**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 a

vector^{6.1}in an dimensionalvector space. That is, each sample is regarded as acoordinatein that space. Avectoris mathematically a singlepointin -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