The Obtainable Uncertainty for the Frequency Evaluation of Tones with Different Spectral Analysis Techniques
Round 1
Reviewer 1 Report
This paper compares uncertainty of estimating the frequencies (and amplitudes and phases in some cases) of the sampled signals composed of a linear combination of pure sinusoids (plus noise in some cases).
I have several concerns with this paper. First of all, the notation in equations and inline math are many times incorrect: sometimes variable names are italicised and sometimes not. They are typeset using different font types and sizes, which is awkward and makes the reading and comprehension difficult. Subscripts have different positions. At first I thought these were just minor problems with the pdf and I kept up reading the paper, but arriving at line 166 and starting to see missing math in text and in figure captions, I stopped reading. This looks like a wrong version of the paper or the paper has been written in a hurry. I have also noticed non-consistent formatting of section headers.
Secondly, the paper compares DFT-based methods for deterministic signals with techniques that require, for example, covariance matrices. Of course a deterministic variable can be seen as a specific case of random variable, but still how those are estimated is not explained. I assume they are not computed analytically, but from the samples signal to be fair with the DFT-based methods. All the more, it is necessary to explain how the estimation impacts the considered uncertainty.
By limiting the n to 1...N, Equation (2) directly implies a rectangular window being applied to the signal but this fact is not further considered. Then it is stated that the harmonic interference is negligible. There is no previous mention that harmonic signals were at all assumed. From the further context it seems the authors meant spectral leakage, but I am not sure. In that case this would mean assuming a very large value of N. In this context talking about real-time processing is unrealistic. At the same time, in line 84 the spectrum W(k) is mentioned, which is not defined.
It is surprising that only secondary references for the MUSIC algorithms are cited and not the original paper: R. Schmidt, "Multiple emitter location and signal parameter estimation," in IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276-280, March 1986.
The abstract and the introduction confusingly use the word "application" sometimes as the field of use and sometimes as the software program. The phrase "number of spectral components to be processed" is unclear. I understand that a signal is being processed and the number of parameters of the underlying sinusoids are to be measured / estimated and the number of them is in concern.
In line 44 you mention "parameters of the signal". I understand that first you need to assume some kind of a model for that signal.
What hypotheses do you mean in "unsatisfied hypotheses" in line 157?
What estimators do you mean in "The estimators may be distorted" in line 158?
Sometimes the samples of the processed signals are called points. Sometimes the samples of the DFT are called bins.
The English of the paper requires a thorough revision. I spotted phrases like "to compute this algorithm", "errors on algorithms" or "with an order of magnitude the same for both the tones for a given algorithm".
All in all, he comparison presented in this paper would be very much interesting if the paper were carefully written. In the present form it requires a thorough revision if not rewriting.
Author Response
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Reviewer 2 Report
Please find attached the review.
Comments for author File: Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
This paper introduces various spectral analysis techniques for signals. This topic is important for signal analysis. However, after carefully reviewing this manuscript, the reviewer has raised the following comments:
- The spectral leakage problems and windowing effects are not discussed in this paper.
- A famous multitaper spectral analysis is missed in this paper. It exhibits superior performance, especially for bio-signals. Please refer to the following recently published work:
- Chien, Y.-R.; Wu, C.-H.; Tsao, H.-W. Automatic Sleep-Arousal Detection with Single-Lead EEG Using Stacking Ensemble Learning. Sensors 2021, 21, 6049. https://doi.org/10.3390/s21186049
- In this work, only sinusoidal signals are considered. Do you consider the spectral analysis for non-stationary signals?
- In Eq. (4), the definition of K is missing.
- In Table II, the ranges of SNR considered are from 10 to 80 dB. The reviewer thinks it is meaningless. The spectral analysis is more difficult when the SNR is low (should consider the negative case, eg: SNR=-20 dB). Also, the caption of a table should place on the top of the table.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
I still think the rigorousness of the manuscript is not sufficient.
In my opinion “Frequency evaluation” in the title is not specific and misleading. The paper does not evaluate frequency, rather is estimates it for the different components of the multi-tone signals. Additionally only that one kind of signals is considered.
The paper should specify from the beginning that only the sampled signals composed of a linear combination of pure sinusoids (plus noise in some cases) will be considered. This is not specified until section 2.
It is still not explained (already indicated in the first round review) how covariance matrices have been estimated in case of MUSIC and ESPRIT. Line 132 mentions R_x is computed using the expectation. Does it mean they were computed analytically? This would make the comparison with DFT-based method completely unfair. So this is an important point. By the way, the dot between x and x^H should be removed as it represents the scalar product.
The paper is very informal in many places, especially in the results and in the conclusions.
Please address each of the comments below and not in groups as you did in the first round.
l. 14 You introduced “embedded software”. What do you mean by that? Why embedded is important in this context?
l. 43 “Components” not defined. Is it the same as “tones”?
l. 52 Phase and amplitude are not parameters of the signal but of the individual tones, and the signal is a linear combination of those tones.
l. 60 “more accurate details” does it mean that the details in the paper are less accurate?
l. 81 “more near tones”, near in what sense?
l. 119 “image frequency” not defined
l. 121 “accuracy” not defined
l. 125, l.282 and other, “performance” not defined. By the way, performance should be singular.
l. 222 and nn. In the case of MUSIC N_s can be estimated from the signal by finding the amount of eigenvalues that are zero (no noise) or cluster together near zero (with noise, depending of the amount of noise)
Figure 2. I was surprised to see that N_s_0 affects the DFT-based methods. Do you use it in any step in the DFT-based methods?
l. 259 What does it mean that algorithm “behaves with respect to a signal”
l.276 – 279, what does it mean that algorithm “gets worse” and “lightly”?
l. 282 How do you define “good performance”?
l. 346-7 What do you mean by “very close tones”? Close in frequency? What do you mean by “one of these significantly lower”? In frequency, in amplitude?
l. 383 “Systematic effects” not defined
l. 388 How much is “decent”?
l. 392 “Elaboration time” is processing time?
l. 393-396 Hybrid solutions are not explained. What do you mean? An example of such approach would help to understand your intention.
The English of the paper requires a thorough revision. I have found this phrasing:
Whom is developing
The English of the paper requires a thorough revision.
“two time”, ith, i^th – should be $i$-th in LaTeX notation
harmonic interference on
on turn
consists on
and if we assume in the shape of white noise
to compute this algorithm (not corrected from the previous revision)
a observation
window greater than two periods. in the context of time it should be longer an not greater.
between the considered methods are shown for the first tone.
vice versa occurs for
Ns0 (no subscript)
10-7 (no superscript)
being fixed the sampling frequency
are run on the samples (very colloquial)
gets worse (very colloquial)
Interpolation (why capitalized?)
l. 303 N is not italicized
and few noise
when the contribute due to
are worse than ESPRIT’s ones -> are worse than those of ESPRIT
Author Response
TE
Author Response File: Author Response.pdf
Reviewer 2 Report
Dear Authors,
The paper has been significantly improved and deserves publication.
A more detailed explanation of the selected methods could be presented. In the case of the interpolated FFT methods the frequency deviation should be expressed explicitly.
Some typos must be corrected as they can confuse the reader:
- In Eq. (12b) please write beta_i instead of delta_i
- Both the second and the third part of Table 1 is for beta_12 = 0.1. I suppose in the third part beta_12 is greater, e.g., beta_12 = 1.
Author Response
The response in the file
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Reviewer 3 Report
In this revision, all concerns have been well addressed or responded. Some comments about the equations have been raised below:
- Eq. (8) is incorrect. The dim. of the eigenvector does not match each other.
- Eq. (10) need to modify. The definition of Φ is missing.
Please check all equations are correctly represented.
Author Response
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