A Novel Algorithm for the Decomposition of Non-Stationary Multidimensional and Multivariate Signals

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors The paper proposes an algorithm for the decomposition of non-stationary multidimensional and multivariate signals. It is a continuation of previous research. The paper is well written, with the source code publicly available on the author's page and the test signals taken from the internet with clear mention of them.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Signal decomposition is a fundamental tool in many research fields, including signal processing, geophysics, astrophysics, engineering, medicine, etc. Breaking complex signals into simpler oscillatory components can improve data processing by revealing the hidden information contained within them. Traditional methods, such as Fourier analysis and wavelet transforms, which are effective in processing one-dimensional stationary signals, struggle with non-stationary data sets. In this paper, the authors present the multidimensional and multivariate fast iterative filtering (MdMvFIF) technique, an innovative method extending FIF to handle data that varies simultaneously in space and time, such as those acquired using sensor arrays. This new algorithm is capable of extracting internal mode functions from complex signals that vary in both space and time, overcoming the limitations found in previous methods. The potential of the proposed method is demonstrated through applications to artificial and real signals, highlighting its flexibility and efficiency in the decomposition of multidimensional and multivariate non-stationary signals. The MdMvFIF method offers a powerful tool for advanced signal analysis in many scientific and engineering disciplines.

 

The article is well structured and written, but it could be improved in some aspects according to the recommendations below in the review.
Some recommendations for improving the manuscript "A novel algorithm for the decomposition of non-stationary multidimensional and multivariate signals":
Please, check if you have followed all the requirements of the manuscript formatting template, for example: "These authors contributed equally to this work." (should this sentence be here?), as well as the sections: "Author Contributions:", "Funding:", "Institutional Review Board Statement:", "Data Availability Statement:", "Conflicts of Interest:" - usually in other MDPI publications, these paragraphs are placed at the end of the article. Please, make sure that for this Journal these paragraphs are at the beginning! Please, check if citing sources (links) in footnotes is permissible!
For lines 52 to 62: when listing different applications of signal decomposition, it is good to cite specific sources for each of the applications (data analysis, image processing, machine learning, geophysics, communications, …, enables efficient data representation, noise reduction, and feature extraction, ….). Currently, this paragraph lacks any citation. Lines 60-62 also need specific citation of the source.
In my opinion, it is good to make a comparative analysis of the methods (or at least to present them in the material with at least 1 sentence for the reader), the methods listed in lines 70-75.
Please, if a given figure/formula/table is not the author's, but is borrowed from a literature source, cite the corresponding source! For example, formula (1) (not only) is well known – please, cite the corresponding source. All the notations in formulas must be explained, for example || ||, etc.
Line 105: "and not as a dimension per se." – what is se?
Lines 138-139: "These filters are …, compactly supported, and widely used in applications" - please, clarify the notation you used, as well as in what applications these filters are widely used?
In my opinion, it is good to edit the figures - they contain a lot of unnecessary "white space", which does not carry information, but increases the volume of the article. For example, for Figure 2 and Figure 10 it is very noticeable (and not only for them).
"The data are shown in Figures 1 and 2." - this text should be moved before placing the relevant figures in the text.
"The data are shown in Figures 6 and 7." - this text should be moved before placing the relevant figures in the text.
According to the authors (lines 232-233): "In this first example, we have a signal containing two non-stationary frequencies active in space and two stationary frequencies active in time." – please, explain the meaning of the variables x,y, which are used in Figure 1 (and so on) and the relationship with the frequencies active in space and frequencies active in time, which are discussed in the above text.
In my opinion, at the moment this section (section 3) is rather informative in nature, there is no analysis of the presented figures – it would be good if these figures were explained in detail for the reader. For example, the coordinates used, the meaning of the different colors (it is true that there is a scale for colors and their numerical correspondence - where can the components mentioned in the main text (such as color, for example) for the specific figure be seen?
Please, specify with what software these figures were obtained - is it MATLAB or other software (the mention of MATLAB in the article is a one-time occurrence, when offering the pseudocode of the algorithms)?!
In Figure 3 and Figure 8, according to the legend, there should be another signal depicted in addition to the red signal - but this signal is not visible in any of the three subfigures! There is something dark in the first subfigure in Figure 8, but whether it is this signal - perhaps it would be good to specify with some sentence (where it is observed).
It would be good for the authors to give an example of what artificial signals could represent in practice!?
How are the points (x, y) = (150, 150), (x, y) = (100, 10), (x, y) = (100, 10), (x, y) = (20, 50), (x, y) = (230, 50) chosen? In any case, the graphs will look different for different points – then how is this different choice of all these points justified?
Why is the presence of two curves not reflected for Figure 13 and Figure 19, which resemble Figure 3 and Figure 8, and in fact the red signals are missing? How is this absence explained?

Percent match: 24% - the percentage of correspondence is not small – the question is up to what percentage is allowed for the specific Journal or Organization?!

By the way, 4 of 22 references belong to the authors - it will be better to increase the number of "foreign" references.

Overall, the article is well-written and structured, but I think that addressing the remarks in the reviews would improve the quality of the publication. In my opinion, it can be accepted and published in the Journal after reflecting on the remarks in the reviews.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper introduces an approach for decomposing spatiotemporal signals into intrinsic mode functions without relying on a priori basis functions. It extends iterative filtering to handle multichannel data varying in both space and time.

I have the following comments:

 

 

1. It is highly recommended that authors add more signal processing examples in the introduction for multidimensional analysis, such as tensors involving joint multidimensional parameter estimation such as angles of arrival and times of arrival of multipath, e.g. [REF01]. Kindly include.
2. Equation (2) needs to be checked again in order to ensure it properly reflects the angular measure between consecutive vectors in $\mathbb{R}^n$. One might need to confirm that no degenerate case arises during computation of arccos.
3. It can be good to explain how the algorithm balances the simultaneous extraction of space-dependent and time-dependent modes. The MdMvFIF method iterates first in space then in time, but it would be informative for this reviewer to show how these consecutive procedures avoid missing mixed-modes that could be neither purely spatial nor purely temporal.
4. Why do the authors choose Fokker–Planck filters specifically, instead of other compactly supported filters, for the multidimensional parts of the decomposition? Please make it clear for readers.
5. Please clarify how the algorithm considers boundary artifacts in higher-dimensional cubes, where simple mirror may not suffice to eliminate discontinuities at edges.
6. To support the computational complexity analysis of m log m , please provide actual CPU time on sample large grids which can be be beneficial to check.
7. Provide an expression for the final stopping criterion in the inner loop for both space and time. For instance $\left\|f_{k+1}-f_k\right\| /\left\|f_k\right\|$  should be taken into consideration to avoid misunderstandings about convergence criteria.
8. In simulations in section 3.1 for artificial examples, the synthetic signals combine nnonstationary frequencies in space with stationary or nonstationary frequencies in time. Can adding amplitude modulation complicate the decomposition further?
9. There are some typos like “turbolence” which should be “turbulence”. Please revise the paper.
10. Please show how the algorithm detects the final residual from the last IMF. In multidimensional data, the number and nature of local extrema can be ambiguous unless a specific definition is given.

 

 

References

[REF01] “Efficient Maximum Likelihood Joint Estimation of Angles and Times of Arrival of Multiple Paths,” 2015 IEEE Globecom Workshops (GC Wkshps), San Diego, CA, USA, 2015, pp. 1-7, doi: 10.1109/GLOCOMW.2015.7414203

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

1. The authors have addressed most of mu comments but are still missing many state of the art multidimensional analysis , such as tensors involving joint multidimensional parameter estimation such as angles of arrival and times of arrival of multipath. Kindly include [REF01].

 

References

[REF01] “Efficient Maximum Likelihood Joint Estimation of Angles and Times of Arrival of Multiple Paths,” 2015 IEEE Globecom Workshops (GC Wkshps), San Diego, CA, USA, 2015, pp. 1-7, doi: 10.1109/GLOCOMW.2015.7414203

Author Response

Following the Reviewer's comment, we added the suggested reference.