###

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,

where

and is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

(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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Spectral Roll-Off

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Differentiation Theorem Dual