## 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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