Author responses in the reply to each reviewer's comments are copied here.

Reviewer 1:

Comments:

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:

Response:

I thank the reviewer for the constructive and thoughtful comments. I agree that the current manuscript needs a clearer focus, a more coherent story, and more actionable conclusions. In the revised manuscript, I will sharpen the central goal, clarify the distinction between transform-based analysis and direct nonlinear time-domain fitting, and revise the title, abstract, discussion, and conclusion accordingly.

1. What is the goal? Is it extraction of model parameters, generation of spectrograms, or deciding when reconstruction methods fail and model fitting is needed?

The main goal is to evaluate simulation-based approaches for analyzing NMR FID signals, especially short or non-stationary signals for which conventional FFT-based spectra may become broad, poorly resolved, or difficult to interpret. The manuscript compares transform-based methods, such as FFT, STFT, and wavelet analysis, with direct nonlinear time-domain fitting of damped sinusoidal models. In the revised manuscript, I will clarify that the central focus is not to replace FFT for routine stationary FID analysis, but to examine when direct fitting or time–frequency spectrograms may provide useful complementary information, especially for short signals, non-stationary signals, or future machine-learning workflows.

1. There are decades of work on spectral fitting, but most work addresses stationary NMR signals. Non-stationary signals are less common and more in need of analysis methods.

I agree with the reviewer. This is an important point, and I will revise the manuscript to emphasize non-stationary or dynamically broadened FID signals more clearly. The revised discussion will better distinguish stationary FID signals, for which conventional Fourier-domain methods are highly effective, from non-stationary signals, where frequency, linewidth, or decay behavior may evolve during acquisition and where STFT, wavelet analysis, or direct time-domain fitting may become useful.

1. The title “Comparing Fourier Transform and Direct Wave Fitting” could be improved because these methods do not do the same thing.

I agree. Fourier transformation and direct time-domain fitting serve different purposes. FFT deterministically transforms sampled time-domain data into a frequency-domain representation, while nonlinear time-domain fitting estimates model parameters and requires choices about the functional form, number of components, initial guesses, and fitting strategy. In the revised manuscript, I will avoid presenting these as equivalent alternatives. I agree that the title should better reflect the manuscript’s scope, and I propose revising it to: “Simulating the Extraction of Signal Parameters and Spectrograms for Non-Stationary NMR Signals.” This title more accurately describes the simulation-based nature and intended focus of the work.

1. There is little discussion of how the number of signals is determined, which is often one of the most difficult aspects of NMR fitting.

This is an excellent point. I agree that determining the number of signal components is a major challenge in time-domain NMR fitting and should be discussed explicitly. In the current implementation, initial frequency guesses can be obtained from FFT peak detection, but this does not solve the general problem of model selection. In the revised manuscript, I will clarify that direct fitting requires an assumed or estimated number of components, and that this number may be obtained from peak detection, prior chemical knowledge, residual analysis, or statistical model-selection criteria such as AIC/BIC, but that a fully automatic and reliable determination remains challenging. I will also clarify that FFT-derived initial guesses are used mainly to reduce fitting time and improve convergence, even when the FFT spectrum itself is too broad or low-resolution for confident interpretation.

1. A possible title is “Extraction of Signal Parameters and Spectrograms for Non-Stationary NMR Signals.”

I thank the reviewer for this helpful suggestion. I agree that this title better reflects the manuscript’s direction. To emphasize that the present manuscript is simulation-based rather than a complete experimental study, I will revise the title to: “Simulating the Extraction of Signal Parameters and Spectrograms for Non-Stationary NMR Signals.”

1. Fourier-domain and time-domain data contain the same information, although some tasks are easier in one domain than the other.

I agree with the reviewer. For a given discrete dataset, the Fourier-domain representation and the original time-domain data contain equivalent information in principle. The practical difference is not information content, but the ease, stability, and conditioning of extracting particular parameters. Frequency-domain analysis is usually faster, cleaner, and better suited for treating localized spectral regions independently. Time-domain fitting, however, may be useful when one is interested in decay rates, phases, or short-time behavior, although it becomes much more computationally demanding and model-dependent. I will revise the conclusion to state this distinction more clearly.

1. The manuscript states that frequency-domain peak fitting and time-domain FID fitting are significantly different, but signal parameters in one domain correspond to parameters in the other.

I agree that the original wording was too strong. I will revise this section to clarify that the two domains contain corresponding information, but that parameter extraction may differ in practice. For stationary, sufficiently long, high-SNR signals, Fourier-domain analysis is generally preferred. For short-lived or non-stationary components, however, the signal may appear as a broad or weak feature in the full Fourier spectrum, while useful information may still be present in early time-domain segments or time–frequency representations. The revised text will therefore emphasize practical differences in parameter extraction rather than implying a fundamental difference in information content.

1. The discussion of collection windows is useful and could be mentioned in the abstract.

I thank the reviewer for this positive comment. I will add a sentence to the abstract noting that the manuscript evaluates the effect of simulated data-collection windows and shows that, under sufficiently sharp and reproducible collection conditions, point-sampled FID simulations can approximate window-integrated signals after amplitude and phase correction. This provides a practical justification for the simplified simulation method used in the manuscript.

1. The statement about zero filling needs correction.

I agree and will revise the manuscript. The original wording incorrectly implied that zero filling adds enough data points for FFT “to work.” I will revise this to state that zero filling is commonly used to interpolate the discrete spectrum, improve visual appearance, and facilitate certain peak-picking or processing algorithms, but it does not add new experimental information or recover missing frequency resolution beyond what is contained in the measured time-domain data.

1. Are there guidelines regarding data size and signal count for choosing the best approach?

I agree that the manuscript should provide more actionable guidance. In the revised conclusion, I will add a practical summary: FFT is preferred for stationary, sufficiently long, high-SNR FID signals; STFT or wavelet methods are useful when non-stationary behavior is expected and the data length is sufficient to support meaningful time-windowing; direct nonlinear time-domain fitting may be useful for short signals or for extracting decay, phase, and frequency parameters when the number of components is small or constrained by prior knowledge. However, direct fitting becomes impractical for large numbers of unknown signals unless additional model-selection, regularization, or prior chemical constraints are introduced. I will also note that a quantitative guideline for the minimum number of points relative to the number of fitted parameters requires further study and will likely depend strongly on signal-to-noise ratio, spectral overlap, model complexity, and prior constraints.

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Reviewer 2:

Comments:

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.

Response:

Thank the reviewer for this clarification. I agree that DFT coefficients can be interpreted mathematically as the least-squares projection of a discrete signal onto an orthogonal Fourier basis, and that FFT is an efficient algorithm for computing these coefficients. At the same time, I would like to clarify the practical distinction emphasized in the manuscript: Fourier-transform-based methods are fundamentally limited by the finite observation window, consistent with the Fourier time–frequency uncertainty relationship (uncertainty principle). For example, when only a partial wave or approximately half a cycle is available, the signal does not contain enough oscillatory information to resolve a narrow frequency component. The resulting Fourier spectrum is necessarily broadened, and zero filling can interpolate the spectrum but cannot recover the missing frequency resolution.

My intended distinction was therefore not to deny the linear-algebra interpretation of DFT as an orthogonal least-squares projection. Rather, this manuscript aimed to distinguish orthogonal transform-based analysis, including DFT/FFT and wavelet transforms, from direct nonlinear time-domain fitting of parametrized exponentially damped sinusoidal models. The latter directly optimizes amplitudes, frequencies, phases, and decay constants, and therefore can sometimes extract useful parameter estimates from very short FID segments, although with greater sensitivity to model assumptions, initial guesses, and local minima.

In the revised manuscript, I will replace broad references to “least-squares fitting” with “direct nonlinear time-domain fitting of damped sinusoidal models” where appropriate. I will also clarify that DFT/FFT can be viewed physically as an inner-product or matched-filter calculation that extracts resonant Fourier components, while mathematically the same operation is equivalent to an orthogonal least-squares projection because of the orthogonality of the Fourier basis. This revised wording better reflects the intended comparison in the manuscript and avoids implying that Fourier or wavelet transforms are unrelated to least-squares projection theory.