Counting up of linear lumped parameter systems and their unified representation

This paper discloses a lack of general theory for lumped parameter systems, containing continuous time systems, discrete time systems, systems with time delay, and sampled data systems. Also it provides a unified basis for linear systems of lumped parameters. The basis includes signal spaces mapped with real time functions and delta function series, operator notations of integral and time delay, modified sampled data, a new method to classify linear systems, and a unified representation of six classes of linear systems of time-invariant lumped parameters. For demonstrative examples of modified sampled data, the conventional sampled data system models of interpolating circuits are revised due to false counterexamples of Shannon's sampling theorem.