Matrix formulation of the discrete Hilbert transform

Abstract

The discrete Hilbert transform (DHT) of a periodic sequence is interpreted as a matrix product \bar{\phi} = \bar{C}\bar{A} . A new singleequation form of the DHT operation for any number of sample points N is shown and is used to establish the unique properties of the coefficient matrix \bar{C} . \bar{C} is shown to be highly symmetric in nature, and the determination of the elements of \bar{C} is shown to require a minimum of computation; i.e., less than N elements need to be computed for an N \times N \bar{C} matrix. For the N even case, the relatively sparse nature of \bar{C} is established; i.e., at least half the elements of \bar{C} are zeros.