Continuous and Discrete-Time Filters: A Unified Operational Perspective

Abstract: This paper presents a unified tutorial treatment of continuous-time and discrete-time linear time-invariant systems, emphasizing their shared dynamical structure and the physical constraints that differentiate their realizations. Rather than introducing new mathematical tools, the manuscript revisits foundational concepts-transfer functions, poles and zeros, impulse responses, and stability-from an operational perspective rooted in practical signal processing and circuit implementation. First-order systems are used as a minimal yet expressive framework to illustrate how integration, differentiation, filtering, and delay manifest across the Laplace and Z domains. Particular attention is given to causality, bandwidth limitations, sampling effects, and the approximation errors inherent in discrete-time representations. The goal is to bridge the gap between formal mathematical descriptions and the intuition required for reliable system design in mixed analog-digital environments.