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
![$ \alpha $](http://www.dsprelated.com/josimages_new/sasp2/img4.png)
![]() |
(B.11) |
Proof:
Taking the Fourier transform of the stretched signal gives
![\begin{eqnarray*}
\hbox{\sc FT}_{\omega}(\hbox{\sc Stretch}_\alpha(x))
&\isdef & \int_{-\infty}^\infty x\left(\frac{t}{\alpha}\right) e^{-j\omega t} dt\qquad\hbox{(let $\tau=t/\alpha$)}\\
&=& \int_{-\infty}^\infty x(\tau) e^{-j\omega (\alpha\tau)} d (\alpha\tau) \\
&=& \left\vert\alpha\right\vert\int_{-\infty}^\infty x(\tau) e^{-j(\alpha\omega)\tau} d \tau \\
&\isdef & \left\vert\alpha\right\vert X(\alpha\omega).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2431.png)
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