Scaling Theorem
The scaling theorem (or similarity theorem) says 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,
![$\displaystyle \zbox {\hbox{\sc Stretch}_\alpha(x) \;\longleftrightarrow\;\left\vert\alpha\right\vert\hbox{\sc Stretch}_{(1/\alpha)}(X)}
$](http://www.dsprelated.com/josimages_new/sasp2/img243.png)
where
![$\displaystyle \hbox{\sc Stretch}_{\alpha,t}(x) \isdefs x\left(\frac{t}{\alpha}\right)
$](http://www.dsprelated.com/josimages_new/sasp2/img244.png)
and
![$ \alpha $](http://www.dsprelated.com/josimages_new/sasp2/img4.png)
![]() |
(3.41) |
Proof: See §B.4.
The scaling theorem is fundamentally restricted to the
continuous-time, continuous-frequency (Fourier transform) case. The
closest we come to the scaling theorem among the DTFT theorems
(§2.3) is the stretch (repeat) theorem
(page ). For this and other continuous-time Fourier
theorems, see Appendix B.
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