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The Fourier transform in analytical science

Nature Reviews Methods Primers volume 4, Article number: 49 (2024) Cite this article

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Abstract

Numerous data analysis methodologies depend on the Fourier transform (FT), especially in analytical chemistry. The FT is a potent and versatile tool, influencing many scientific disciplines. Despite its prominence, the FT is often an enigma for many. In response, this Primer aims to provide an all-encompassing elucidation of the FT for readers not well versed in advanced mathematics. The article explores the theoretical underpinnings of the FT, alongside practical applications, to demystify the fundamental concepts of the method. Its utility is demonstrated through diverse examples, such as mass spectrometry, NMR, infrared spectroscopy and other analytical techniques. Potential extensions of the FT are explored, including potential future developments.

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Fig. 1: A complex plane representation, showing the real (horizontal axis) and imaginary (vertical axis) parts.
Fig. 2: Examples of overlap integrals between a main sine wave s (solid line) and an additional one (dashed line).
Fig. 3: The real-time signal with three components and its Fourier transform.
Fig. 4: Relations between the symmetries in the signal (left) and the properties of the Fourier transform spectrum (right).
Fig. 5: Pairs of functions and their Fourier transform.
Fig. 6: Effect of apodization.

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Code availability

The code used to generate the figures can be found at https://github.com/delsuc/Fourier-Transform-Review.

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Acknowledgements

The authors thank P. Kern for correcting the text and helpful advice on its content.

Author information

Authors and Affiliations

  1. I.G.B.M.C., Illkirch, France

    Marc-André Delsuc

  2. Université de Strasbourg, Strasbourg, France

    Marc-André Delsuc

  3. CNRS, Paris, France

    Marc-André Delsuc

  4. Department of Chemistry, University of Warwick, Coventry, UK

    Peter O’Connor

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  2. Peter O’Connor
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Contributions

Introduction (M.-A.D. and P.O.C.); Experimentation (M.-A.D. and P.O.C.); Results (M.-A.D. and P.O.C.); Applications (M.-A.D. and P.O.C.); Reproducibility and data deposition (M.-A.D. and P.O.C.); Limitations and optimizations (M.-A.D. and P.O.C.); Outlook (M.-A.D. and P.O.C.); overview of the Primer (M.-A.D. and P.O.C.).

Corresponding author

Correspondence to Marc-André Delsuc.

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Nature Reviews Methods Primers thanks Javier Segurado and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Glossary

Apodization

Also known as windowing. Consists of multiplying the signal before Fourier transform by a usually decaying adequate function to improve the signal-to-noise ratio. This enables correction of truncation artefacts and improvement of peak line shapes.

Convolution

An integral transformation that combines two independent functions into a resulting function and appears in many aspects of the spectral analysis.

Discrete Fourier transform

A linear algebra transformation that can be represented by a square matrix. It shares many properties with the analytical Fourier transform operation applied to continuous functions.

Exponential decay

A signal decaying with an exponential dependency to a characteristic time s(t) \(\propto \) exp(−t/τ).

Fast discrete Fourier transform

An algorithm that performs the discrete Fourier transform computation more rapidly than the naive matrix approach, enabling processing of large vectors if their size is decomposable into small prime numbers.

Fellgett effect

Also known as multiplex advantage. Improvement of the signal-to-noise ratio observed for stationary signals.

Fourier series

A series of sinusoidal functions that can approximate any periodic function.

Full width at half maximum

A standardized measure of the width of a spectral peak obtained by computing its width at 50% of the maximum point of the peak.

Gate function

Also known as the box-car function. A function null everywhere except in a continuous interval [a, ..., b] in which it is equal to 1. Used to express the measurement process of a signal during this interval. Its Fourier transform is the sinc function.

Gaussian decay

A signal decaying with a Gaussian dependency to a characteristic time τ: s(t) \(\propto \) exp(−t2/τ2).

Sampling criterion

Also known as the Nyquist criterion. Determines the highest frequency faithfully sampled by a given periodic sampling.

Signal-to-noise ratio

Ratio of the intensity of a frequency signal in the spectrum to the background noise observed in the same spectrum, measured either from its standard deviation or by peak-to-peak extension.

Sinc function

Also referred to as a sine cardinal function. A function defined as sinc x = sin x/x with a characteristic bell shape around zero and oscillating tails decaying away from the centre. Its Fourier transform is the gate function.

White noise

A random signal, with no characteristic frequency and a flat frequency spectrum.

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Delsuc, MA., O’Connor, P. The Fourier transform in analytical science. Nat Rev Methods Primers 4, 49 (2024). https://doi.org/10.1038/s43586-024-00326-2

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