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 Tutorial Home 1. What are Time Scales? 2. Definition of a Time Scale 3. Notation 4. Differentiation and Integration on Time Scales 5a. Operations Applet 6a. Applet Examples TimeScales.org
 1. What are Time Scales? The theory of time scales was introduced in Stefan Hilger's 1988 Ph.D. dissertation and subsequent landmark paper as a way to unify the seemingly disparate fields of discrete dynamical systems (i.e. difference equations) and continuous dynamical systems (i.e. differential equations). Although these two fields are similar in certain instances while very different in others, Hilger reasoned that there must be an underlying mathematical structure to explain when and why the theories coalesce or differ. Time scale systems might best be understood as the continuum bridge between discrete time and continuous time systems. Signals and systems are taught in today‘s undergraduate engineering curriculum in two distinct phases: discrete and continuous. The richness in dual properties of the two domains commonly prompts simultaneous presentation of the two theories to the student. Some of the dual properties of continuous and discrete time systems are listed in the following table. As we continue with this tutorial, the meaning of the entries in this table will become clearer.   