On the energy concentration property for zero-phase sequences

One of the main properties of minimum-phase sequences is that they concentrate their energy towards the origin. Recently it has been shown that the causal part of a zero-phase sequence is a minimum-phase sequence, thus a zero-phase sequence might also concentrate its energy towards the origin. In this paper we shall discuss how to slightly modify the partial energy of zero-phase sequence such that the modified partial energy should always satisfy the energy concentration property. The windowed partial energy is introduced and few properties are presented. Finally the modified rectangular window, the damped window and the Gaussian window are proposed for modifying the partial energy.