Allpass Problems

1. The BiQuad Allpass Section
   1. Show that every second-order filter having transfer function
      
      is a unit-gain allpass filter. That is, show that , for all and . (Typically, and are chosen such that the filter is stable, but this is not necessary for the result to hold.)
   2. Find the zeros of the filter as a function of the poles. In other words, given two poles, what is the rule for placing the zeros in order to obtain an allpass filter?

   3. Find the phase response of the zeros in terms of the phase response of the poles.

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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.