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.

Practice

1. What does it mean for a system to be stable in terms of its poles on the pole-zero plot?

easy

A. All poles must lie in the left half of the complex plane (negative real parts).

B. All poles must lie on the imaginary axis.

C. All zeros must lie in the right half of the complex plane.

D. Poles and zeros must be equal in number.

Solution

1. Step 1: Understand pole location and stability

   Stability depends on poles, not zeros. Poles with negative real parts mean the system's response decays over time.

2. Step 2: Identify correct pole location for stability

   Poles in the left half-plane (negative real parts) ensure the system is stable.

3. Final Answer:

   All poles must lie in the left half of the complex plane (negative real parts). -> Option A

4. Quick Check:

   Stable system = poles with negative real parts [OK]

Hint: Stable system means poles left of imaginary axis [OK]

Common Mistakes:

• Confusing zeros with poles for stability
• Thinking poles on imaginary axis are stable
• Assuming zeros affect stability directly

2. Which of the following is the correct Python syntax to plot poles and zeros using the scipy.signal library?

easy

A. signal.pzmap(system, plot=True)

B. signal.pzplot(system)

C. signal.pzplot(system, plot=True)

D. signal.pzmap(system, plot=False)

Solution

1. Step 1: Recall scipy.signal function for pole-zero plot

   The function to plot poles and zeros is pzmap, not pzplot.

2. Step 2: Check correct usage with plotting enabled

   Using pzmap with plot=True plots the pole-zero map correctly.

3. Final Answer:

   signal.pzmap(system, plot=True) -> Option A

4. Quick Check:

   Use pzmap with plot=True for pole-zero plot [OK]

Hint: Use pzmap with plot=True to plot poles and zeros [OK]

Common Mistakes:

• Using non-existent function pzplot
• Forgetting plot=True argument
• Confusing function names

3. Given the poles of a system as poles = [-1+2j, -1-2j, 0.5], what can you say about the system's stability?

medium

A. The system is stable because poles are complex conjugates.

B. The system is stable because all poles have negative real parts.

C. The system is marginally stable because poles are on the imaginary axis.

D. The system is unstable because one pole has a positive real part.

Solution

1. Step 1: Analyze the real parts of each pole

   Poles are -1+2j (real part -1), -1-2j (real part -1), and 0.5 (real part 0.5).

2. Step 2: Determine stability from pole real parts

   Since one pole has positive real part (0.5), the system is unstable.

3. Final Answer:

   The system is unstable because one pole has a positive real part. -> Option D

4. Quick Check:

   Positive real pole means unstable system [OK]

Hint: Check if any pole has positive real part for instability [OK]

Common Mistakes:

• Ignoring positive real part pole
• Assuming complex conjugates guarantee stability
• Confusing marginal stability with instability

4. The following Python code attempts to plot poles and zeros but gives an error:

from scipy import signal
system = signal.TransferFunction([1, 3], [1, 2, 1])
signal.pzplot(system)

What is the error and how to fix it?

medium

A. TransferFunction requires discrete=True for pzplot to work.

B. TransferFunction arguments are reversed; swap numerator and denominator.

C. Function pzplot does not exist; use pzmap(system, plot=True) instead.

D. Import statement is wrong; use import scipy.signal as signal.

Solution

1. Step 1: Identify the function causing error

   The function pzplot does not exist in scipy.signal, causing the error.

2. Step 2: Use correct function to plot poles and zeros

   Use pzmap(system, plot=True) to plot poles and zeros correctly.

3. Final Answer:

   Function pzplot does not exist; use pzmap(system, plot=True) instead. -> Option C

4. Quick Check:

   Use pzmap, not pzplot, for pole-zero plots [OK]

Hint: Use pzmap with plot=True, not pzplot [OK]

Common Mistakes:

• Using non-existent pzplot function
• Swapping numerator and denominator unnecessarily
• Wrong import statement causing confusion

5. You have a system with poles at -0.5, -1+1j, -1-1j and zeros at 0, -2. Which statement best describes the system's stability and response?

hard

A. The system is marginally stable because of zeros at zero.

B. The system is stable with oscillatory decay due to complex poles.

C. The system is unstable because zeros are on the right half-plane.

D. The system is unstable due to poles with positive real parts.

Solution

1. Step 1: Check pole locations for stability

   All poles have negative real parts (-0.5, -1), so system is stable.

2. Step 2: Analyze pole types for response behavior

   Complex conjugate poles (-1±1j) cause oscillations that decay over time.

3. Step 3: Consider zeros' effect on stability

   Zeros do not affect stability directly; zeros at 0 and -2 are not in right half-plane.

4. Final Answer:

   The system is stable with oscillatory decay due to complex poles. -> Option B

5. Quick Check:

   Negative real poles + complex = stable oscillatory decay [OK]

Hint: Stable if poles negative; complex poles cause oscillations [OK]

Common Mistakes:

• Confusing zeros with poles for stability
• Thinking zeros at zero cause instability
• Ignoring oscillatory nature of complex poles