Discrete signal scaling is a basic operation of central importance that poses several challenges, which are usually addressed as an interpolation problem. Our approach, which does not involve interpolation in the conventional sense, provides a novel and unique perspective to this problem, and may be useful in a wide variety of theoretical and practical contexts. We also believe that the introduction of hyperdifferential operator theory to the problem of defining a discrete operation might also open up new research directions for other operations that are important for signal analysis and processing.
Signal scaling is a fundamental operation of practical importance in which a signal is made wider or narrower along the coordinate direction(s). Scaling, also referred to as magnification or zooming, is complicated for signals of a discrete variable since it cannot be accomplished simply by moving the signal values to new coordinate points. Most practical approaches to discrete scaling involve interpolation. We propose a new approach based on hyperdifferential operator theory that does not involve conventional interpolation. This approach provides a self-consistent and pure definition of discrete scaling that is fully consistent with discrete Fourier transform theory. It can potentially be applied to other problems in signal theory and analysis such as transform designs. Apart from its theoretical elegance, it also provides a basis for numerical implementation.