Matrix formulation of the discrete Hilbert transform

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.