Instantaneous Frequency Estimation of FM Signals under Gaussian and Symmetric α-Stable Noise: Deep Learning versus Time–Frequency Analysis

Round 1

Reviewer 1 Report (Previous Reviewer 3)

Current abstract has too many flaws.

1) The term ANN is not clear. Typically, ANN is used as a terminology that subsumes CNN. Please change ANN to a more precise, commonly used term which is what the authors used in this paper. Maybe multilayer perceptron (MLP)? Related to this, I'm not sure what the authors want to indicate with ANN -- is it few layer? or the architecture that composed of fully connected layer unlike CNN? this should be clarified and authors need to find a proper term.

2) In Abstract, line 20~23 is unclear in this sense. Moreover, ANN and CNN is general deep learning modeling type which is not designed for FE and SE, while current abstract reads as if ANN and CNN are meant to be designed for FE and SE.

3) Also, in line 23, are you sure that CNN has been 'proved' to work better than ANN? Since I haven't seen any precise theoretical proof on this, though there are many attempts on this in various viewpoints such as manifold learning and statistical learning theory, I'm afraid that this term is too strong. Maybe changing to 'empirically proved' would work. 

4) DCNN has not been detailed in the abstract.

5) Abstract typically does not use past tense.

Author Response

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Reviewer 2 Report (New Reviewer)

The manuscript is an overview of the instantaneous frequency estimation methods. The paper is interesting. However, some concerns have to be addressed:

1. Page 2, lines 50-51, please define acronyms RCTSL, A&M, HAQSE

2. The Introduction is full of content but is not shaped like a typical introduction because the literature review is excessive. I suggest dividing the Introduction onto two sections: Introduction and State-of-te-art (or something like this).

3. Please stick to assumed terminology, symbols, and acronyms. Page 4, Line 156 - the authors used "alpha" instead of its symbol. Page 5, line 185, the abbreviation IF is given on the 5th page. It can be done earlier. Page 5, line 194 - "linear FM," ten lines earlier, there is an acronym LFM. Page 5, line 197, the abbreviation FE was defined earlier. Page 5, line 202, RADAR and SONAR are capitalized, previously written in lowercase. Page 6, line 233, LFM multiply defined. From this perspective, the manuscript is chaotic. I suggest the authors carefully review the text. There are many such problems.

4. The description needs to be more consistent, e.g. references to equation (16) at the beginning of the article (Page 5, line 208). This makes some sections difficult to read.

5. Equation (1) - It would be good to explain why the initial phase was omitted. 

6. I suggest putting Sections 4, 5, and 6 into a single one with subsections. Is this description needed in this article? Noise definitions are widely described in the literature. It would be enough to add references. 

7. Equation (20), please remove the dot after A or change it to a multiplication sign. Also, phi is defined as the instantaneous phase. I suppose it should be a function of time, not a scalar as it is. 

8. In my humble opinion, a very detailed description is unnecessary in section 7.2. Especially the explanation of file storage and folder organization.

9. Page 12, line 391. Please change 1e-4 to a more elegant form. The same at page 13, lines 466 and 467. Please unify the decimal notation.

10. There is a lack of space between "60,90" on page 13, line 469.

11. Please rephrase the 9th point of the Algorithm (3).

12. The paper deeply investigates all considered problems. However, the definition of the STFT in Section 8 is missing. I suggest putting the definition.

13. Equation (23) is not the best IF estimator, especially for multicomponent signals. In general, there are dedicated IF estimators in the time-frequency domain that allows for obtaining a two-dimensional estimate. If the paper is a review of different methods, it should at least mention such techniques (e.g., D.J. Nelson Instantaneous Higher Order Phase Derivatives). The estimator (23) can also be biased for a badly conditioned window. This problem is not investigated in the work.

14. It is a pity that the analysis of real-life signals was not included.

15. Can the proposed algorithm work with complex signals? Please clarify.

16. I wonder whether the research was appropriately designed. If only LFM signals are analyzed, why the authors used the STFT for IF estimation? Pure LFMs are better distributed using the Wigner-Ville distribution. 

17. I do not understand why Appendix (B) is given. This (and a few other sections mentioned previously) is well described in the literature and is excessive here. 

18. What is the purpose of using the Hilbert transform in Algorithm (4)? Is it easier to analyze only positive frequencies? If real signals are analyzed, the negative part of the f axis is excessive. 

19. Last but not least. I suggest changing the paper category to "review" or removing excessive descriptions of obvious methods and phenomena. Please see: https://www.mdpi.com/about/article_types

Author Response

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Round 2

Reviewer 2 Report (New Reviewer)

Thank you for addressing my comments. The paper can be published in the current form.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

In this manuscript, the authors propose CNN-based frequency estimation and slope estimation methods considering different noise conditions. In general, the results of this manuscript are relatively solid. My comments are as follows.

1. It is suggested to clarify the main contributions in the introduction.

2. In Section 2, the problem is formulated in a ambiguous way. It is better to combine the problem with instantaneous frequency and noise condition by using specific models.

3. The detailed descriptions of AWGN, SASN, machining learning, deep learning are not suggested. A better presentation is to jointly describe them, and to point out the abilities of machining learning and deep learning in dealing with the FE task under AWGN and SASN.

4. Figure 1 and Figure 2 are common knowledge in the domain of deep learning.

5. I wonder whether the analyses of results are sufficient.

Author Response

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Reviewer 2 Report

A novel approach to employ artificial neural networks to estimate the instantaneous frequency has been developed. My comments are as follows:

1) Introduction: The introduction needs improvement with detailed discussion of recent methods and limitations of these methods and how this study fills the research gap. Currently, few methods are discussed but limitations of these methods are not discussed.

2) A lot of unnecessary background material regarding machine learning methods and metrics for their evaluation is added. E.g. authors are addressing a regression problem so measures such as precision, recall and not relevant to the given problem. 

3) The description of the proposed method needs improvement for better understanding and reproducibility of results. It is mentioned on line#334 that noisy signals are converted into 2D images, but it is not described how this conversion is performed.

4) Authors need to add time-frequency images of signals considered in the Results section for better readability of the paper.

5) It seems that the method is just applicable to mono-component signals. This constraint needs to be mentioned and it would be good if the authors explain how authors intend to address this limitation in future studies.

6) A number of very effective instantaneous frequency estimation algorithms have been developed specifically for mono-component or LFM signals, e.g. Viterbi algorithm, Fractional Fourier-based approaches. Authors need to make an appropriate comparison with relevant methods. Currently, the performance comparison is being made with a simple peak detection method which is not an accurate method.

Author Response

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Reviewer 3 Report

I like the authors' idea of using DNN for instantaneous frequency estimation. Also, it seems that the authors have nice experimental results. However, current description of the paper totally conceals nice achievement of the authors. I strongly recommend to make a thorough review for all the equations to match with common usages in most scientific papers. Hope some of my comments can be useful for revising the manuscript.

0. Objective function for DL models are not clear. What is the objective function and how the loss is defined? This is the most important thing yet I could not find it. Please write 'mathematical equations' for both objective function and loss function, and describe how DL model is trained.

1. Eq 1) f_i(t) -- what does 'i' stand for? Does it came from 'i'nstantaneous frequency estimation? It would be better to use 'IT' instead of 'i' since 'i' feels like some variable.

2. Eq 2) \theta should depend on t. Please change to \theta(t)

3. Eq 3) \alpha being linear modulation index conflicts with usage of \alpha for the alpha-stable noise. Please change alpha in eq 3.

4. Eq 7) it's very hard to see notations using p(n) for pdf. p(n) is used for pmf.  

5. Eq 8), you cannot simply write 'L = length(x)' -- this is not a code but a paper.

6. above eq 10) there is a typo -- 'AWG' noise.

7. Eq 11), \beta has never been used but used in the following paragraph. Please make sure why we need \beta.

8. Line 258, I think it's \alpha >2 .

9. Line 268, is N_S the random variable? It feels like variance of the noise not random variable. Please change this. Also, u,v \in \mathcal{U} doesn't make any sense. At least you need to change \in with \sim. Also, sometimes, random variables are in capital letter, while sometimes in lower letter. Please make everything to be consistent. You can try using some bold letter for random variable at least.

10. Eq 16), you have one more parentheses.

11. In 20), what does N_SS stands for? 

12. Description from 298~304 makes no sense. It's not related to stochastic gradient descent. Maybe conjugate gradient descent, but this is not stochastic gradient descent. And, I've never seen calling stochatic gradient descent as SCG. People call it SGD.

13. Line 321. DL is a subset of deep machine learning? Makes no sense. At least you need to remove 'deep'. And '(CNN)' should go right after convolutional neural network not after deep learning.

14. Eq 24) doesn't seem like a mathematical equation at all. Also, how come 'Inp' has some covariates? In other words, how 'Inp' is defined? Also, notation of 'Inp' is really not common in scientific papers, also for 'act_fun'. Please follow usual notations with most of the papers as these kinds of things are very very nicely described in tons of papers.

15. Eq 32) this is not entropy but cross-entropy. Using 'Entropy' as function name is not good in this regard. Also, echoing my previous comments, please avoid using this kind of texts as function name. 

Sorry for the massive comments but I think these kinds of things are minimal requirements for the scientific paper. Thank you.

Author Response

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Round 2

Reviewer 1 Report

The authors have made revisions according to the previous comments. I have no comment.

Author Response

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Reviewer 2 Report

The authors have not properly addressed my comments. 

There is still a lot of redundant detail regarding machine learning methods in the paper, and the authors have failed to clarify how they transform a 1 D signal into a 2D signal e.g. what type of TFD is used and what are its parameters.

Author Response

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Reviewer 3 Report

Thanks for addressing my comments.