Stability analysis (pole-zero plot) in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding system stability from pole locations

Consider a discrete-time system with poles at 0.5, 0.8 + 0.3j, and 1.1. Which statement correctly describes the system's stability?

A The system is stable because all poles are inside the unit circle.

B The system is marginally stable because all poles lie on the unit circle.

C The system is unstable because one pole lies outside the unit circle.

D The system is stable because poles on the unit circle do not affect stability.

💡 Hint

Recall that for discrete-time systems, stability requires all poles to be strictly inside the unit circle.

Poles inside the unit circle mean the system's response decays over time. A pole at 1.1 lies outside the unit circle, causing instability.

❓ data_output

intermediate

Number of poles inside the unit circle

Given the poles of a system as [-0.9, 0.7 + 0.6j, 0.7 - 0.6j, 1.0], how many poles lie strictly inside the unit circle?

A 1

B 2

C 4

D 3

💡 Hint

Calculate the magnitude of each pole and count those with magnitude less than 1.

The pole at 1.0 lies exactly on the unit circle and is not inside. The other three poles have magnitudes less than 1.

❓ Predict Output

advanced

Output of pole-zero plot code snippet

What does the following Python code output?

import numpy as np
poles = np.array([0.5, 0.8 + 0.3j, 1.1])
inside_unit_circle = np.sum(np.abs(poles) < 1)
print(inside_unit_circle)

A 2

B 3

C 1

D 0

💡 Hint

Check the magnitude of each pole and count how many are less than 1.

Poles 0.5 and 0.8+0.3j have magnitudes less than 1, so count is 2.

❓ visualization

advanced

Interpreting a pole-zero plot for stability

You see a pole-zero plot with poles marked as 'x' and zeros as 'o'. Poles are at 0.9, -0.95, and 1.05 on the real axis. Which statement is true about the system's stability?

A The system is unstable because a pole lies outside the unit circle.

B The system is stable because all poles are inside the unit circle.

C The system is marginally stable because poles lie on the unit circle.

D The system is stable because zeros outside the unit circle do not affect stability.

💡 Hint

Focus on the poles' locations relative to the unit circle.

The pole at 1.05 lies outside the unit circle, causing instability regardless of zeros.

🚀 Application

expert

Determining system stability from transfer function poles

Given the transfer function H(z) = (z - 0.5) / (z^2 - 1.2z + 0.36), determine the system's stability by analyzing the poles.

Signal Processing

import numpy as np
coefficients = [1, -1.2, 0.36]
poles = np.roots(coefficients)
stable = all(np.abs(poles) < 1)
print(f"Poles: {poles}")
print(f"Is system stable? {stable}")

A Poles are [0.6, 0.6], system is unstable.

B Poles are [0.6, 0.6], system is stable.

C Poles are [0.9, 0.4], system is stable.

D Poles are [1.2, -0.3], system is unstable.

💡 Hint

Find roots of denominator polynomial and check if their magnitudes are less than 1.

The roots of z^2 - 1.2z + 0.36 are both 0.6, which are inside the unit circle, so the system is stable.