NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 similarity theorem is
fundamentally restricted to the continuous-time case. It
says that if you ''stretch'' a signal by the factor 
in the time domain, you ''squeeze'' its Fourier transform
by the same factor in the frequencydomain.
This is such a fundamental Fourier relationship, that we include it
here rather than leave it out as a non-DFT result.
The closest we came to the similarity theorem among the DFT
theorems was the the Interpolation Theorem. We found that
''stretching'' adiscrete-time signal by the integer factor
(filling in between samples with zeros) corresponded to the
spectrum beingrepeated
times around the unit circle. As a result, the ''baseband'' copy of
the spectrum ''shrinks'' in width (relative to 
) by
the factor . Similarly, stretching a signal usinginterpolation
(instead of zero-fill) corresponded to the repeated spectrum with
all spurious spectral copies zeroed out. The spectrum of the
interpolated signal can therefore be seen as having been stretched
by the inverse of the time-domain stretch factor. In summary, the
Interpolation DFT Theorem can be viewed as the discrete-time
counterpart of the similarity Fourier Transform (FT) theorem.
Theorem: For all continuous-time functions 
possessing a Fourier transform,
where
and
is any nonzero real number (the abscissa scaling factor).
Proof:
The absolute value appears above because, when 
, 
, which brings out a minus sign in front of the
integral from 
to 
.
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