NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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DFT Math Outline
In summary, understanding the DFT takes us through the following topics:
- Complex numbers
- Complex exponents
- Why ?
- Euler's formula
- Projecting signals onto signals via the inner product
- The DFT as the coefficient of projection of a signal onto sinusoids
- The IDFT as a weighted sum of the projections
- Various Fourier theorems
- Elementary time-frequency pairs
- Practical spectrum analysis in Matlab
We will additionally discuss practical aspects of working with sinusoids, such as decibels (dB) and display techniques.