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'' and amplify 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,
![]() |
(B.9) |
where
![]() |
(B.10) |
and

![]() |
(B.11) |
Proof:
Taking the Fourier transform of the stretched signal gives

The absolute value appears above because, when
,
, which brings out a minus sign in front of the
integral from
to
.
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Shift Theorem
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Differentiation Theorem Dual