## 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

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