  Signal scaling is a fundamental operation of practical importance in which a
signal is enlarged or shrunk in the coordinate direction(s). Scaling or
magnification is not trivial for signals of a discrete variable since the
signal values may not fall onto the discrete coordinate points. One approach is
to consider the discretely-spaced values as the samples of a signal of a real
variable, find that signal by interpolation, scale it, and then re-sample.
However, this approach comes with complications of interpretation. We review a
previously proposed alternative and more elegant approach, and then propose a
new approach based on hyperdifferential operator theory that we find most
satisfactory in terms of obtaining a self-consistent, pure, and elegant
definition of discrete scaling that is fully consistent with the theory of the
discrete Fourier transform.
