**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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

For a length complex sequence , , the

discrete Fourier transform(DFT) is defined by

We are now in a position to have a full understanding of the transformkernel:

The kernel consists of samples of a complex sinusoid at discrete frequencies uniformly spaced between and the sampling rate. All that remains is to understand the purpose and function of the summation over of the pointwise product of times each complex sinusoid. We will learn that this can be interpreted as aninner productoperation which computes thecoefficient of projectionof the signal onto the complex sinusoid . As such, , the DFT at frequency, is a measure of the amplitude and phase of the complex sinusoid at that frequency which is present in the input signal . This is the basic function of all transform summations (in discrete time) and integrals (in continuous time) and their kernels.