All-pass matrices, the positive real lemma, and unit canonical correlations between future and past

Abstract

Stationary vector ARMA processess x( t), t = 0, ±1, ±2,…, that are full rank are considered. It is shown, with E( z) denoting the ‘symbol’ of the canonical correlation operator, that the number of zeros (counting multiplicities) on | z| = 1 of the transfer function, V( z), from innovations to outputs = ( change in arg detE(z) around |z| = 1) 2π = the number of unit Hankel singular values of E(s) . It follows that the number of unit canonical correlations between the future x( t), t > 0 and the past x( t), t ≤ 0 is equal to the number of zeros V( z) on | z| = 1, which is the result of Hannan & Poskitt 1986. This note, however, proves this result directly from the properties of E( z), which Hannan & Poskitt suggested should be possible. It is also shown that the result can be obtained using an entirely state space approach via the positive real lemma.