The Discrete Fourier Transform (DFT) Derived

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. The Discrete Fourier Transform (DFT) Derived

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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The Discrete Fourier Transform (DFT)

Given a signal $x(\, the spectrum is defined by

\

or, as is most often written
\

That is, the $k$th sample $X(\ of the spectrum of $x$ is defined as the inner product of $x$ with the $k$th DFT sinusoid $s_k$. This definition is $N$ times the coefficient of projection of $x$ onto $s_k$, i.e.,
\

The projection of $x$ onto $s_k$ itself is
\

The inverse DFT is simply the sum of the projections:
\

or, as we normally write,
\

In summary, the DFT is proportional to the set of coefficients of projection onto the sinusoidal basis set, and the inverse DFT is the reconstruction of the original signal as a superposition of its sinusoidal projections.



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