Stability analysis (pole-zero plot) in Signal Processing - Step-by-Step Execution

Concept Flow - Stability analysis (pole-zero plot)

Define system transfer function

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Calculate poles and zeros

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Plot poles and zeros on complex plane

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Check pole locations

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Inside unit circle?

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System is stable

Start by defining the system, find poles and zeros, plot them, then check if poles lie inside the unit circle to decide stability.

Execution Sample

Signal Processing

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

b = [0.5, -0.5]
a = [1, -0.8]
z, p, k = tf2zpk(b, a)
plt.scatter(np.real(z), np.imag(z), marker='o', label='Zeros')
plt.scatter(np.real(p), np.imag(p), marker='x', label='Poles')
plt.gca().add_artist(Circle((0,0),1,color='r',fill=False,ls='--'))
plt.legend()
plt.show()

This code finds poles and zeros of a system and plots them on the complex plane with the unit circle.

Execution Table

Step	Action	Poles (p)	Zeros (z)	Check Pole Radius	Stability Decision
1	Define numerator coefficients b=[0.5, -0.5]	-	-	-	-
2	Define denominator coefficients a=[1, -0.8]	-	-	-	-
3	Calculate zeros and poles using tf2zpk	p=[0.8]	z=[1.0]	-	-
4	Calculate radius of pole |0.8|	0.8	-	0.8 < 1	-
5	Since all poles have radius < 1	-	-	-	System is stable
6	Plot poles (x) and zeros (o) on complex plane	p=[0.8]	z=[1.0]	-	-
7	Draw unit circle for reference	-	-	-	-
8	End of analysis	-	-	-	-

💡 All poles have magnitude less than 1, so system is stable.

Variable Tracker

Variable	Start	After Step 3	After Step 4	Final
b (numerator coeffs)	[ ]	[0.5, -0.5]	[0.5, -0.5]	[0.5, -0.5]
a (denominator coeffs)	[ ]	[1, -0.8]	[1, -0.8]	[1, -0.8]
z (zeros)	[ ]	[1.0]	[1.0]	[1.0]
p (poles)	[ ]	[0.8]	[0.8]	[0.8]
pole radius	[ ]	[ ]	0.8	0.8

Key Moments - 3 Insights

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

Because poles inside the unit circle mean the system's output will not grow unbounded over time, indicating stability (see Step 4 and 5 in execution_table).

What is the difference between poles and zeros on the plot?

Poles (marked with 'x') are points where the system's response can become infinite, zeros (marked with 'o') are points where the output is zero; stability depends on poles only (see Step 6).

Why is the unit circle drawn on the plot?

The unit circle helps visually check if poles lie inside it, which is the key criterion for stability (see Step 7).

Visual Quiz - 3 Questions

Look at the execution_table at Step 4, what is the radius of the pole?

A 1.0

B 0.5

C 0.8

D 1.2

💡 Hint

Check the 'Check Pole Radius' column at Step 4 in execution_table.

At which step does the system get declared stable?

A Step 3

B Step 5

C Step 6

D Step 7

💡 Hint

Look at the 'Stability Decision' column in execution_table.

If a pole had radius 1.2, what would change in the execution_table?

A Stability Decision would say 'System is unstable'

B Zeros would move outside unit circle

C Poles would be marked with 'o' instead of 'x'

D No change in stability decision

💡 Hint

Refer to Step 5 where stability depends on pole radius.

Concept Snapshot

Stability analysis uses poles and zeros of a system.
Poles are roots of denominator; zeros are roots of numerator.
Plot poles (x) and zeros (o) on complex plane.
If all poles lie inside unit circle (radius < 1), system is stable.
Unit circle is drawn as reference on plot.
Stability depends only on poles, not zeros.

Full Transcript

We start by defining the system's numerator and denominator coefficients. Then we calculate the zeros and poles using a function. We plot these on the complex plane, marking zeros with circles and poles with crosses. We draw the unit circle to check visually. The key step is checking the radius of each pole. If all poles have radius less than 1, the system is stable. This is because poles inside the unit circle mean the system's output will not grow without bound. The zeros do not affect stability but are shown for completeness. The plot and radius check together give a clear stability analysis.