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

(B.11) |

*Proof:*Taking the Fourier transform of the stretched signal gives

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