Notes on the minimum-energy delay property of impulse-response sequences of minimum-phase transfer functions
Y. Inouye · IEEE Transactions on Circuits and Systems · 1987
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 sequenceH_khaving the same magnitude|H(e^{j\omega})|, the partial energy\epsilon(m)defined by\epsilon(m) = \sum_{k=0}^{m}|H_k|^2is maximum for allm {\geq} 0when the rational transfer functionH(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 classH^2, and then is verified for the matrix-valued transfer functions of the Smirnov classN^+. 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 forL^2class functions.