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 having the same magnitude , the partial energy defined by is maximum for all when the rational transfer function 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 , and then is verified for the matrix-valued transfer functions of the Smirnov class . 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 class functions.