مرکز منطقه ای اطلاع رساني علوم و فناوري - Notes on the minimum-energy delay property of impulse-response sequences of minimum-phase transfer functions

DocumentCode :

1196368

Title :

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

Author :

Inouye, Yujiro

Volume :

Issue :

fYear :

1987

fDate :

2/1/1987 12:00:00 AM

Firstpage :

188

Lastpage :

190

Abstract :

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.