# Signals as Vectors

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

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

We define the following as equivalent:

where is the th sample of the signal (vector) . From now on, unless specifically mentioned otherwise, all signals are length .

Subsections

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