Comparing Fourier Transform and Direct Wave Fitting in NMR FID Data Analysis

Chen, Jixin

doi:10.5194/mr-2026-8

Preprints

https://doi.org/10.5194/mr-2026-8

Preprints

01 Jun 2026

| 01 Jun 2026

Status: this preprint is currently under review for the journal MR.

Comparing Fourier Transform and Direct Wave Fitting in NMR FID Data Analysis

Jixin Chen

Abstract. This report compares various simulation and data analysis methods for free induction decay (FID) signals in Nuclear Magnetic Resonance (NMR) Spectroscopy. The methods discussed include discrete fast Fourier transformation (FFT), least squares fitting (LSF), short-time Fourier transformation (STFT), and wavelet transformation. NMR is a widely used technique for elucidating the chemical composition and structure of molecules. It measures the nuclear magnetic spin frequencies of different atoms in a sample, producing signals that are waves over the time domain. This report employs Gaussian wave frequency simulations to model the interference and dephasing among adjacent frequencies of each Gaussian chemical shift. Typically, FIDs are analysed using the FFT algorithm, although least squares fitting and machine learning algorithms are also employed. The simulations indicate that STFT can generate spectrograms that are useful for further neural network training, while LSF is effective in processing signals around or even below one wave cycle.

Received: 14 May 2026 – Discussion started: 01 Jun 2026

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RC1:
'Comment on mr-2026-8', Anonymous Referee #1, 17 Jun 2026 reply
The author presents results of applying Fourier and other reconstruction and fitting methods suggested as suitable for time domain data with non-stationary signals. The author emphasizes use of their previously-published “JCFit” approach to fit time domain data. Only a limited amount of simulated data is evaluated, using modest numbers of points (~2028) and a small number of signals.
There are useful ideas in this work, and also enthusiastic scholarship, and the manuscript deserves refinement. That said, in its current form, it seemed hard to find a coherent story or an actionable conclusion. Here is some discussion that could lead to a stronger manuscript:
What is the goal? Is it extraction of model parameters, or is it generation of spectrograms? Is it to be able to decide when reconstruction methods cannot yield a useful spectrogram, so that model fitting must be used?

There are decades of work on spectral fitting in the time domain and frequency domain, but most all of it is directed at the predominant form of NMR experiments, which generate stationary signals (that is, signals whose frequencies and decay envelopes do not change during measurement). Application to non-stationary signals, and approaches for analyzing this kind of data are less common, and more in need of analysis methods.

The title “Comparing Fourier Transform and Direct Wave Fitting” could be improved upon, because these two methods do not do the same thing. The Fourier transform deterministically generates an estimated spectrum from time domain data without other input, whereas time domain fitting produces estimated signal parameters, it requires decision-making about the functional form, and in practice, model fitting of sinusoids often requires non-deterministic methods to avoid local minima (the author’s JCFit method is an example of this).

While there is much discussion of spectral fitting and model functional forms, there seems to be no discussion whatsoever about determining the number of signals to model, and this is often the most difficult aspect of an NMR fitting task. Surprisingly, this discussion also seems to be missing from the original JCFit publication. So for example, in Section 3.2 JCFit can “fit FID using the initial guess obtained from the FFT results”, does this mean “we did peak detection on a spectrum, decided what features were valid peaks, and used this information to fit a time domain model”?

Considering the points above, perhaps a more suitable title and focus would be something like “Extraction of Signal Parameters and Spectrograms for Non-Stationary NMR Signals”.

Speaking specifically about the Fourier transform, the resulting spectrum has exactly the same information content as the original time domain data, and if the number of signals is known, along with their functional forms, the exact same information can be extracted from the two cases, regardless of the number of points. To be sure though, some tasks are easier to apply in one domain than in the other. In the time domain, all signals can potentially extend throughout the entire measured data, whereas in the frequency domain, signals will tend to be localized. This means that signal analysis in the frequency domain has the advantage that it can typically treat small spectral regions independently of each other, whereas in time domain methods, even if frequency band selection is employed, the entire length of the time-domain data must be considered.

(Page 2, ~ line 45) “There are significant differences between wave and peak data fitting. The frequency domain peak fitting is to decouple overlapped peaks due to peak broadening, while the time domain FID fitting is to obtain the chemical shift peaks with corrected peak heights and phases.” As discussed above, while applications in one domain might be easier than in another, the information content in the two domains is comparable, and signal parameters in one domain have corresponding parameters in the other.

Section 3.1: this seldom-seen discussion of collection windows is excellent, and has a practical conclusion for generating simulated data. Perhaps it could be mentioned in the abstract.

(Page 11 ~line 230) “When the length of data points is short and even compatible with or shorter than one wave cycle, zero fillings are added to get enough data points for FFT to work”. This is a misunderstanding. Zero filling produces a result where the real and imaginary parts of the measured time domain data are encoded together in the real part of the resulting spectrum, and also redundantly encoded in the imaginary part of the resulting spectrum. This is why it’s appropriate to analyze the real part of a properly-processed spectrum without the need to consider or retain the imaginary part. Beyond the first zero fill, no new information is added, although the result might be easier to analyze visually or with a particular algorithm.

Regarding actionable conclusions: methods such as short-time Fourier transform (STFT) require dividing the time domain data into overlapping subsets, which can’t easily be done for very short time domain data, such as in the indirect dimensions of higher-dimensional data, which might only have ~16 points. Furthermore, as the author notes, the reconstruction methods presented are not ideal for measurements that only capture a small fraction of one cycle, whereas the fitting approach might not be practical for large numbers of signals. Are there guidelines regarding data size and signal count for determining the best way to approach a given case?

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Citation: https://doi.org/10.5194/mr-2026-8-RC1
RC2: 'Comment on mr-2026-8', Anonymous Referee #2, 18 Jun 2026 reply

This reviewer is unable to discern any novel results in this manuscript. Furthermore, the manuscript is highly misleading and fails to grasp the important distinctions among methods. The discussion of "least squares" methods implies they are a separate class from the discrete Fourier transform or the discrete wavelet transform. The discrete Fourier transform is a "least squares" solution to the problem of expanding a time series in a discrete Fourier basis. There happens to be a particularly efficient method for computing the coefficients, and there is a unique solution, because the Fourier basis is orthogonal. The same holds for the discrete wavelet transform -- it is a least-squares solution to an expansion in an orthogonal wavelet basis. Least squares solutions using other models -- for example exponentially decaying sinusoids -- don't have such efficient methods for computing the coefficients, nor do they typically have a unique solution without some additional regularization criterion (as exponentially decaying sinusoids form an "over complete" basis). It's noteworthy that the orthogonality of wavelet bases precludes perfectly symmetric wavelets (although nearly-symmetric symlets have been developed) which makes them less than ideal for fitting symmetric NMR lineshapes. All of these observations are reflected in more than five decades of results on signal processing of time series.

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Citation: https://doi.org/10.5194/mr-2026-8-RC2

Jixin Chen

Model code and software

jcNMR Jixin Chen https://github.com/nkchenjx/jcNMR

Jixin Chen

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Short summary

This study compares different ways to simulate NMR raw data and analyze such data. The results show that direct signal fitting can recover useful information from very short or weak signals where traditional methods struggle, helping improve future artificial intelligence tools for chemical analysis.


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