Reverse Curve Fitting Approach for Quantitative Deconvolution of Closely Overlapping Triplets in Fourier Transform Nuclear Magnetic Resonance Spectroscopy Using Odd-Order Derivatives

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper magnetochemistry-3645653 Reverse curve fitting approach for quantitative deconvolution of closely overlapping triplets in Fourier transform nuclear magnetic resonance spectroscopy using odd-order derivatives, it was an interesting paper to be read, with promising results to solve some (really) closely overlapped triplets. In spectroscopy this is an important problem and over the time gain many solutions. I was really excited to learn about your approach, but at the end I have to admit that there are many unclear points, unclear terms and unclear procedures. Also, I have to confess that I was expected to a more automatically procedure, but form I could understand (not everything) your interventions (not mentioned explicitly) was assumed. Here I have to ask you (additional to my many comments), how much time had required the entire procedure? From a practical point of view, I have to ask myself if it will be to acquire 10 NMR spectra per day, can I apply this procedure for all of them? Additionally, it will be an important discussion related to gain and loss when compared your approach to a classical one (in terms of time and precision). Finally, I have to congratulate you for your procedure, and I hope that you will dedicate some time to improve the clarity of your explanations. I consider that it is important, and can gain in application if it could be made more automatically and it can be performed into a reasonable time. The results are believable, and therefore my recommendation to the editor was that the manuscript could be published, once will gain in clarity. Below you have some of my observations.
1.    Abstract, “The deconvolution can be refined progressively by tuning the peak positions and peak widths.” – Please be more specific. Manual or automatic? For the readers this information is important. 
2.    Abstract (line 19): “results” is unclear! It is a result (better spectrum or equivalent) from another measurement with increased resolution?
3.    Abstract “This new approach is also applicable to other FT spectroscopies for the deconvolutions.” – Please rephrase. 
4.    Introduction, line 26: “Its line shape is a convolution result of a sampled signal with a window function.”- I would say that it is more than that: complex interactions between nuclear spins, and relaxation times (which are not mentioned at all in your paper).
5.    Line 32 “High performance NMR spectrometers may not resolve well the overlapping peaks in biomolecular researches”. Some readers may ask: how about low performance NMR spectrometers? I think that the sense is “Nor even high-performance NMR spectrometers may not resolve well…” Please clarify!
6.    Line 37-38 “overlapping band of a Gaussian profile with a Lorentzian profile [13], the overlapping degree γ between two adjacent FT peaks” the use of “band”, “profile” and “peaks” may create confusion. Please clarify the sentence!
7.    Line 50 “algorithm.10” 10 at superscript! Should be a reference?
8.    The discussion started in line 36 up to line 54, needs a deeper discussion. Also, an introductory sentence in the problem will be welcomed. 
9.    Line 56. “the same decay process” – for an inexperienced reader this may create confusion. It will be a god chance to introduce (briefly) the transverse (or spin-spin) relaxation time (and/or relaxation process) T2 and the relationship with the full width at half height (FWHM).
10.    Line 57: “Because of these features, we initiated a reverse curve fitting procedure…” It is not cause and effect! Because => “Grace” or “thanks to” etc. You “proposed” a (new) procedure and you demonstrated! Please change the words “because” and “initiated” to reflect better the novelty!
11.    Line 60 “inverse FT (deconvolution)”. Not all the time the inverse Fourier Transform is identical with deconvolution. Please discuss! And please specify: the inverse FT of what?
12.    Line 69 in which sense: the Proton NMR 300 MHz signals of Ethylbenzene “were adopted”? Moreover, for clarity please use either the “time domain signals” or “frequency domain spectrum”!
13.    Equation 2! Please be carefully with the subscript indices. My guess is that here it is a problem and therefore the equation may look incorrect. Also please define A. Probably the amplitude of a signal!
14.    Equation (4) and explanation. “the slope” (of a curve/function) is the first derivative. Discussing about “nth-order derivative” expressed as a linear function in terms of K and B most probably K->K(n) and B->B(n). Or with other words K an B should depend on the “nth-order of the derivative”. 
15.    In section 2.2 a graphical representation of a single peak and this successive (odd-number) derivate will be welcomed. 
16.    Up to section 3. Results. The noise it is not mentioned. Will have a great influence on the numerical derivative of a spectrum. Please discuss about noise handling in Abstract, Introduction and Methods section!
17.    The term “signal decay coefficient” is unclear. Moreover, I don’t see where this information (devised to be 628.319 (2π*100 Hz)) will have an impact!
18.    Lines 103-104. “sampling rate 8 times the Nyquist frequency”. It is less informative. I see in Figure 1 a range of 90 to 120 ppm. Much better is to provide the number of points in this range then one can estimate the spectral resolution (in ppm).
19.    Figure 1. Vertical axis. For many readers where 0.9 is it is not important. But may be important to see where 0 or 0.5 or 1.0 or 1.5 is! Please change the label accordingly. The human brain is more comfortable if one could identify 0 and 1.
20.    Line 111. “There are several advantages in having a reference peak in the deconvolution”. Not an observation. Just that it is important to be mentioned. Thank you!
21.    Line 116 “After being apodized by a 3-term Blackman-Harris window, FWHM”. This is the first time that you mention this procedure!!! It should be mentioned also before. How are you doing? Usually, in NMR spectroscopy the apodization procedure is applied on time domain! Please provide more details!
22.    Lines 119-120 “The most common noises appeared in FT spectroscopy are white noises. It is necessary to denoise well in advance of the spectral deconvolution.” – Indeed! As I mentioned before, you should discuss this problem before. Here please indicate an efficient “denoising procedure” (Kauppinen’s band-pass filtering technology?) and describe shortly the principle and availability (Python?).
23.    Figure 3 and text. Fitting equation y = - 627.265x+0.8093. Since y is the logarithm of magnitude it is adimensional. Then, the product 627.265x will have also no dimension, and since x represents the time (s) then 627.265 has a measurement unit of 1/second = s^-1. Please use in Figure legend and in text the measurement unit when you refer to “decay coefficient = 627.265” (e.g. in line 131).
24.    Section “3.2. Partial Curve Matching Strategy and Reverse Curve Fitting Procedure”. I read it at least 4 time, and it is steel unclear. You introduced many particular concepts, poorly explained. For example, in Fig. 2b I may understood the left and right primary maximum but in Fig. 4a, 4b and 4c it becomes unclear. There are too many low defined concepts, that I don’t know how to tell you that in this part the confusion is almost total. Please rewrite this section. Please make sure that the reader all the time understood in which space you are working (frequency, time, etc). Please make sure that you clearly explain when it is a transform from which space to which space it is performed. Please use the full name of the concept. For example “matched omega1”. Named as matched peak centered in omega1. Please state clearly the space for matching procedure. And as an advice: It would be much better if you describe the concept on a simulated spectrum without noise. Then the additional oscillation will not interfere with the concept. And then you may discus the influence of the noise. It is unclear that the arrow from Fig. 4 should point to a red dot or to a position on D(3) curve! In particular something is wrong with this sentence “The middle peak ω3 displayed in Figure 4(d) after dismembering the two matched edge 182 peaks, ω2 and ω4.”
25.    Line 230. “We previously manipulated a 300 MHz NMR magnitude-230 mode spectrum of Ethylbenzene [11].”? In which sense, you manipulated? And all the time please specify the nuclear spin (e.g. 1H or 13C), since you are using both!
26.    Line 240. “The acquired raw signal was reprocessed by adding 8 zero-fillings”! Please be more specific. In which sense “8 zero-fillings” 8 points? Or more likely 8 times (2048) up to 16k?
27.    Line 261. “The deconvolution of Figure 8 (after removing ortho- & para-protons)”. The deconvolution is applied on a convoluted curve, e.g. a spectrum not on a figure. In Figure it is presented the result. Please change accordingly!
28.    Figure 9. When indicating a precise value (with a green arrow) please use a vertical line. Use oblique line to point to a curve! Also change the horizontal labeling such that not 7.19 and 7.21 (etc) to be presented but instead, 7.18, 7.20, 7.22, etc.
29.    Many parts from discussion would be more welcomed in Introduction. 
30.    The authors should discus and compare their procedure with a classical one. In terms of resolution, accuracy, automatically/manual procedure, time consuming. 
31.    Finally, you used more that 20 times the term “reverse curve fitting” algorithm, procedure, technique, etc. But in none of this case it was not clear what is the curve that was fitted! And the space (in frequency, time, interferogram?). Please, at least once, preferable in introductory part, be specific with this aspect!

Author Response

Dear Reviewer:

Our manuscript has been revised according to your insightful observations. We made a lot of modifications to elucidate our method more understandable and enhanced the application and discussion.  Appreciate very much for your time and professional advice. Please see our reply letter in the attached file.

Best regards!

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The publication "Reverse curve fitting approach for quantitative deconvolution of closely overlapping triplets in Fourier transform nuclear magnetic resonance spectroscopy using odd-order derivatives" by Shu-Ping Chen et al. describes a challenging and relevant topic in this field, along with a well-described theoretical approach.

Overall, it is a well-written publication that could be accepted for publication in Magnetochemistry. There are only some minor observations. For instance:

1. Please add some real-life demonstrations of how the small molecule analysis approach would work in everyday situations. Using ethylbenzene is too simplistic.
2. What are the limitations regarding the NMR observed nuclei? Is it possible to deconvolute extremely broad signals, such as those from 11B NMR spectroscopy?

Best regards.

Author Response

Dear Reviewer:

Thanks for your professional comments to our manuscript. A new deconvolution example of Tetraphenyl porphyrin was added in "Results" section according to your proposal. Your concern about broad peaks in 11B NMR spectroscopy was discussed in the attached file and you can also refer to deconvolution and spectral background of Tetraphenyl porphyrin in the manuscript.

Best regards!

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Peer Review Report:

Manuscript Title: Reverse curve fitting approach for quantitative deconvolution of closely overlapping triplets in Fourier transform nuclear magnetic resonance spectroscopy using odd-order derivatives

Authors: Chen et al.

Summary

This manuscript presents a new deconvolution strategy referred to as “reverse curve fitting,” designed to resolve closely overlapping spectral peaks, particularly in FT-NMR and FT-IR spectroscopy. The approach relies on using zero-crossing points in odd-order derivatives (mainly the third derivative) to estimate peak positions, followed by a "partial curve matching" (PCM) technique to quantify intensities. The method then iteratively subtracts each matched peak to simplify the spectrum—an approach the authors term “dismembering.”

The authors validate this technique using both simulated data (13C NMR triplet with varying overlap degrees) and real experimental spectra (Ethylbenzene for NMR and 1,1-Dichloroethene for FT-IR). A key claim is that the method can handle peaks with different widths and is particularly effective when peak overlap is significant (γ < 1.0).

Major Comments

1. Novelty and Comparison with Existing Methods

The reverse curve fitting concept is interesting, but it bears similarities to known iterative deconvolution techniques. It would be helpful for the authors to more clearly articulate what sets this approach apart—especially in comparison to standard iterative least-squares fitting methods. What unique advantages does the PCM + dismembering strategy provide? A brief comparative discussion (even if qualitative) would help readers situate this work within the broader deconvolution landscape.

2. Sensitivity and Robustness of the PCM Approach

The PCM technique—matching a small discreet number of points (e.g., 10) around the peak maximum in the third-order derivative—is central to this method. That said, its robustness to noise, especially in low SNR regions or crowded spectra, isn’t fully addressed. How sensitive are the results to the number and position of those points? And how was “10” selected? A short explanation or sensitivity analysis (even qualitative) would be valuable. Additionally, a clearer description of the least-squares matching process used to refine the peaks would help improve understanding of the PCM step.

3. Determining the Number of Peaks

The manuscript describes accurate peak count estimation as “a matter of prime importance,” but the methods section doesn’t clarify how this is achieved. Is the number of peaks assumed to be known ahead of time? Or does the method allow for some estimation or validation of peak count during processing? Since this is often one of the most challenging parts of deconvolution, more detail here would be very helpful.

4. Peak Width Estimation

The Ethylbenzene example nicely illustrates the ability to deal with peaks of varying widths, but some important details are missing. Specifically, how are initial decay coefficients or width parameters estimated in the absence of clean reference peaks? The authors mention "rough" estimation followed by refinement, but in more complex spectra, this step may not be straightforward. A short explanation or guidance on this initial guess would strengthen the methods section.

5. User-Defined Parameters and Automation

While Table 2 gives a clear stepwise overview of the workflow, it’s not clear how much of the process is automated versus requiring manual intervention. Parameters like derivative order, PCM point selection, or iteration stopping criteria appear to be user-defined. Some discussion of how sensitive the method is to these choices—and whether any of them can be standardized or automated—would make the approach more accessible, particularly for users who are not experts in spectral analysis.

Minor Comments

Line 69: The reported Python version “3.14.10” seems incorrect. Python versions generally follow a major.minor.patch format (e.g., 3.10.14). Please double-check.

Equation 2: Formatting needs attention—please correct the use of subscripts and superscripts.

Abbreviations: The abbreviation list is helpful, but make sure all abbreviations (e.g., FT-IR) are either included in the list or defined on first use.

Equation 6: It seems this should read ω₀ = –B/K. Please verify.

Table Numbering: There appear to be two separate tables labeled “Table 3.” Please correct the numbering for clarity and consistency.

Overall Assessment

This work introduces a promising strategy for addressing the long-standing challenge of deconvoluting highly overlapped peaks in FT-NMR and FT-IR data. The combination of derivative-based peak detection and iterative refinement offers a potentially valuable alternative to traditional methods. The real-data examples are particularly convincing.

To enhance the impact of this paper, I recommend the authors clarify how their approach compares to existing methods, provide more detail on the PCM procedure and its parameter sensitivity, explain the handling of unknown peak counts, and address the minor formatting and numbering issues.

With these revisions, this manuscript would make a strong contribution to the field of spectral analysis and quantitative NMR/IR deconvolution.

Author Response

Dear Reviewer:

Appreciate very much for you insightful comments to our manuscript and many of them are very critical to the deconvolution studies. We have paid attentions in the data points, peak widths and peak numbers in the revised manuscript. It is impossible to know in advance how many peaks in a closely or tightly overlapping band. Our approach provided a useful solution to test "purity" of a deconvoluted peak and help o find true peak number in the overlapping band. Please see our reply details in attached file. Thanks for catching several important typos.

Best regards!

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you for considering my observations.

Author Response

Dear reviewer:

Appreciate your positive feedbacks! Thank you for your so many valuable suggestions to strengthen our manuscript including the scientific writings.

Best regards!

Sincerely

Shu-Ping Chen

Reviewer 3 Report

Comments and Suggestions for Authors

1. Overall Goodness-of-Fit Metric: While the "Relative Residue %" (Equation 7, Page 8) is used for assessing individual peak deviations after dismembering cycles, the manuscript would still benefit from an overall goodness-of-fit metric for the entire reconstructed band (i.e., the sum of all deconvoluted peaks) compared to the original experimental spectrum (or the denoised spectrum). Metrics like the sum of squared residuals, R-squared, or a chi-squared value for the complete fit would provide a global measure of how well the deconvolution model represents the experimental data, complementing the individual peak assessments. 

2. Guidance on Setting the "Analytical Threshold":The concept of an "analytical threshold" is mentioned in the general protocol (Table 1, Step 2, Page 3) and in the discussion of the Ethylbenzene spectrum (Page 10, regarding the tiny unknown peak: "not worthy to deconvolute further because its filtered intensity was below our analytical threshold (<4% of ω2 intensity in Table 3)"). Could the authors provide more guidance on how this threshold is determined? Is it based on the signal-to-noise ratio (SNR) of the spectrum, a percentage of the major peak intensity, or another objective criterion? Adding a brief discussion on establishing this threshold would enhance the practical applicability of the method.

Comments on the Quality of English Language

Minor corrections; 

1. Page 1, Line 38 (Introduction): "Nor even high-performance NMR spectrometers may not resolve well..." This is a double negative. Consider rephrasing to "Even high-performance NMR spectrometers may not resolve..." or "Not even high-performance NMR spectrometers can resolve...
2. Page 6, Line 208: "The 10 data points should be disposed within a less overlapping portion." "Disposed" could be clearer; perhaps "selected from" or "located within

3. Page 7, Lines 218-220: "The right primary maximum of edge peak ω4 should be matched appropriately (no need to get fitted utterly because it is not a single peak) by manually tuning..." The parenthetical explanation is a bit informal for a scientific paper. It could be integrated more formally, e.g., "...should be matched appropriately by manually tuning... noting that a perfect fit is not expected at this stage as it is not yet an isolated peak."

Author Response

Dear Reviewer:

We appreciate your positive feedbacks and thanks for your kind advices in the scientific writings!

The manuscript has been revised and updated according to your new comments. Please see the revision details in attached file.

Best regards!

Sincerely

Shu-Ping Chen

Author Response File: Author Response.pdf