Length Three FIR Loop Filter

The simplest nondegenerate example of the loop filters of §6.8 is the three-tap FIR case ( ). The symmetry constraint leaves two degrees of freedom in the frequency response:^10.1

If the dc gain is normalized to unity, then , and there is only one remaining degree of freedom which can be interpreted as a damping control. As damping is increased, the duration of free vibration is reduced at all nonzero frequencies, and the decay of higher frequencies is accelerated relative to lower frequencies, provided

In this coefficient range, the string-loop amplitude response can be described as a ``raised cosine'' having a unit-magnitude peak at dc, and minimum gains at plus and minus half the sampling rate ( ).

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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.