On the input output behavior of the Hilbert transform

Abstract

Of course we have h x ( k ) = ('HS)(k), where S is the Delta signal, i.e.,In this paper the behavior of the Hilbert transform is investigated. It is well known that the Hilbert transform is stable with respect to the energy norm. However it is shown in the paper that the causal form of the Hilbert transform is not stable with respect to the energy norm.The frequency response H~ of the Hilbert transform is defined by 00 H.tl(eJ") = x h x ( I ) e -J L W . (4) l=-00 We have -j o < w < 7 r j -7 r < w < o . 0 w = -7 r , O , 7 r