NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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For a length 
complex sequence 
, 
, thediscrete Fourier transform (DFT) is defined
by
We are now in a position to have a full understanding of the
transform kernel:
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 an inner
product operation which computes the coefficient of
projection of 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.
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