Scaling Theorem
The
scaling theorem (or
similarity theorem) provides
that if you horizontally ``stretch'' a
signal by the factor

in the time domain, you ``squeeze'' its
Fourier transform by the same
factor in the
frequency domain. This is an important general Fourier
duality relationship.

Theorem: For all continuous-time functions

possessing a Fourier
transform,
where
and

is any nonzero
real number (the abscissa stretch factor).
Proof:
Taking the Fourier transform of the stretched signals gives
The absolute value appears above because, when

,

, which brings out a minus sign in front of the
integral from

to

.
The scaling theorem is fundamentally restricted to the
continuous-time, continuous-frequency (Fourier transform) case.
The closest we came to the scaling theorem among the
DFT
theorems was the
stretch theorem (§
7.4.10). We found that
``stretching'' a
discrete-time signal by the integer factor

(filling in between samples with zeros) corresponded to the
spectrum being
repeated 
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 using
interpolation (instead of zero-fill)
corresponded to the same repeated
spectrum with the 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 stretch theorem for DFTs can be
viewed as the discrete-time, discrete-frequency counterpart of the
scaling theorem for Fourier Transforms.
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