the original Lamb wave signal and the reconstructed Lamb wave signal, respectively. It can be seen that their time-frequency energy distributions are consistent. The maximum energy distribution is about 4.5 μs, which implies that the separated A

_{1}mode and S

_{1}mode are consistent with the original single modes.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**(

**a**) The compensated multimodal Lamb wave signal at the distance of 2.7 cm; (

**b**) separated A

_{1}mode time domain waveform using the proposed method; (

**c**) separated A

_{1}mode time domain waveform using the time-frequency blind signal separation method; (

**d**) amplitude spectrum of A

_{1}mode separated using the proposed method; (

**e**) amplitude spectrum of A

_{1}mode separated using the time-frequency blind signal separation method; (

**f**) time-frequency distribution of A

_{1}mode separated using the proposed method; (

**g**) time-frequency distribution of A

_{1}mode separated using the time-frequency blind signal separation method.

**Figure 5.**(

**a**) The compensated multimodal Lamb wave signal at the distance of 2.7 cm; (

**b**) separated S

_{1}mode time domain waveform using the proposed method; (

**c**) separated S

_{1}mode time domain waveform using the time-frequency blind signal separation method; (

**d**) amplitude spectrum of S

_{1}mode separated using the proposed method; (

**e**) amplitude spectrum of S

_{1}mode separated using the time-frequency blind signal separation method; (

**f**) time-frequency distribution of S

_{1}mode separated using the proposed method; (

**g**) time-frequency distribution of S

_{1}mode separated using the time-frequency blind signal separation method.

**Figure 6.**(

**a**) The reconstructed multimodal signal using the time-frequency blind signal separation method and the original signal at the distance of 1.6 cm; (

**b**) the reconstructed multimodal signal using the proposed method and the original signal at the distance of 1.6 cm; (

**c**) the time-frequency distribution of the original signal at the distance of 1.6 cm; (

**d**) the time-frequency distribution of the reconstructed signal at the distance of 1.6 cm.

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**MDPI and ACS Style**

Chen, X.; Ma, D.
Mode Separation for Multimodal Ultrasonic Lamb Waves Using Dispersion Compensation and Independent Component Analysis of Forth-Order Cumulant. *Appl. Sci.* **2019**, *9*, 555.
https://doi.org/10.3390/app9030555

**AMA Style**

Chen X, Ma D.
Mode Separation for Multimodal Ultrasonic Lamb Waves Using Dispersion Compensation and Independent Component Analysis of Forth-Order Cumulant. *Applied Sciences*. 2019; 9(3):555.
https://doi.org/10.3390/app9030555

**Chicago/Turabian Style**

Chen, Xiao, and Dandan Ma.
2019. "Mode Separation for Multimodal Ultrasonic Lamb Waves Using Dispersion Compensation and Independent Component Analysis of Forth-Order Cumulant" *Applied Sciences* 9, no. 3: 555.
https://doi.org/10.3390/app9030555