Lagrange Frequency Response Magnitude Bound

The amplitude response of fractional delay filters based on Lagrange interpolation is observed to be bounded by 1 when the desired delay lies within half a sample of the midpoint of the coefficient span [502, p. 92], as was the case in all preceeding examples above. Moreover, even-order interpolators are observed to have this boundedness property over a two-sample range centered on the coefficient-span midpoint [502, §3.3.6]. These assertions are easily proved for orders 1 and 2. For higher orders, a general proof appears not to be known, and the conjecture is based on numerical examples. Unfortunately, it has been observed that the gain of some odd-order Lagrange interpolators do exceed 1 at some frequencies when used outside of their central one-sample range [502, §3.3.6].

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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.