## 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,

(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