Split-Spectrum Processing with Raised Cosine Filters of Constant Frequency-to-Bandwidth Ratio for L(0,2) Ultrasonic Guided Wave Testing in a Pipeline

Round 1

Reviewer 1 Report

The authors proposed an advanced technique to improve the signal interpretation for L(0,2) guided wave propagating in a pipeline. Experimental validation is presented. The paper is suitable for Applied Sciences in general since it combines advanced algorithm and its application for the NDE problem for pipes.

The following points are to be dealt with before publication:

- The algorithm is not described well enough. For instance, the flowchart exposing the proposed algorithm might be added.

- It is not clear which transform is used by the authors. The reviewer is not familiar with the Chript transform mentioned by the authors, so it must be explicitly described.

- The importance of Section 2.2 is doubtful in the present form. Equations (2) - (7) seem not important for further treatment in the manuscript. 

Author Response

Reviewer #1: The authors proposed an advanced technique to improve the signal interpretation for L(0,2) guided wave propagating in a pipeline. Experimental validation is presented. The paper is suitable for Applied Sciences in general since it combines advanced algorithm and its application for the NDE problem for pipes.

The following points are to be dealt with before publication:

Q1: The algorithm is not described well enough. For instance, the flowchart exposing the proposed algorithm might be added.

A1: Thank you for the suggestion. When I tried to describe the algorithm, I just wanted to visualize the steps of the algorithm and ignored the flow chart, which was an oversight on my part. According to your suggestion, I have added the flowchart exposing the proposed algorithm (Page 3, annotation A1), which is shown as follows:

Figure 1. Schematic diagram of split spectrum processing (SSP).

I sincerely hope that this figure will enable reviewers and readers to understand the principles of the algorithm more directly.

Q2: It is not clear which transform is used by the authors. The reviewer is not familiar with the Chript transform mentioned by the authors, so it must be explicitly described.

A2: I am sorry that I mistakenly wrote Chriplet transform as Chript transform, it is really my fault. And I apologize for not explaining the Chriplet transform in the manuscript, the reason why Chriplet transform was not explicitly described is that I think only the time-frequency analysis by Chriplet transform cannot accurately identify defect signals and it is not the focus of my study. But, according to your suggestion, I realize that it is quite necessary to explain the Chriplet transform.

Firstly, I would like to explain why I applied Chriplet transform to ultrasonic guided wave (UGW) signals. For the dispersion and multi-mode of UGW, the time-frequency analysis needs to consider the separation of UGWs with different modes, but the conventional time–frequency representation solutions like short-time Fourier transform and Wigner Ville distribution may not achieve good UGW mode separation results, because multiple modes usually occupy the same frequency band and intersect each other in time. The chirplet transform, a generalization of both the wavelet and the short-time Fourier transform, enables the extraction of components of a signal with a particular instantaneous frequency and group delay, which can effectively distinguish different modes of UGWs, and there are a lot of studies that report on this [1-4].

Secondly, in order to illustrate the advantages of Chritpt transform, I used part of Kim's research results in "Mode separation and characterization of torsional guided wave signals reflected from defects using chirplet transform" [2], as shown in Figure 2 and 3.

Figure 2. Screenshots of Kim’s paper.

It can be found form Figure 3b that the STFT cannot separate the overlapping multimodal reflections from the notch, which are smeared together in the spectrogram. The effectiveness of Chriplet transform is evident from Figure 3c, where clearly the individual overlapping modes are resolved. The chirplet transform was able to resolve the overlapping T(0,1) and F(1,2) mode reflections from the defect. Also the transform could determine the F(2,2) mode reflection which cannot be revealed using the STFT.

     

Figure 3. (a) Time traces for the reflection signal from the defect and spectrogram ofthe signal by using (b) STFT and (c) chirplet transform.

Thirdly, I would like to introduce Chriplet transform briefly. The chirplet transform was introduced as a generalized time–frequency representation by Mann and Haykin [5]. The basis function can be adjusted by means of shifting, shearing, and scaling operators, resulting in a five-dimensional parameter space for the energy density which comprises projections of the respective densities obtained from a short-time Fourier transform (time and frequency shift) and a wavelet transform (time shift and scaling); a comparison is shown in Figure 4.

Fig. 4(a) represents the generation of the STFT and wavelet transforms by introducing the operators which allow arbitrary time and frequency shifts, and scale. Modulation with a chirplet in the time domain (Qq) shears the time–frequency atom in the frequency direction and modulation with a chirp in the frequency domain (Pp) shears in the time direction. The shear parameters p and q determine the slope of the semi-axes of the chirplet atom ellipses.

 

     

Figure 4. Representation of time–frequency atoms of a TFR (a) STFT and wavelet transform (b) chirplet transform..

The standard definition of the chirplet transform is given by the inner product of a basis function g(t) and the signal x(t):

(1)

where * denotes the complex conjugate. The basis function g(t) and its Fourier transform G(ω) belong to a family of chirp signals c(t) and C(w):

		(2)
		(3)

where the operators T_t0, F_ω0, S_s, Q_q, and P_p act in the manner represented in Table 1 on the window function or its Fourier transform, respectively.

Table 1. Operations for time–frequency atoms for the chirplet transform..

Operation		Transformation
Time shift Tt0
Frequency shift Fω0
Scaling Ss
Frequency shear Qq
Time shear Pp

Finally, I would like to repeat that the Chriplet transform is not my research focus, and the purpose of applying Chriplet transform in the manuscript is to perform time-frequency analysis of the UGW signal, and it is not enough to identify the defect signals for the coherent noise. Therefore, it is necessary to apply split-spectrum processing (SSP) to improve the signal interpretation accuracy of UGW testing, which is my research focus. In addition, I have already added the description of Chriplet transform in manuscript (Page 6, annotation A9).

Q3: The importance of Section 2.2 is doubtful in the present form. Equations (2) - (7) seem not important for further treatment in the manuscript.

A3: Thank you for your comment. Section 2.2 describes the signal recombination methods, which is the last step of split-spectrum processing (SSP), and it's really important. The improvement of SSP in the manuscript is filter bank design not signal recombination methods, so I didn't explain too much in Section 2.2. However, the effects of these signal recombination methods are compared in Section 4 and 5, which can be seen from Figure 8 and 14 in the manuscript, and I have chosen Polarity Thresholding (PT) and Polarity Thresholding with Minimization (PTM) as the signal recombination methods in Section 6.

Therefore, Equations (2) - (7) are not particularly important for the further treatment in the manuscript, but it is necessary to describe the signal recombination methods by several equations, and I have tried to describe them as short as possible (Page 4 and 5, annotation C3). So I sincerely hope that Section 2.2 can be kept in the manuscript.

Author Response File: Author Response.docx

Reviewer 2 Report

In the paper, a modification of the split spectrum processing for the enhancement of the damage-reflected ultrasonic guided wave signal detection is realized. The performance of the proposed approach is illustrated over numerical and experimental data, and its high performance is confirmed. In general, the reviewed manuscript is within the scope of the Applied Sciences, and the obtained results might be of potential interest for the scholars from signal processing and non-destructive communities. 

Before being accepted, it might be suggested to the authors to introduce certain improvements to the paper.

The main question is related to the complexity of the provided numerical and experimental examples. As it could be judged from the paper, the main output of the proposed signal processing technique is the possibility to detect clearly the time-of-arrival (ToA) of the reflected signals. However, for all the presented examples, even conventional time-frequency analysis provides rather good resolution, and spots corresponding to L(0,2) reflection are clearly visible (e.g., in Figures 7,12 and 13). The authors should compare the results of ToA detection directly from time-frequency analysis with the results obtained with the proposed technique. Moreover, from the provided examples the particular benefit of such a sophisticated signal processing technique is questionable. Therefore, more complex cases should be considered at least in numerical examples (recent work of S.K. Pedram might be advised as a good example - https://doi.org/10.3390/s20174759).

Some minor comments regarding the text of the manuscript (as they appear throughout it):
1. What does the term "Chript transform" mean? Is this a widespread term?
2. Lines 127-128: What do B_f and B mean here?
3. What are the mechanical properties (i.e., elastic constants and density) of the steel pipe used in simulations and experiments?
4. Line 149: relative references to the paper and/or code repository with "PC_disp" should be provided.
5. Line 180: How these UGW signals are synthesized?
6. Line 271: What does "r" mean here?
7. How the authors could explain that theoretical group-delay curves in lower plots of Figure 12 do not match with spots of time-frequency analysis applied to the simulated signals?
8. How the authors could explain that there is no spot in the time-frequency plot (the case of 3 defects) which corresponds to L(0,2) reflection? Such reflection is clearly visible at ~ 1.65 ms in upper right plot.
9. Some additional proofreading could be necessary. There are certain amount of sentences where the subject is missing (e.g., line 169 and further on).

Author Response

Reviewer #2: In the paper, a modification of the split spectrum processing for the enhancement of the damage-reflected ultrasonic guided wave signal detection is realized. The performance of the proposed approach is illustrated over numerical and experimental data, and its high performance is confirmed. In general, the reviewed manuscript is within the scope of the Applied Sciences, and the obtained results might be of potential interest for the scholars from signal processing and non-destructive communities. 

Before being accepted, it might be suggested to the authors to introduce certain improvements to the paper.

Q1: The main question is related to the complexity of the provided numerical and experimental examples. As it could be judged from the paper, the main output of the proposed signal processing technique is the possibility to detect clearly the time-of-arrival (ToA) of the reflected signals. However, for all the presented examples, even conventional time-frequency analysis provides rather good resolution, and spots corresponding to L(0,2) reflection are clearly visible (e.g., in Figures 7,12 and 13). The authors should compare the results of ToA detection directly from time-frequency analysis with the results obtained with the proposed technique. Moreover, from the provided examples the particular benefit of such a sophisticated signal processing technique is questionable. Therefore, more complex cases should be considered at least in numerical examples (recent work of S.K. Pedram might be advised as a good example - https://doi.org/10.3390/s20174759).

A1: Thank you for your valuable comment, and this is what we would like to do in our future work. I admit that the signals in my manuscript are not very complicated compared with those of Pedram’s work (the signals concluded the reflections of corrosion, weld and crack) in 2020, but it doesn’t mean the signals in my manuscript are meaningless. As I wrote in the manuscript, “the ultrasonic guided wave (UGW) nondestructive testing would suffer from the poor signal interpretation accuracy caused by the coherent noise which is related to the dispersion, multi-mode and mode conversion”. On this premise, we would like to give wrong conclusion when we know nothing about the defect information.

Next, I would like to explain with the experimental UGW signals in the manuscript, and the signals are shown in Figure 5. When the real defect number is 1 (upper left), it is easy to locate the defect if we know the number of defects in advance. But if we know nothing about the defect number, we would like to give wrong conclusion especially when there are multiple defects (lower right). In the lower right plot, if we don’t know the specific defect number in advance, it is hard to determine the real defect number and position due to the influence of coherent noise, so this is the reason why we use SSP which can just keep the defect signals to process UGW signals.

Figure 5. Experimental UGW signals with multi-defect (1≤ defect number ≤3, 60°≤ α ≤300°).

Finally, I would like to thank you for your comment again, your suggestion about the applying SSP to complicated UGW signals is really what we’re going to do next. Now limited by equipment and pipelines, we are not yet able to do relevant experiments, but we have contacted the manufacturer to help us in October. At that time, I will make further improvements to the SSP and use more complex UGW signals.

Q2: What does the term "Chript transform" mean? Is this a widespread term?

A2: I am sorry that I mistakenly wrote Chriplet transform as Chript transform, it is really my fault. And I apologize for not explaining the Chriplet transform in the manuscript, the reason why Chriplet transform was not explicitly described is that I think only the time-frequency analysis by Chriplet transform cannot accurately identify defect signals and it is not the focus of my study. But, according to your suggestion, I realize that it is quite necessary to explain the Chriplet transform.

I have already added the description of Chriplet transform as short as possible in manuscript (Page 6, annotation B8), and the details are as follows:

“The Chriplet transform, a generalization of both the wavelet and the short-time Fourier transform, can extract signal components with specific instantaneous frequency and group delay, and overcome the shortcomings of conventional time–frequency rep-resentation solutions like short-time Fourier transform and Wigner Ville distribution that may not achieve good UGW mode separation results, because multiple modes usually occupy the same frequency band and intersect each other in time. The standard definition of the chirplet transform is given by the inner product of a basis function g(t) and the signal x(t) :

		(4)
		(5)

Where * denotes the complex conjugate, g(t) and G(ω) are basis function and the Fourier transform of basis function. T_t0, F_ω0, S_s, Q_q, and P_p are the time shift, frequency shift, scaling, frequency shear and time shear operators, and the details are shown in Table 2. c(t) is one of chirp signal family.

The basis function can be adjusted by the operators of shifting, scaling, and shearing, leading to the generation of five-dimensional space which combines the projections of short time Fourier transform (time and frequency shift) and a wavelet transform (time shift and scaling).”

 

Table 2. Operations for time–frequency atoms for the chirplet transform.

Operation		Transformation
Time shift Tt0
Frequency shift Fω0
Scaling Ss
Frequency shear Qq
Time shear Pp

Considering the similarity between your comment and reviewer 1's Q2, I would like not to explain the Chriplet transform in detail. If you want know more about the details of Chriplet transform, I would like to suggest that you can move to Page 2 (Reviewer 1, Q2).

Q3: Lines 127-128: What do B_f and B mean here?

A3: I would like to apologize that I didn’t explain B and Bf clearly. B is the total processing bandwidth of signal, and Bf is the filter bandwidth. I have added these in Section 2.1 (Page 3, annotation B2).

Q4: What are the mechanical properties (i.e., elastic constants and density) of the steel pipe used in simulations and experiments?

A4: I am really sorry I didn’t provide the mechanical properties of pipe in manuscript, and I have added these in Section 5 (Page 10, annotation B11). “The pipe material is low carbon steel, and the density, Young's modulus and poisson's ratio are 7.89 g/cm³, 200 GPa and 0.27 respectively.”

Q5: Line 149: relative references to the paper and/or code repository with "PC_disp" should be provided.

A5: According to your comment, I have added the reference about "PC_disp" (Page 5, annotation B5), which is “Seco F., Martín J. M., Jiménez A., Pons J. L., Calderón L., Ceres R. PCDISP: a tool for the simulation of wave propagation in cylindrical waveguides[C]. 9th International Congress on Sound and Vibration. 2002 ”

Q6: Line 180: How these UGW signals are synthesized?

A6: Use Matlab to open “PC_disp” package, there is a program called “pcsignalpropagation”, and this program can be used to generate the synthesized UGW signals (Figure 5). In this program, we can set the number of points of the time window (should be a power of two) Nt, length of the time window for the signal (s) T, position (m) z0, time vector t, central frequency f0, number of cycles ncyc, excitation signal u0, circumferential order n and module m.

Figure 5. Screenshot of “pcsignalpropagation.m” in “PC_disp” package.

Take L(0,2) UGW as an example, let me show how I use this program to generate the synthesized UGW signals.

Firstly, enter the values of the above parameters in the code, and the parameters of L(0,2) UGW can be shown in Table 3.

Secondly, run the program, and the results are shown in Figure 6.

Thirdly, repeat the above steps to get L(0,1), F(n,1), L(0,2) and F(n,3) (1≤n≤4) UGW synthesized signals respectively, and accumulate these signals in the time domain.

In this way, the UGW signals are synthesized.

 

 

 

Table 3. Parameters of L(0,2) UGW.

Parameter	Value	Parameter	Value
number of points of the time window Nt	1024	number of cycles ncyc	5
length of the time window (s) T	2000e-6	excitation signal u0	signalbank('sinemod3',t,[0,ncyc,f0]) (sinemod3 means Hanning window)
position (m) z0	0.8	circumferential order n	0
time vector t	0:T/(Nt-1):T	module m	2
central frequency f0	110e3

Figure 6. Results of “pcsignalpropagation.m” in “PC_disp” package.

Q7: Line 271: What does "r" mean here?

A7: I am sorry that I didn’t explain “r” clearly, r is the radius of the pipe, and I have added this in the manuscript (Page 10, annotation B12).

Q8: How the authors could explain that theoretical group-delay curves in lower plots of Figure 12 do not match with spots of time-frequency analysis applied to the simulated signals?

A8: Thank you for your comment.

Firstly, I would like to emphasize that the theoretical group velocity curves are obtained by dividing the group velocity data of Figure 3 in the manuscript by the propagation distance, so it is obvious that this method is not the best choice to match the theoretical group velocity curves and spots of time-frequency analysis. But, it still can show some useful information and reference value around the central frequency.

Secondly, if we want to get the best match of theoretical group velocity curves and spots of time-frequency analysis, I would like to recommend using 2D-FFT. Before using 2D-FFT, it is necessary to get the data set containing the temporal and spatial (along the axis) samples of UGW signals on the outer surface. The data set is first transformed into the k- f plane using a 2-D FFT, then the result is converted from the wavenumber domain (k) to the group velocity domain (Cg) by using the relation Cg=dw/dk and interpolations. For example, Panda et.al. [6] got the temporal and spatial data by B-scan (Figure 7), the 2D-FFT was applied, and the results which were compared with predicted dispersion curves of plate modes obtained using DISPERSE software are shown in Figure 7b. It is clearly that 2D-FFT can match the theoretical group velocity curves and spots of time-frequency analysis.

 

(a) B-scan results                            (b) 2D-FFT results

Figure 7. Results of Panda’s paper.

Finally, although Figure 12 does not achieve a good match, it can still indicate the existence of modes other than L(0,2), and this is the message I want to convey. It is precisely because of the existence of modes other than L(0,2), the defect signal would like to be confused with coherent noise when we don’t know the defect information, and it is necessary to apply SSP to solve this problem. Therefore, I think the mismatch in Figure 12 would not affect the content of manuscript. But, I would like to thank you for your comment again, and I will choose the more accurate method to match the theoretical group velocity curves and spots of time-frequency analysis in my future work.

Q9: How the authors could explain that there is no spot in the time-frequency plot (the case of 3 defects) which corresponds to L(0,2) reflection? Such reflection is clearly visible at ~ 1.65 ms in upper right plot.

A9: In Figure 13, the 3^rd defect reflection is obvious in upper right plot but not obvious in the time-frequency plot, and this would be caused by the following reasons:

1. The 3^rd defect reflection is weaker compared with the 1^st and 2^nd defect reflections, and it is overlapped with the coherent noise;
2. The time-frequency analysis based on Chriplet transform may be not suitable to this condition where the UGW signals contain 3 defects.

Thank you for your comment. In addition, I would like to clarify that the time-frequency analysis in the manuscript is not my research focus, and the intension of time-frequency analysis is that it may be not enough to just apply time-frequency analysis when we want to distinguish the defects signals. Hence, it is necessary to introduce SSP to process the UGW signals, and SSP do show the effects of improving the signal resolution.

Q10: Some additional proofreading could be necessary. There are certain amount of sentences where the subject is missing (e.g., line 169 and further on).

A10: Sorry for this low-level error, and I have proofread the full text based on your comments.

Author Response File: Author Response.docx

Reviewer 3 Report

This paper shows an interesting and effective way to improve the signal detection of guided waves. However, some points are unclear; questions and comments are below:

1)      Section 2.2: Please add more detailed information about the six signal recombination methods. Information about the features and advantages of each method will help in understanding.

2)      Line 149: Is “PC_disp” a name of software for the calculation of theoretical velocities of guided wave? Please add more description or citations.

3)      Line 169: Could you explain more about “normalized displacement”? In addition, Figure 4 is difficult to understand (I think the graphs of L(0,1) and L(0,2) is important but it is difficult to find them).

4)      Figure 6(b): Is this also time–frequency diagram? Where can I find information on frequencies?

5)      Equation (8) and (9): What is the difference between S (amplitude of signal) and D (defect echo), and how did you determine the N (the root mean square of whole the signal for 1 ms?)

6)      Figure 12: Does the upper figure (which vertical axis shows “Angle”) show the amplitude of the signals at each angle point? In addition, it looks like the defect echo is divided into two signals (one is at approximately 1.0 ms and another is at 1.5 ms). Please add more description about the signals.

7)      Figure 16: Does the upper left figure include four results? It is difficult to distinguish between the four graphs. I recommend to reconsider how to display the graphs (how about displaying seven figures side by side?)

Author Response

Reviewer #3: This paper shows an interesting and effective way to improve the signal detection of guided waves. However, some points are unclear; questions and comments are below:

Q1: Section 2.2: Please add more detailed information about the six signal recombination methods. Information about the features and advantages of each method will help in understanding.

A1: According to your comment, I have added the description of six signal recombination methods (Page 4 and 5, annotation C3).

“

(2)

The output signal of NORM-MIN is the normalized minimum value of all sub-band signals y_i(t) (i=1,2,…,N) at time t, which is suitable for the case where the defect signal amplitude is significantly higher than the coherent noise level, otherwise the defect signal would not be completely preserved.

(3)

MEAN retains the average value of all sub-band signals at each time t, it can reduce the coherent noise level by averaging as the noise value varies in different sub-band signal. However, it is similar to NORM-MIN, which would get great result at the condition that he coherent noise level is much lower than defect signal amplitude. So NORM-MIN and MEAN would not suitable to the high dispersive signals.

(4)

Each sub-band signal are multiplied by each other to generate the FM output signal, and this method could enhance the defect with large amplitude and reduce the noise level with low amplitude, which implies FM has good processing effect on low dispersive signals.

(5)

The output signal of PT is the unprocessed signal x(t) at time t if all the sub-band signals are negative or positive, otherwise it is zero. Thus, PT can eliminate the coherent noise which is highly sensitive to the frequency and has different sign in different sub-band signal. Although the requirements of PT are not as strict as those of the above methods, the defect signal amplitude should not lower than noise level.

		(6)
		(7)

Both PTM and SPT are the combination of the minimization and PT, and the difference between they and PT is that the minimization of all sub-band signals at time t is retained when all the sub-band signals are negative or positive. N+ and N- are the total positive and negative numbers of sub-band signal group. However, PTM and SPT may reduce the defect signal amplitude while reducing the noise level, which would influence the further quantitative analysis of defects.”

Q2: Line 149: Is “PC_disp” a name of software for the calculation of theoretical velocities of guided wave? Please add more description or citations.

A2: According to your comment, I have added the reference about "PC_disp" (Page 5, annotation C4), which is “Seco F., Martín J. M., Jiménez A., Pons J. L., Calderón L., Ceres R. PCDISP: a tool for the simulation of wave propagation in cylindrical waveguides[C]. 9th International Congress on Sound and Vibration. 2002 ”

Q3: Line 169: Could you explain more about “normalized displacement”? In addition, Figure 4 is difficult to understand (I think the graphs of L(0,1) and L(0,2) is important but it is difficult to find them).

A3: The normalized displacement distributions of Figure 4 in the manuscript indicate the wave structure of UGWs with different mode, and they can be obtained by “pcwaveform.m” of the PC_disp package (Figure 8). Through this program, the displacement and stress components of UGWs with different mode in the axial, circumferential and radial directions can be obtained. Take L(0,2) UGW at 110 kHz as an example, the output results are shown in Figure 9.

Figure 8. Screenshot of pcwaveform.m.

Figure 9. Results of pcwaveform.m.

Both the normalized displacement and stress components can show the difference among UGWs with different mode, and I choose to use normalized displacement for comparison in my manuscript. It can be found from Figure 9 that axial displacement is dominant in L(0,2) UGW, so I choose axial displacement curves as the simulated UGW signal in Section 5, which is also the reason why I choose the normalized displacement. Besides, the normalized displacements of Figure 4 in the manuscript can illustrate the difference among UGWs with different mode, which is a common analysis in the UGW testing.

In addition, I am really sorry that I didn’t make Figure 4 clearly, I have modified it (Page 6, annotation C7), which is shown as follows:

Figure 10. Modified version of Figure 4 in the manuscript.

Q4: Figure 6(b): Is this also time–frequency diagram? Where can I find information on frequencies?

A4: I apologize for writing the title of Figure 6b as the time-frequency diagram, and Figure 6b shows time domain UGW synthesized signal at different propagation distance. I have changed the title of Figure 6b to “time domain diagram of UGW synthesized signal at different propagation distance” (Page 7, annotation C10).

Q5: Equation (8) and (9): What is the difference between S (amplitude of signal) and D (defect echo), and how did you determine the N (the root mean square of whole the signal for 1 ms?)

A5: In Equation 8 and 9, S means the maximum amplitude of the whole signal, which is also the end-reflected echo amplitude of the signals in the selected period, D is the amplitude of the defect echo. N is the coherent noise level calculated by the root mean square of the signal, and it is exactly the root mean square of the whole signal not the signal during 1 ms. In addition, the calculation of D is a reference to the Pedram’s paper [7], and the description is as follows:

Figure 11. Screenshot of Pedram’s paper.

Q6: Figure 12: Does the upper figure (which vertical axis shows “Angle”) show the amplitude of the signals at each angle point? In addition, it looks like the defect echo is divided into two signals (one is at approximately 1.0 ms and another is at 1.5 ms). Please add more description about the signals.

A6: Figure 12 shows the amplitude of the signals at each angle point. I am sorry that I didn’t label the coherent noise in Figure 12, and this would be the reason that makes you think there are two defect signals, and the modified Figure (Page 10, annotation C13) are shown as follows:

Figure 11. Modified version of Figure 12 in the manuscript.

Q7: Figure 16: Does the upper left figure include four results? It is difficult to distinguish between the four graphs. I recommend to reconsider how to display the graphs (how about displaying seven figures side by side?)

A7: In order to make the upper left figure of Figure 16 more obvious, I have amplified the defect signals (Page 13, annotation C14), and it is shown as follow:

Figure 12. Modified version of Figure 16 in the manuscript.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The authors have corrected the manuscript enough, so I recommend it for publication after careful editing of English language.

Besides, I strongly intend to use term Chirplet transform (not "Chriplet transform").

Author Response

Dear Editors and Reviewers:

Thank you for your letter and the comments concerning our manuscript entitled “Split-spectrum processing with raised cosine filters of constant frequency-to-bandwidth ratio for L(0,2) ultrasonic guided wave testing in pipeline” (applsci-1824674). Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. Revised portion are marked in yellow in the paper. The corrections in the paper and the responds to the comments are as follows:

Reviewer #1: The authors have corrected the manuscript enough, so I recommend it for publication after careful editing of English language. Besides, I strongly intend to use term Chirplet transform (not "Chriplet transform").

Response to Reviewer #1: Thank you for your recognition of my last response and revised manuscript, which means a lot to me. And according to your comment, I have proofread the manuscript carefully, and replaced “Chriplet transform” with “Chirplet transform”. I hope the above changes are satisfactory to you.

The corrections in the paper and the responds to the comments are described above, we have tried our best to improve the manuscript and hope the correction will meet with approval.

Thank you for reviewing our manuscript, may happiness and health with you always.

 

Author Response File: Author Response.docx

Reviewer 2 Report

The authors have carefully responded to my comments. The paper could be accepted.

Author Response

Dear Editors and Reviewers:

Thank you for your letter and the comments concerning our manuscript entitled “Split-spectrum processing with raised cosine filters of constant frequency-to-bandwidth ratio for L(0,2) ultrasonic guided wave testing in pipeline” (applsci-1824674). Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. Revised portion are marked in yellow in the paper. The corrections in the paper and the responds to the comments are as follows:

Reviewer #2: The authors have carefully responded to my comments. The paper could be accepted.

Response to Reviewer #2: Thank you for your recognition of my last response and revised manuscript, which means a lot to me. In addition, I would like to thank you for your previous suggestions and opinions, which are of great help to improve the quality of articles and help me write.

The corrections in the paper and the responds to the comments are described above, we have tried our best to improve the manuscript and hope the correction will meet with approval.

Thank you for reviewing our manuscript, may happiness and health with you always.

 

 

Author Response File: Author Response.docx