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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