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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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