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.

信号缩放是具有实际重要性的基本操作，其中使信号沿着坐标方向变宽或变窄。对于离散变量的信号，缩放（也称为放大或缩放）很复杂，因为无法简单地通过将信号值移动到新的坐标点来完成缩放。离散缩放的最实用方法涉及插值。我们提出了一种基于超微分算子理论的新方法，该方法不涉及常规插值。这种方法提供了独立的，纯粹的离散缩放定义，与离散傅里叶变换理论完全一致。它可以潜在地应用于信号理论和分析中的其他问题，例如变换设计。除了理论上的优雅之外，