Spectral Roll-Off

Definition: A function is said to be of order if there exists 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.,

Proof: See §B.2.

---

Previous: Scaling Theorem
Next: Spectral Interpolation

Order a Hardcopy of Spectral Audio Signal Processing

---

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.