Continuous-Time Fourier Theorems

Selected Fourier theorems for the continuous-time case are stated and proved in Appendix B. However, two are sufficiently important that we state them here.

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.

Spectral Roll-Off

Definition: A function is said to be of order if there exist and some positive constant such that for all .

Theorem: (Riemann Lemma): If the derivatives up to order of a function exist and are of bounded variation, then its Fourier Transform is asymptotically of order , i.e., (3.42)

Proof: See §B.18.

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Spectral Interpolation
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Fourier Theorems for the DTFT