Pole-zero analysis for stability in Signal Processing - Mini Project: Build & Apply

Pole-zero analysis for stability

📖 Scenario: Imagine you are working with a simple digital filter in signal processing. You want to check if this filter is stable by looking at its poles and zeros.In real life, engineers check filter stability to make sure signals don't get distorted or blow up.

🎯 Goal: You will create a dictionary of filter coefficients, set a stability threshold, find poles and zeros, and finally check if the filter is stable.

📋 What You'll Learn

Create a dictionary called filter_coeffs with keys 'b' and 'a' for numerator and denominator coefficients.

Create a variable called stability_radius and set it to 1.

Use scipy.signal.tf2zpk to find poles and zeros from filter_coeffs['b'] and filter_coeffs['a'].

Check if all poles have absolute value less than stability_radius to determine stability.

Print poles, zeros, and is_stable.

💡 Why This Matters

🌍 Real World

Engineers use pole-zero analysis to check if digital filters will behave well and not cause signal distortion or instability.

💼 Career

Understanding filter stability is important for roles in signal processing, audio engineering, communications, and control systems.

1

Create filter coefficients dictionary

Create a dictionary called filter_coeffs with keys 'b' and 'a'. Set filter_coeffs['b'] to the list [0.0675, 0.1349, 0.0675] and filter_coeffs['a'] to the list [1.0, -1.1430, 0.4128].

Signal Processing

# Create the filter_coeffs dictionary with keys 'b' and 'a'
# Your code here

Hint

Use a dictionary with keys 'b' and 'a' and assign the exact lists as values.

2

Set stability radius

Create a variable called stability_radius and set it to 1.

Signal Processing

filter_coeffs = {'b': [0.0675, 0.1349, 0.0675], 'a': [1.0, -1.1430, 0.4128]}
# Set stability_radius to 1
# Your code here

Hint

Just assign the number 1 to the variable stability_radius.

3

Find poles and zeros

Import scipy.signal and use scipy.signal.tf2zpk with filter_coeffs['b'] and filter_coeffs['a'] to find zeros, poles, and gain. Then create a variable is_stable that is True if all poles have absolute value less than stability_radius, otherwise False.

Signal Processing

filter_coeffs = {'b': [0.0675, 0.1349, 0.0675], 'a': [1.0, -1.1430, 0.4128]}
stability_radius = 1
# Import scipy.signal and find zeros, poles, gain
# Then check if all poles are inside the stability_radius
# Your code here

Hint

Use tf2zpk to get zeros, poles, gain. Use a generator expression with all() to check poles.

4

Print poles, zeros, and stability

Print the variables poles, zeros, and is_stable each on a separate line.

Signal Processing

import scipy.signal

filter_coeffs = {'b': [0.0675, 0.1349, 0.0675], 'a': [1.0, -1.1430, 0.4128]}
stability_radius = 1

zeros, poles, gain = scipy.signal.tf2zpk(filter_coeffs['b'], filter_coeffs['a'])
is_stable = all(abs(pole) < stability_radius for pole in poles)
# Print poles, zeros, and is_stable
# Your code here

Hint

Use three print statements to show poles, zeros, and stability result.