Unified framework for linear scale invariant signals, systems, and transforms: A tutorial

This paper presents a unified framework for linear scale invariant signals, systems, and transforms from a system theoretic perspective. The work is the scale counterpart of the theory related to linear shift invariant systems and transforms. Similar to Fourier and Laplace transforms that are used to study linear shift or time invariant systems, Mellin transform is used to study scale invariant systems. However, unlike the shift invariant theory, the theory related to scale invariant systems and transforms has so far not been presented with a unified approach. In this work, we present this theory from signal processing viewpoint, where we present the development of scale invariant transform as a systematic progression from scale series for scale periodic signals to scale invariant transform for scale aperiodic signals. We also present a few examples to illustrate the utility of the presented theory.