Fourier workshop

These notes originate from a workshop I used to teach as part of year-3 lab for MPhys students. This doesn't exist any more, and the content has been subsumbed in a number of other lectures and lab modules. I've decided to leave these notes online as a summary of Fourier transforms.

Second-year maths touches on Fourier series and transformations, but given that the technique is so widespread and important in experimental physics, we need to expand on this further. This workshop is a three-day block course which reiterates the maths behind the Fourier transform, shows the implications for physics applications arising from the fact that measured data sets aren't continuous functions, and looks at the way Fourier transforms are implemented in computing.

Contents of the workshop

• Fourier series
• Fourier transformation
• Symmetry: even, odd, real, imaginary functions
• Fourier theorems
• Power spectra and Parseval's theorem
• Transforming in practice - Free induction decays in NMR
• FT by eye
• FT of digitised data - sampling artifacts and Nyquist frequency
• Fast Fourier Transform - the FFT algorithm
• Mathcad's FT functions
• Correcting for experimental artefacts: deconvolution

Please note that theses notes are meant to help you revise. However, I may include additional material in the workshops or not cover all of the material listed here.

Further reading

• WH Press, SA Teukolsky, WT Vetterling, BP Flannery; Numerical Recipes; Cambridge University Press ²1992, ch. 12.0-2 and 13.0-1
  This book is available for a number of programming languages. It is probably the best known source of scientific algorithms. The introductory sections in each chapter are very concise and understandable explanations of the mathematical background. I find the Recipes' coverage of FT better than that in most dedicated Fourier theory books I have seen. Any of the copies available in the Physical Sciences Library will do.
• ML Boas; Mathematical Methods in the Physical Sciences; New York: Wiley ²1983, ch. 15.4-5.
  My favourite maths textbook, although its discussion of Fourier transforms is based on the preceding chapter on Laplace transforms, which we haven't done in year-2 maths. If you're prepared to go the extra mile and read up on both, it's certainly worthwhile. There are a few copies in the Physical Sciences Library.
• M Cartwright; Fourier Methods for Mathematicians, Scientists and Engineers; New York: Ellis Horwood 1990
  This book lives up to its claim and provides all the mathematical rigour alongside plenty of examples from science and engineering, although reading it would be a lot easier if the diagrams were properly labelled. [Physical Sciences Library]
• DC Champeney; Fourier Transforms in Physics; Bristol: Hilger 1985
  There are a couple of copies of this booklet in the Physical Sciences Library. It is quite useful because it has loads of worked examples based on physical applications. [Physical Sciences Library]