Peak locations in all-pass signals - the Makhoul Conjecture Challenge

The Makhoul Conjecture Challenge (Makhoul, 2000) has been published. To answer was the question whether or not the location of the peak of a digital stable all-pass filter lies in [0,2p-1], where p is the order of the all-pass filter. We construct numerous counter-examples, prove a new theorem stating that there is at least an upper bound on the order of p^3/2 for the location of the peak, and discuss the algebraic structure of all-pass filters and their impulse responses. The paper is heavily based on an experimental approach