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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 $ \alpha $ 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 $ x(t)$ possessing a Fourier transform,

$\displaystyle \zbox {\hbox{\sc Stretch}_\alpha(x) \;\longleftrightarrow\;\left\vert\alpha\right\vert\hbox{\sc Stretch}_{(1/\alpha)}(X)}


$\displaystyle \hbox{\sc Stretch}_{\alpha,t}(x) \isdefs x\left(\frac{t}{\alpha}\right)

and $ \alpha $ is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

$\displaystyle \zbox {x\left(\frac{t}{\alpha}\right) \;\longleftrightarrow\; \left\vert\alpha\right\vert\cdot X(\alpha\omega)}$ (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 $ W(\omega)$ is said to be of order $ 1/\omega^{n+1}$ if there exist $ \omega_0$ and some positive constant $ M<\infty$ such that $ \left\vert W(\omega)\right\vert<M/w^{n+1}$ for all $ \omega > \omega_0$ .

Theorem: (Riemann Lemma): If the derivatives up to order $ n$ of a function $ w(t)$ exist and are of bounded variation, then its Fourier Transform $ W(\omega)$ is asymptotically of order $ 1/\omega^{n+1}$ , i.e.,

$\displaystyle W(\omega) = {\cal O}\left(\frac{1}{\omega^{n+1}}\right), \quad(\hbox{as }\omega\to\infty)$ (3.42)

Proof: See §B.18.

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Spectral Interpolation
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Fourier Theorems for the DTFT