Start from the governing relation
\[x[n]=e^{snT_s}\Rightarrow z=e^{sT_s}\]
This is the smallest equation that preserves the physics needed for the formula.
Choose the model state or domain.
FORMULA / 003
Z-transform: foundation of discrete systems
Interactive derivation module for Z-transform: foundation of discrete systems.
\[X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n}\]
Connects the formula to engineering interpretation, limits, and computation.
Problem Definition
Problem statement and variable definition
Sampled firmware sees only x[n], not the continuous waveform between samples.
Core variable legend
| Symbol | Quantity | Unit | Meaning |
|---|---|---|---|
| $z$ | Complex discrete frequency | 無 | Discrete-time growth and rotation factor. |
| $T_s$ | Sampling period | s | Time between samples. |
| $n$ | Sample index | 無 | Discrete time counter. |
Analytical Derivation
Model assumptions and analytical derivation
Model simplifications
| Item | Assumption |
|---|---|
| Assumption 1 | Linear time-invariant or locally linear behavior is assumed when using transfer functions. |
| Assumption 2 | Parameters are treated as constant over the analysis interval. |
| Assumption 3 | Numerical plots are educational approximations, not full circuit or field solvers. |
Apply engineering assumptions
\[x[n]=e^{snT_s}\Rightarrow z=e^{sT_s}\]
The model is simplified so the dominant mechanism is visible.
Drop second-order parasitics unless they are the topic.
Rearrange to the working formula
\[H(z)\text{ stable}\Rightarrow |p_i|<1\]
The final form is the one used for design, simulation, or measurement review.
Algebraic rearrangement.
parameter $\to0$dominant term remainsThe formula reduces to its simplest physical behavior.
parameter $\to\infty$parasitics and implementation limits dominateThe closed-form equation becomes a warning rather than a full implementation model.
Parametric Verification
Parametric simulation and numerical verification
Use the sliders to change the physical parameters and watch the formula expose its behavior. The metric panel keeps sample count and compute latency visible so the visualization remains an engineering instrument, not only a picture.
INTERACTIVE MODULE
Discrete pole radius across time
Engineering Application
Engineering application and implementation notes
Design rules
- Check units before trusting numeric output.
- Sweep the key parameter through the critical region, not only the nominal value.
- Validate the simplified formula against measurement or simulation before committing hardware.
Core algorithm
python
import numpy as np
def z_pole_from_s(pole_s, ts):
return np.exp(pole_s * ts)