Pole-zero analysis for stability in Signal Processing - Step-by-Step Execution

Concept Flow - Pole-zero analysis for stability

Define system transfer function H(z)

↓

Find poles and zeros of H(z)

↓

Plot poles and zeros on z-plane

↓

Check pole locations relative to unit circle

↓

Determine system stability

↓

Stable if all poles inside unit circle

↓

Unstable if any pole on or outside unit circle

This flow shows how to analyze a system's stability by finding poles and zeros, plotting them, and checking if poles lie inside the unit circle.

Execution Sample

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import tf2zpk

b = [0.5, -0.5]
a = [1, -0.8]
zeros, poles, _ = tf2zpk(b, a)

This code finds the zeros and poles of a system defined by numerator b and denominator a coefficients.

Execution Table

Step	Action	Input	Output	Notes
1	Define numerator coefficients b	[0.5, -0.5]	b = [0.5, -0.5]	Represents zeros polynomial
2	Define denominator coefficients a	[1, -0.8]	a = [1, -0.8]	Represents poles polynomial
3	Calculate zeros and poles using tf2zpk	b, a	zeros = [1.0], poles = [0.8]	Zeros at z=1, poles at z=0.8
4	Plot poles and zeros on complex plane	zeros, poles	Plot with zeros as 'o', poles as 'x'	Visual check of locations
5	Check if poles inside unit circle	poles = [0.8]	0.8 < 1 (inside unit circle)	System is stable
6	End	-	-	Poles inside unit circle, stability confirmed

💡 Execution stops after confirming all poles are inside the unit circle, indicating stability.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	Final
b	undefined	[0.5, -0.5]	[0.5, -0.5]	[0.5, -0.5]	[0.5, -0.5]
a	undefined	undefined	[1, -0.8]	[1, -0.8]	[1, -0.8]
zeros	undefined	undefined	undefined	[1.0]	[1.0]
poles	undefined	undefined	undefined	[0.8]	[0.8]

Key Moments - 3 Insights

Why do we check if poles are inside the unit circle?

Because the system is stable only if all poles lie strictly inside the unit circle, as shown in step 5 of the execution_table.

What does a zero at z=1 mean for the system?

A zero at z=1 means the system output is zero at that frequency, but it does not affect stability, which depends only on poles (see step 3).

Can a pole on the unit circle be stable?

No, poles on or outside the unit circle cause instability or marginal stability; stability requires poles strictly inside, as confirmed in step 5.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what are the calculated poles?

A [0.5]

B [1.0]

C [0.8]

D [-0.8]

💡 Hint

Check the 'Output' column in step 3 of the execution_table.

At which step do we confirm the system is stable?

A Step 5

B Step 3

C Step 2

D Step 4

💡 Hint

Look for the step where poles are compared to the unit circle in the execution_table.

If a pole was at 1.2 instead of 0.8, what would change in the execution_table?

A Step 3 would show zeros changed

B Step 5 would show pole outside unit circle and system unstable

C Step 4 plot would not change

D No change in stability conclusion

💡 Hint

Consider the 'Check if poles inside unit circle' action in step 5.

Concept Snapshot

Pole-zero analysis checks system stability by:
- Finding poles and zeros from transfer function
- Plotting them on the complex z-plane
- Stability requires all poles inside the unit circle
- Zeros do not affect stability
- Poles on or outside unit circle mean instability

Full Transcript

Pole-zero analysis for stability involves defining the system's transfer function with numerator and denominator coefficients. We calculate zeros and poles using a function like tf2zpk. Then, we plot these on the complex plane to visualize their locations. The key step is checking if all poles lie inside the unit circle; if yes, the system is stable. Zeros do not affect stability. If any pole lies on or outside the unit circle, the system is unstable. This process helps us understand and ensure system stability in signal processing.