Continuous and Discrete-Time Filters: A Unified Operational Perspective

TL;DR

This paper unifies the analysis of continuous and discrete-time linear systems by highlighting their shared structures, stability, and transfer function representations, bridging the gap between the two formalisms.

Contribution

It provides a unified operational framework connecting continuous and discrete-time systems through transfer functions, pole-zero analysis, and discretization methods.

Findings

Unified treatment of CT and DT systems emphasizing modal structure.

Illustration of the correspondence between s-plane and z-plane regions.

Analysis of discretization and sampled data implementation methods.

Abstract

Continuous time (CT) and discrete time (DT) linear time invariant (LTI) systems are commonly introduced through distinct mathematical formalisms, which can obscure their underlying dynamical equivalence. This tutorial presents a unified treatment of firstorder CT and DT systems, emphasizing their shared modal structure and stability properties. Beginning with transfer functions and pole zero representations in the Laplace domain, canonical first order low pass and high-pass dynamics are examined from an operational perspective. The discussion then transitions to discrete-time sequences and the Z transform, highlighting geometric sequences as eigenfunctions of DT systems and establishing the correspondence between the left half of the s plane and the interior of the unit circle in the z plane. Practical discretization and sampled data implementations are analyzed to illustrate how continuous time dynamics are reinterpreted through recursion and accumulation in discrete time realizations. By maintaining structural symmetry between domains, the manuscript consolidates established concepts into a coherent framework linking mathematical representation, physical realizability, and implementation.