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The Discrete Fourier Transform (DFT)
Given a signal
, its DFT is defined
by^{6.3}
where
or, as it is most often written,
We may also refer to
as the
spectrum of
, and
is the
th
sample of the
spectrum at frequency
.
Thus, the
th sample
of the
spectrum of
is defined
as the
inner product of
with the
th DFT
sinusoid . This
definition is
times the
coefficient of projection of
onto
,
i.e.,
The projection of
onto
is
Since the
are
orthogonal and span
, using the main
result of the preceding chapter, we have that the inverse DFT is
given by the sum of the projections
or, as we normally write,

(6.1) 
In summary, the DFT is proportional to the set of coefficients of
projection onto the sinusoidal basis set, and the inverse DFT is the
reconstruction of the original signal as a superposition of its
sinusoidal projections. This basic ``architecture'' extends to all
linear orthogonal transforms, including wavelets, Fourier transforms,
Fourier series, the discretetime Fourier transform (DTFT), and
certain shorttime Fourier transforms (STFT). See Appendix B
for some of these.
We have defined the DFT from a geometric signal theory point of view,
building on the preceding chapter. See §7.1.1 for
notation and terminology associated with the DFT.
Previous: An Orthonormal Sinusoidal SetNext: Frequencies in the ``Cracks''
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.