This text is designed to accompany your study of introductory digital signal processing.^{1} It’s an eccentric piece of notsorigorous literature with a preoccupation for explaining things using interactive visualizations, animations and sound.^{2} My goal is to explain the Discrete Fourier Transform using a miniature curriculum which leverages your ability to learn concepts and absorb information visually instead of linguistically.^{3} My hope is that these glyphs become slightly more comprehensible and slightly less intimidating after reading the subsequent 30 or so pages.^{4}
$$ \mathrm{DFT}[k] = \sum_{n=0}^{N1} \mathrm{x}[n] \cdot
e^{\varphi\mathrm{i}} \\
where \quad \varphi = k \frac{n}{N} 2\pi
$$


Equation 1. The Discrete Fourier Transform 
Figure 1. Visualizing the Composition of a Complex Signal 
 David Foster Wallace, Everything and More^{5}
If you’re really serious about learning the theory and practice of digital signal processing, you should not use this website as your sole resource. Wherever a compromise was necessary, I’ve erred in favor of simplicity over completeness, fun and engagement over rigor. This text is a piece of popscience and you should treat it as such.
If you want a solid introduction to digital signal processing, I strongly suggest that you read Understanding Digital Signal Processing by Richard J Lyons. Lyons writes with a great deal of empathy for the reader, patiently explicating concepts that other authors might deem to be “obvious”. Paolo Prandoni and Martin Vetterli periodically offer a course on digital signal processing through Coursera. Their course is free, and the lectures are really quite lovely. Martin and Paolo have also written Signal Processing for Communications, which is freely available online.^{6} Julius O. Smith III has an encyclopedic catalog of DSP writings freely available on his site. Smith’s online text, The Mathematics of the Discrete Fourier Transform is of particular interest. Dan Ellis posts all of his teaching materials online for free and open access. Geoff Martin’s delightful online book is recommended if you’re interested in a less formal, but still very thorough resource on mathematics and DSP.
Contact me on twitter if you have any comments, questions, or complaints about this site. If you encounter an error in the mathematics or a technical issue with the presentation, please file this as an issue on the github site.