Stability analysis (pole-zero plot) in Signal Processing - Mini Project: Build & Apply

Stability Analysis Using Pole-Zero Plot

📖 Scenario: Imagine you are an engineer working on a digital filter for audio processing. You want to check if your filter is stable by looking at its poles and zeros on the complex plane.

🎯 Goal: You will create a simple digital filter using its numerator and denominator coefficients, then plot its poles and zeros to analyze stability.

📋 What You'll Learn

Create numerator and denominator coefficient lists for the filter

Define a threshold radius for stability check

Calculate poles and zeros using the coefficients

Plot the poles and zeros on the complex plane

Print whether the filter is stable based on pole locations

💡 Why This Matters

🌍 Real World

Engineers use pole-zero plots to check if digital filters will behave well without causing unwanted oscillations or instability in audio, communications, or control systems.

💼 Career

Understanding filter stability is essential for signal processing roles in industries like telecommunications, audio engineering, and embedded systems design.

1

Create filter coefficients

Create a list called numerator with values [0.0675, 0.1349, 0.0675] and a list called denominator with values [1.0, -1.1430, 0.4128] representing the filter coefficients.

Signal Processing

# Create numerator and denominator coefficient lists
# Your code here

Hint

Use Python lists with the exact values given for numerator and denominator.

2

Set stability radius threshold

Create a variable called stability_radius and set it to 1.0 to represent the unit circle radius for stability check.

Signal Processing

numerator = [0.0675, 0.1349, 0.0675]
denominator = [1.0, -1.1430, 0.4128]
# Set stability radius threshold
# Your code here

Hint

Set stability_radius to 1.0 as a float value.

3

Calculate poles and zeros

Import numpy as np and use np.roots() to calculate zeros from numerator and poles from denominator.

Signal Processing

numerator = [0.0675, 0.1349, 0.0675]
denominator = [1.0, -1.1430, 0.4128]
stability_radius = 1.0
# Import numpy and calculate zeros and poles
# Your code here

Hint

Use np.roots() to find roots of numerator and denominator lists.

4

Plot poles and zeros and check stability

Import matplotlib.pyplot as plt. Plot zeros as blue circles and poles as red crosses on the complex plane. Draw a unit circle with radius stability_radius. Then print "Stable filter" if all poles are inside the unit circle, otherwise print "Unstable filter".

Signal Processing

import numpy as np

numerator = [0.0675, 0.1349, 0.0675]
denominator = [1.0, -1.1430, 0.4128]
stability_radius = 1.0

zeros = np.roots(numerator)
poles = np.roots(denominator)
# Import matplotlib, plot poles and zeros, draw unit circle, and print stability
# Your code here

Hint

Use plt.scatter() for poles and zeros, plt.Circle() for the unit circle, and check if all poles have magnitude less than stability_radius.