Explanatory Material (many from www.fftw.org)

Tutorials and introductions to Fourier transforms and FFTs, in no particular order.

• The Mathworld (Mathmatica) page on Fourier Transforms
• The Wikepedia page on Fourier Transforms, often a good place to start. Note that the definition differs from what is used in 360 and crystallography in general.
• There is a survey and history of FFT algorithms and related information in the free Wikipedia collaborative encyclopedia.
• Numerical Recipes, which is readable on-line (with a special plugin, unfortunately), has a decent introduction to Fourier transforms, DFTs, and FFTs (albeit somewhat dated).
• For a good description of the FFT literature c. 1997 (with many references), see C. S. Burrus's "Notes on the FFT" (our mirror of the now-missing original).
• The Scientist's and Engineer's Guide to Digital Signal Processing is an entire DSP book free online by Steven W. Smith.
• Another book online: Mathematics of the Discrete Fourier Transform (DFT)—With Music and Audio Applications, by Julius O. Smith III.
• The Picture Book of Fourier Transforms by Kevin Cowtan gives an interesting graphical tutorial on the interpretation of 2D FFT output, with a special emphasis on crystallography. There is also a tutorial on Fourier transforms, the convolution theorem, and other material.
• An"intuitive explanation of Fourier theory" by Steven Lehar.
• DFT and FFT Introduction by Paul Bourke, describing the discrete Fourier transform in one and two dimensions in terms of the continuous transform, with examples of the transforms of various functions. Also has introductions to digital filters, image filtering, and other related topics.
• A chapter on the history of Fourier's theorem, from the charming book Trigonometric Delights (readable online) by Eli Maor.
• A biography of Jean Fourier can be found at the excellent MacTutor History of Mathematics Archive; there is also another biography of Jean Fourier on Wikipedia.
• The FFT Demystified is a site by Adrian Hey covering many introductory and not-so-introductory aspects of FFT algorithms (original has disappeared; now mirrored by the Internet Archive.
• The book Numerical Computing with MATLAB online has a tutorial on Fourier analysis based on Matlab's fft function (which uses FFTW).
• A mostly non-mathematical introduction to Fourier transforms at the site of a DSP company with the (hopefully non-descriptive) name of Bores.
• DSP Dimension, by Stephan M. Bernsee, contains tutorials and other links for Fourier analysis and DSP, focusing on audio processing.
• dspGuru contains various tutorials, FAQs, and other information related to digital signal processing (and FFTs).
• DSPRelated.com is another site collecting DSP links and discussion groups.
• A "graphical interpretation" of the DFT and FFT, by Steve Mann.
• Scanned copies of the original Cooley & Tukey FFT paper, placed online by Charles Chiu
• Integrate your function times a complex exponential... Sing along with "Fourier's Song."
• The FFT Laboratory site has a neat Java applet for experimenting with the DFT.