Continuous and Discrete-Time Filters: A Unified Operational Perspective
Luca Giangrande
TL;DR
The paper addresses the fragmentation between continuous-time and discrete-time signal processing by presenting a unified framework based on first-order LTI dynamics. It develops CT transfer functions and impulse responses, then systematically maps them to the DT domain using the Z-transform, DTFT, and DFT, highlighting pole-zero mappings, stability on the unit circle, and discretization rules such as backward-Euler and bilinear transforms. The key contributions include an operational, intuition-building treatment of CT and DT systems, explicit relations between $H(s)$ and $H(z)$, and modular discretized architectures for sampled-data filters that preserve causal, stable behavior. The work provides practical guidance for mixed analog-digital design, showing how CT intuition translates into robust DT implementations across sampling, delay, and implementation constraints.
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.