Notes on the minimum-energy delay property of impulse-response sequences of minimum-phase transfer functions

In the scalar case, it is widely known that the impulseresponse sequence of a minimum-phase transfer function possesses the minimum-energy delay property, i.e., on the set of an impulse-response sequence
H_k
having the same magnitude
|H(e^{j\omega})|
, the partial energy
\epsilon(m)
defined by
\epsilon(m) = \sum_{k=0}^{m}|H_k|^2
is maximum for all
m {\geq} 0
when the rational transfer function
H(z)
is minimum phase [1]. In this brief, it is shown that the minimum-energy delay property is valid in the matrix case. This is proved first for the matrix-valued transfer functions of the Hardy class
H^2
, and then is verified for the matrix-valued transfer functions of the Smirnov class
N^+
. We shall see that the minimum-energy delay property is proved by using the inner-outer (or all-pass and minimum-phase) factorization of transfer functions and the Parseval identity for
L^2
class functions.