When NMR Signals Won’t Sit Still: Simulations Test Fourier, Wavelets, and Direct Fitting

Nuclear magnetic resonance spectroscopy has spent more than half a century relying on a single, remarkably effective mathematical trick: the fast Fourier transform, or FFT, which converts the wiggling time-domain signal of a sample into the familiar frequency-domain spectrum that chemists read like a fingerprint. But the FFT has an Achilles heel. It assumes the signal it is analyzing is stationary, meaning its essential properties do not change while the data are being collected. In real experiments, particularly in solid-state NMR, surface chemistry, and catalysis, peaks can broaden, decay, and drift during acquisition. A new simulation study by Jixin Chen of Ohio University, published in the journal Magnetic Resonance, systematically compares how well the FFT and three rival approaches handle these awkward, non-stationary free induction decay signals, and the results offer a practical roadmap for anyone pushing NMR into harder territory.

The free induction decay, or FID, is the damped oscillation that a sample emits after being excited by a radiofrequency pulse. Each chemically distinct atomic nucleus contributes a sinusoidal component whose frequency encodes its chemical shift and whose decay rate reflects its local environment. Chen simulated FIDs built from exponentially damped sinusoids, from peaks with Gaussian-distributed frequencies, and from peaks whose widths grow over time, mimicking dynamic broadening processes such as molecular diffusion. The simulations used a 500 MHz proton NMR model spanning chemical shifts from minus 10 to 10 ppm with a fine 0.001 ppm resolution, and realistic noise was added to both the timing and the signal amplitude. All computations ran in MATLAB on an ordinary laptop, and the source code has been released publicly on GitHub, making the entire benchmark reproducible by any laboratory.

One of the study’s most technically interesting findings concerns how the act of data collection itself is simulated. In a real spectrometer, each data point is not an instantaneous sample but the integral of the signal over a short acquisition window, typically far shorter than the sampling interval. Integrating these windows exactly is computationally expensive, especially for spectra with many peaks. Chen showed mathematically that integrating a sine or cosine over a sharp-edged window simply produces another sinusoid of the same frequency with a shifted phase and altered amplitude. As a result, when the collection window is sufficiently sharp and reproducible, point sampling gives essentially the same answer as window integration, once amplitude and phase corrections are applied. Simulations comparing one-point collection with 90-nanosecond windows showed no appreciable difference, a result that slashes the computational cost of generating large batches of synthetic FIDs.

The window analysis also produced a practical engineering guideline. For a 500 MHz instrument, the simulations indicated that a ramp time of less than one nanosecond at the edges of the collection window is needed to amplify all chemical shifts of interest uniformly, a demanding figure given that a 500 MHz oscillation completes a full cycle in just 2 nanoseconds. The quality of the final signal, beyond simple amplitude noise, was found to depend mainly on the accuracy of the starting time point and the repeatability of the window length, with timing errors below 10 picoseconds required to sustain a signal-to-noise ratio above 500. These are exactly the kinds of numbers that instrument designers and simulation builders need when deciding how faithfully a virtual spectrometer must mimic a physical one.

Chen’s simulations of broadened peaks revealed a subtle but important physical effect that anyone simulating NMR signals ignores at their peril. When a chemical shift peak consists of many closely spaced frequencies, those components start in phase, interfere constructively, and then gradually drift out of synchronization as the signal decays. This dephasing, combined with ordinary damping, dramatically shortens the measurable FID. When the peak width itself grows over time, as in a diffusion-like broadening process modeled with a broadening rate expressed in ppm squared per second, the FID decays even faster. Notably, the FFT spectrum of such a signal reflects the average peak width of the useful portion of the signal, meaning that a naive analysis could misattribute dynamic broadening to static line shapes. Peaks that are far apart in chemical shift, the simulations confirmed, generally do not interfere with one another, which simplifies analysis of well-separated resonances.

Against this simulated backdrop, Chen pitted four analysis methods against one another. The FFT emerged, unsurprisingly, as the fastest and most reliable tool for stationary, sufficiently long, high-signal-to-noise data. But the study also quantified a familiar frustration: zero filling, the common practice of appending zeros to a short FID before transforming, improves the visual interpolation of the spectrum but adds no new information and cannot recover the frequency resolution that was lost when the acquisition was cut short. When the observation window contains less than one full oscillation cycle of the slowest frequency, the Fourier-domain feature becomes intrinsically broad and poorly resolved, no matter how elegantly the spectrum is dressed up afterward.

This is where direct nonlinear time-domain fitting shows its promise. Rather than transforming the data, this approach fits a parametrized model of damped sinusoids directly to the raw FID, estimating amplitudes, frequencies, phases, and decay constants in one step. Chen tested a jump-chain fitting algorithm, called JCFit, which was recently developed to navigate the notoriously bumpy landscape of wave-fitting problems. Instead of following local gradients, the algorithm scans each parameter by simulation in a randomized order, searching exponentially far from the initial guesses and thereby dodging many of the local minima that trap conventional Gauss-Newton or Levenberg-Marquardt optimizers. In the simulations, JCFit fitted FIDs containing only 20 or 10 data points, shorter than a single cycle of the slowest frequency, within a few seconds, and recovered chemical shifts and amplitudes of most isolated peaks to within about 1 percent of the values obtained from much longer signals.

The triumph comes with caveats that the study is careful to spell out. A closely spaced doublet near 4 ppm remained unresolved even by direct fitting, with chemical shift errors of roughly 0.2 ppm, mirroring its poor behavior in the Fourier analysis. The reliability of time-domain fitting depends strongly on choosing the right model, supplying good initial estimates, constraining the number of signal components, and achieving adequate signal-to-noise. For complex molecules with many overlapping and unknown components, the fitted solution can be non-unique or poorly conditioned, and fully automatic determination of the number of components remains an unsolved challenge. The sensible workflow, Chen suggests, is hybrid: use fast Fourier analysis and prior chemical knowledge to generate initial frequency estimates, then refine with model fitting when the signal is short and simple enough for the approach to be trustworthy.

The final piece of the study looks toward machine learning. Short-time Fourier transforms, wavelet transforms, and a deep-learning variant of the STFT were all applied to simulated FIDs to generate two-dimensional time-frequency spectrograms, the same kind of representation that powers speech recognition. These methods slice the FID into overlapping windows and transform each piece, revealing how individual frequencies decay and interfere over time. All three sacrificed frequency resolution compared with the full FFT, as expected, with the sharp-edged STFT windows preserving the best resolution, the softer Gaussian wavelets trading resolution for reduced edge artifacts, and the MATLAB deep-learning STFT implementation performing worst and lacking support for complex-valued input. Yet the spectrograms capture exactly the time-dependent behavior that the FFT averages away, and the study argues that high-quality simulated spectrograms, generated efficiently thanks to the point-sampling shortcut, could train neural networks to classify chemicals rapidly, much as machines now classify human speech. For a technique as foundational as NMR, that would be a genuinely transformative upgrade, and this simulation framework provides the testing ground to get there.

Subject of Research: Comparative simulation of Fourier, wavelet, and nonlinear fitting methods for non-stationary NMR free induction decay signals

Article Title: Simulating the extraction of signal parameters and spectrograms for non-stationary NMR signals

Article References: Chen, J. (2026). Simulating the extraction of signal parameters and spectrograms for non-stationary NMR signals. Magnetic Resonance, 7(2), 125-134. https://doi.org/10.5194/mr-7-125-2026

Image Credits: AI Generated

DOI: 10.5194/mr-7-125-2026

Keywords: NMR spectroscopy, free induction decay, fast Fourier transform, short-time Fourier transform, wavelet transform, nonlinear fitting, signal processing, spectrograms, machine learning, chemical shift, zero filling, simulation