Makhoul's conjecture for p = 2

The IEEE Signal Processing Society (2000) offered a prize of $1000 for proving or disproving Makhoul's conjecture, which says that, given a causal all-pass digital signal x/sub n/ of order p, with nonzero x/sub 0/, the location of the peak of x/sub n/ always lies between n = 0 and n = 2p-1. The case of p = 1 is trivial, and no further progress had been made in 25 years until Lertniphonphun, Rajagopal, and Wenzel gave counter examples for large p. In this paper, Makhoul's conjecture is proven for p = 2. It is also shown that the conjecture fails dramatically in the case of complex coefficients.