  Linear canonical transforms (LCTs) are of importance in many areas of science
and engineering with many applications. Therefore a satisfactory discrete
implementation is of considerable interest. Although there are methods that
link the samples of the input signal to the samples of the linear canonical
transformed output signal, no widely-accepted definition of the discrete LCT
has been established. We introduce a new approach to defining the discrete
linear canonical transform (DLCT) by employing operator theory. Operators are
abstract entities that can have both continuous and discrete concrete
manifestations. Generating the continuous and discrete manifestations of LCTs
from the same abstract operator framework allows us to define the continuous
and discrete transforms in a structurally analogous manner. By utilizing
hyperdifferential operators, we obtain a DLCT matrix which is totally
compatible with the theory of the discrete Fourier transform (DFT) and its dual
and circulant structure, which makes further analytical manipulations and
progress possible. The proposed DLCT is to the continuous LCT, what the DFT is
to the continuous Fourier transform (FT). The DLCT of the signal is obtained
simply by multiplying the vector holding the samples of the input signal by the
DLCT matrix.
