Stability of Discrete-Time Systems
Given Poles
z = 0 , z = -0.9 , z = 0.9
What is Stability?
A discrete-time system is BIBO Stable (Bounded Input Bounded Output) if every bounded input produces a bounded output.
Stability Condition:
For a discrete-time LTI system:
All poles must lie inside the unit circle.
|z| < 1
Unit Circle in Z-Plane
The unit circle is a circle centered at the origin with radius = 1.
- Inside circle → Stable
- On circle → Marginally Stable
- Outside circle → Unstable
Check Each Pole
| Pole | Magnitude | Inside Unit Circle? |
|---|---|---|
| z = 0 | |0| = 0 | Yes |
| z = -0.9 | |-0.9| = 0.9 | Yes |
| z = 0.9 | |0.9| = 0.9 | Yes |
Conclusion
All poles lie inside the unit circle → The system is STABLE.
Steps to Check Stability
- Find the poles of the system.
- Calculate magnitude of each pole.
- Compare magnitude with 1.
- If all |z| < 1 → Stable.
Examples
| Poles | Result |
|---|---|
| 0.5 , -0.3 | Stable |
| 1 , -0.5 | Marginally Stable |
| 1.2 , 0.4 | Unstable |
Practice Problems
Determine stability for the following poles:
- z = 0.2 , 0.3
- z = 1 , -1
- z = 1.3 , 0.7
- z = -0.8 , 0.6
Z-Plane Visualization
The poles (0, −0.9, 0.9) are plotted below.
Quick Exam Rule
|z| < 1 → Stable
|z| = 1 → Marginally Stable
|z| > 1 → Unstable
|z| = 1 → Marginally Stable
|z| > 1 → Unstable