Signals as Vectors

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Signals as Vectors

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

For the DFT, all signals and spectra are length $N$. A length $N$ sequence$x$ can be denoted by $x(n)$, $n=0,1,2,\, where $x(n)$ may be real ($x\) or complex ($x\). We now wish to regard $x$ as avector $\6.1 in an $N$ dimensional vector space. That is, each sample $x(n)$ is regarded as a coordinate in that space. A vector $\ is mathematically a single point in $N$-space represented by a list of coordinates $(x_0,x_1,x_2,\ called an $N$-tuple. (The notation $x_n$ means the same thing as $x(n)$.) It can be interpreted geometrically as an arrow in $N$-space from the origin $\ to the point $\.

We define the following as equivalent:

\

where $x_n \ is the $n$th sample of the signal (vector) $x$. From now on, unless specifically mentioned otherwise, all signals are length $N$.


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