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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University