⚡ electrical engineering

Continuous and Discrete-Time Filters: A Unified Operational Perspective

This tutorial unifies the mathematical formalisms of continuous and discrete-time LTI systems by demonstrating their shared modal structure and stability properties, thereby establishing a coherent framework that links Laplace and Z-domain representations through the correspondence between the s-plane and z-plane.

Original authors: Luca Giangrande

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luca Giangrande

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot how to react to the world. You have two ways to describe how it should behave:

The "Smooth Movie" View (Continuous-Time): The robot sees the world as a smooth, flowing movie where everything happens instantly and continuously. This is how the real, physical world works (like water flowing in a river).
The "Flipbook" View (Discrete-Time): The robot sees the world as a series of still photos taken one after another. It only knows what happened in the last photo and needs to guess what's happening now. This is how computers and digital chips work.

The Problem:
Usually, engineers teach these two views as if they are completely different languages. One uses "differential equations" (math for smooth flows), and the other uses "difference equations" (math for step-by-step jumps). This makes it hard to see that they are actually describing the same underlying behavior, just with different rules.

The Solution (This Paper):
Luca Giangrande's paper is like a translator's guide that shows how these two views are actually twins. He focuses on the simplest possible filters (the building blocks of signal processing) to prove that they share the same DNA.

Here is the breakdown using everyday analogies:

1. The Two Worlds: Smooth vs. Stepped

Continuous-Time (The River): Imagine a river flowing. If you drop a leaf in, it moves smoothly. In math, we use the Laplace Transform to describe this. The "poles" (which determine if the system is stable or chaotic) are like the depth of the river. If the pole is in the "left half" of the map, the river flows calmly to a stop (stable).
Discrete-Time (The Staircase): Now imagine the river is frozen into a staircase. You can only move from one step to the next. In math, we use the Z-Transform. Here, the "poles" are like the distance between steps. If the pole is inside a "unit circle" (a safe zone on the map), the system is stable.

The Big Reveal: The paper shows that the "left half" of the smooth river map is mathematically identical to the "inside of the circle" in the staircase map. They are just different ways of drawing the same safety zone.

2. The Two Main Characters: Low-Pass and High-Pass

The paper focuses on two simple characters that appear in both worlds:

The Low-Pass Filter (The "Leaky Bucket"):
- What it does: It lets slow changes through but blocks fast, jittery noise.
- Analogy: Imagine a bucket with a small hole in the bottom. If you pour water in slowly, it fills up (the signal passes). If you splash water in violently (noise), the hole drains it away before it builds up.
- In the paper: Whether it's a smooth bucket (Continuous) or a bucket that gets filled and drained one cup at a time (Discrete), the math shows they both act as a "weighted average." They remember the past but care more about the recent past.
The High-Pass Filter (The "Speed Bump"):
- What it does: It blocks slow, steady changes but lets fast, sudden changes through.
- Analogy: Imagine a speed bump on a road. A car driving at a steady, slow speed (DC) just rolls over it without noticing. But a car that suddenly accelerates or brakes (a fast change) hits the bump hard.
- In the paper: The paper explains that a digital high-pass filter is just a "speed bump" made of math. It takes the current photo, subtracts the previous photo, and shows you the difference. If nothing changed, the result is zero. If something changed fast, the result is big.

3. The Magic Trick: Turning Smooth into Stepped

How do we turn a smooth river into a staircase? The paper discusses Discretization.

The Analogy: Imagine you are taking a video of a runner. If you take a photo every second, you get a smooth-looking movie. If you take a photo every hour, the runner looks like they are teleporting.
The Paper's Insight: You can't just copy-paste the math from the smooth world to the stepped world. You have to "translate" the poles.
- If a smooth system has a pole at a certain speed, the digital version needs a pole at a specific "step size" to behave the same way.
- The paper shows that if you get this translation wrong, your digital filter might become unstable (explode) or act weirdly, even if the smooth version was perfect.

4. Why This Matters

The author argues that engineers often treat digital filters as if they are just "approximations" of analog ones. They are not.

The Metaphor: It's like saying a digital photo is just a "bad copy" of a painting. No, a digital photo is a different medium with its own rules, strengths, and weaknesses.
The Takeaway: By understanding that the "smooth river" and the "stepped staircase" are structurally identical twins, engineers can design better systems. They can take a concept from the physical world (like a capacitor) and build a digital version that behaves exactly the same way, without the "glitches" that happen when you try to force one language into the other.

Summary

This paper is a unified field theory for filters. It tells us:

Don't be scared of the math: Whether you are dealing with smooth waves or digital steps, the core rules of stability and memory are the same.
Poles are the heartbeat: In both worlds, the location of the "poles" tells you if the system will settle down or go crazy.
Digital is not just a copy: Digital filters reinterpret physical laws through the lens of "accumulation" (adding up steps) and "recursion" (using the past to predict the future).

In short, the paper bridges the gap between the analog world (where things flow) and the digital world (where things count), showing that they are speaking the same language, just with different accents.

1. Problem Statement

Continuous-time (CT) and discrete-time (DT) linear time-invariant (LTI) systems are traditionally taught and analyzed using distinct mathematical formalisms: differential equations and the Laplace transform for CT, versus difference equations and the Z-transform for DT. While this separation is mathematically convenient, the author argues that it obscures the underlying dynamical equivalence between the two domains.

Engineers frequently transition between CT and DT representations when designing filters and control loops. However, this separation often leads to:

Misinterpretation of physical behaviors (e.g., stability, causality, and memory) when translating designs.
Unexpected system behaviors when moving from analog to digital implementations.
A lack of intuitive understanding of how continuous dynamics (integration/differentiation) are reinterpreted through recursion and accumulation in discrete systems.

The paper addresses the need for a unified framework that links mathematical representation, physical realizability, and implementation structure across both domains, specifically focusing on first-order systems as a minimal yet expressive model.

2. Methodology

The paper employs a comparative analytical approach, systematically mapping CT concepts to their DT counterparts to highlight structural symmetries.

Unified Operational Perspective: The analysis begins with CT transfer functions and pole-zero representations in the Laplace domain, then transitions to DT sequences and the Z-transform.
Eigenfunction Analysis: The core methodology relies on identifying the eigenfunctions of the systems.
- In CT, complex exponentials ( $e^{st}$ ) are the eigenfunctions of differential operators.
- In DT, geometric sequences ( $c^n$ ) are the eigenfunctions of difference operators.
Modal Decomposition: The paper analyzes system responses by decomposing them into modal contributions (poles and zeros), demonstrating how the location of poles dictates stability and transient behavior in both domains.
Discretization Strategies: The author examines practical conversion methods, specifically the backward-Euler (backward difference) and bilinear (Tustin) transforms, to show how continuous poles map to the discrete plane.
Modular Implementation: The paper constructs filter architectures using basic building blocks (unit delays, summation, scaling, and feedback) to illustrate how CT dynamics are physically realized in DT.

3. Key Contributions

A. Unification of Stability and Dynamics

The paper establishes a rigorous correspondence between the stability criteria of both domains:

CT: Stability requires poles to lie in the Left Half Plane (LHP) of the $s$ -plane (negative real part).
DT: Stability requires poles to lie inside the unit circle of the $z$ -plane ( $|z| < 1$ ).
Mapping: The transformation $z = e^{sT_s}$ maps the LHP to the interior of the unit circle, preserving the concept of exponential decay (CT) as geometric decay (DT).

B. Reinterpretation of Integration and Differentiation

Integration:
- CT: An ideal integrator has a pole at $s=0$ .
- DT: An ideal accumulator (discrete integrator) has a pole at $z=1$ . The paper shows that a "leaky integrator" (low-pass filter) in DT is a direct analog to a damped CT integrator, where the pole is moved inside the unit circle to ensure stability.
Differentiation:
- CT: An ideal differentiator has a zero at $s=0$ but is physically unrealizable due to infinite gain at high frequencies.
- DT: A discrete differentiator is a two-tap FIR filter ( $1 - z^{-1}$ ). The paper demonstrates that a stable discrete high-pass filter (differentiator) is realized by combining a differentiator with a low-pass pole (leaky accumulation), effectively smoothing the derivative to ensure physical realizability.

C. Physical Realizability and Parasitics

The paper emphasizes that "ideal" mathematical models (infinite gain, zero delay) are physically impossible. It introduces:

Properness: The constraint that the number of zeros cannot exceed the number of poles ( $m \le n$ ) to ensure finite high-frequency gain.
Parasitic Effects: How real-world components (capacitance, inductance) introduce non-dominant poles that limit bandwidth and prevent infinite gain, effectively acting as low-pass filters on differentiators.
Delay: The inclusion of propagation delay ( $e^{-s\tau_d}$ ) in CT and its discrete equivalent, showing how delay enforces strict causality.

D. Modular Filter Architectures

The author provides block diagrams for first-order DT systems constructed from unit delays and arithmetic operations:

FIR Differentiator: A zero at $z=1$ (DC) creates a high-pass characteristic.
IIR Accumulator: A pole at $z=1$ creates an ideal integrator (unstable in open loop).
Leaky Integrator/High-Pass: Moving the pole inside the unit circle creates stable, realizable filters that approximate CT behavior.
Oscillators: A pole at $z=-1$ creates a discrete oscillator at the Nyquist frequency, illustrating how negative poles induce sign alternation.

4. Results

Structural Symmetry: The paper successfully demonstrates that the mathematical structure of first-order CT and DT systems is identical when viewed through the lens of pole-zero constellations and modal decomposition.
Frequency Response Mapping: The analysis confirms that the frequency response of a DT system is a periodic repetition of the CT response, restricted to the Nyquist range ($0$ to $F_s/2$ ). It highlights aliasing as the mechanism where frequencies above Nyquist fold back, and spectral leakage as a consequence of finite observation windows.
Discretization Accuracy: The backward-Euler method is shown to preserve stability (mapping LHP to the unit circle) but introduces frequency warping. The paper illustrates how the choice of discretization method affects the fidelity of the filter's cutoff frequency and phase response.
Transient Behavior: The paper visualizes how CT impulse responses (exponential decay) translate to DT impulse responses (geometric sequences), and how step responses in DT (ramps) correspond to CT ramps, provided the sampling rate is sufficient.

5. Significance

Pedagogical Value: The paper provides a cohesive framework for teaching signal processing, helping students and engineers bridge the gap between analog theory and digital implementation without treating them as disjoint subjects.
Design Robustness: By unifying the perspective, engineers can better predict how a filter designed in the continuous domain will behave when sampled. It prevents common pitfalls, such as assuming a digital differentiator behaves exactly like an analog one without considering stability and aliasing.
Implementation Insight: The focus on "operational perspective" (how systems are built from delays and sums) aids in the design of mixed-signal and fully digital systems, particularly in fields like control theory and particle physics (the author's domain at CERN).
Scalability: While focused on first-order systems, the paper argues that these principles of modal decomposition, pole mapping, and stability constraints naturally extend to higher-order, multirate, and noise-shaped systems, making the unified approach a foundational tool for advanced signal processing.

In conclusion, the paper asserts that CT and DT filters are not merely approximations of one another but are reinterpretations of the same dynamical laws under different operational constraints (sampling and accumulation). Recognizing this continuity is essential for robust system design.