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Exploring Fourier Transform Techniques with Mathcad

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Abstract

These four Mathcad documents illustrate different aspects of the Fourier transform. They explore the source of the Nyquist frequency and the dependence of the resolution of the spectrum on sampling. In FTIntro.mcd, students achieve a qualitative understanding of the Fourier transform through experimentation with graphic representations of data and their transforms. In Part 1 of FTIntro.mcd, students sample a continuous function and learn how aliasing can occur; in Part 2, a vector of a more complex waveform illustrates that the frequencies contained in the waveform appear as spectral lines in the transform. Exercises permit further exploration of the Nyquist frequency and illustrate the relationship of resolution to number of samples. At the end, the students perform an integral transform of the same waveform. The second document, FT2IR.mcd, demonstrates how spectral information is coded in an interferogram and how the interferometric data limits resolution in the spectrum. In FT2IR.mcd, the inverse FT is used to generate an interferogram from a spectrum with two notches representing absorption. The third document, FT3Pulse.mcd, demonstrates that a short sample of a wave must contain a range of frequencies. In the fourth document, FT4FreeIndDecay.mcd, free induction decay (FID) and its Fourier transform are explored and related to NMR signal analysis. The entire set of documents contains many student exercises and hands-on activities for learning the concepts presented.

Figure 1. A simulated free induction decay (FID) and the frequency spectrum of the FID.

Downloads

Version	Download
Mathcad	FTIntro.mcd
Mathcad	FT2IR.mcd
Mathcad	FT3Pulse.mcd
Mathcad	FT4FreeindDecay.mcd

Version	Download
Mathcad	FTIntro.pdf
Mathcad	FT2IR.pdf
Mathcad	FT3Pulse.pdf
Mathcad	FT4FreeindDecay.pdf
Instructor's Notes	FourierNotes.pdf

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Commentary

Other Information

Keywords

Domain: Analytical Chemistry; Laboratory Instruction; Physical Chemistry;

Pedagogy: Computer-Based Learning;

Topics: Fourier Transform Techniques; Mathematics / Symbolic Mathematics; NMR Spectroscopy;

JCE Citation