Pole-zero analysis for stability in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a pole in the context of pole-zero analysis?

A pole is a value of the complex variable where the system's transfer function becomes infinite. It represents natural frequencies where the system's output can grow very large.

beginner

What does a zero represent in pole-zero analysis?

A zero is a value of the complex variable where the system's transfer function becomes zero. It indicates frequencies that the system blocks or cancels out.

intermediate

How do poles relate to system stability?

For a system to be stable, all poles must lie inside the unit circle in the z-plane (discrete systems) or in the left half of the s-plane (continuous systems). Poles outside these regions cause instability.

intermediate

Why is the unit circle important in discrete-time pole-zero analysis?

The unit circle defines the boundary for stability in discrete-time systems. Poles inside the unit circle mean the system's output will not grow unbounded over time.

advanced

What happens if a pole lies exactly on the unit circle in discrete-time systems?

If a pole lies exactly on the unit circle, the system is marginally stable. The output neither grows nor decays but can oscillate indefinitely.

Where must all poles lie for a discrete-time system to be stable?

A Anywhere in the complex plane

B Outside the unit circle

C On the unit circle

D Inside the unit circle

For discrete-time systems, stability requires all poles to be inside the unit circle in the complex plane.

What does a zero of a system's transfer function indicate?

A Frequency where output is zero

B Frequency where output is infinite

C Frequency where input is zero

D Frequency where system is unstable

Zeros are frequencies where the system's output becomes zero, effectively blocking those frequencies.

In continuous-time systems, where must poles lie for stability?

A Right half of the s-plane

B On the imaginary axis

C Left half of the s-plane

D Anywhere in the s-plane

Continuous-time systems are stable only if all poles lie in the left half of the s-plane.

What is the effect of a pole on the unit circle in a discrete system?

A System is marginally stable

B System is unstable

C System is stable

D System output is zero

A pole on the unit circle means the system is marginally stable, causing sustained oscillations.

Which of the following best describes a pole-zero plot?

A Plot showing input vs output

B Graph of poles and zeros in the complex plane

C Time-domain signal plot

D Frequency response magnitude plot

A pole-zero plot shows the locations of poles and zeros in the complex plane, helping analyze system behavior.

Explain how pole locations affect the stability of a discrete-time system.

Think about how the system's output behaves over time depending on pole positions.

Describe the difference between poles and zeros in a system's transfer function.

Consider what happens to the system output at poles and zeros.

Practice

1. What does the location of poles inside the unit circle on a pole-zero plot indicate about a system's stability?

easy

A. The system is stable

B. The system is unstable

C. The system is marginally stable

D. The system has no poles

Solution

1. Step 1: Understand pole location and stability

   Poles inside the unit circle mean the system's response decays over time.

2. Step 2: Relate pole location to system behavior

   Since poles are inside the unit circle, the system output will not grow unbounded.

3. Final Answer:

   The system is stable -> Option A

4. Quick Check:

   Poles inside unit circle = stable system [OK]

Hint: Poles inside unit circle mean stability [OK]

Common Mistakes:

• Confusing poles inside with outside the unit circle
• Thinking zeros affect stability directly
• Ignoring the unit circle boundary

2. Which of the following is the correct Python syntax to plot poles and zeros using matplotlib for a system with poles at [0.5+0.5j, 0.5-0.5j] and zeros at [0.8]?

easy

A. plt.scatter([0.5, 0.5], [0.5, -0.5], marker='x', color='r'); plt.scatter([0.8], [0], marker='o', color='b')

B. plt.plot([0.5, 0.5], [0.5, -0.5], 'ro'); plt.plot([0.8], [0], 'bo')

C. plt.scatter([0.5, 0.5], [0.5, -0.5], marker='o', color='b'); plt.scatter([0.8], [0], marker='x', color='r')

D. plt.plot([0.5, 0.5], [0.5, -0.5], 'bx'); plt.plot([0.8], [0], 'ro')

Solution

1. Step 1: Identify correct markers for poles and zeros

   Poles are usually marked with 'x' in red, zeros with 'o' in blue.

2. Step 2: Match syntax with matplotlib scatter function

   plt.scatter([0.5, 0.5], [0.5, -0.5], marker='x', color='r'); plt.scatter([0.8], [0], marker='o', color='b') uses plt.scatter with correct markers and colors for poles and zeros.

3. Final Answer:

   plt.scatter([0.5, 0.5], [0.5, -0.5], marker='x', color='r'); plt.scatter([0.8], [0], marker='o', color='b') -> Option A

4. Quick Check:

   Correct markers and colors for poles and zeros [OK]

Hint: Poles = red 'x', zeros = blue 'o' in plots [OK]

Common Mistakes:

• Swapping markers for poles and zeros
• Using plt.plot with wrong format strings
• Plotting zeros at wrong coordinates

3. Given the poles at [0.9, 1.1] and zeros at [0.5], what will be the stability status of the system?

medium

A. Stable because one pole is inside the unit circle

B. Stable because zeros are inside the unit circle

C. Marginally stable because poles are on the unit circle

D. Unstable because one pole is outside the unit circle

Solution

1. Step 1: Check each pole's location relative to the unit circle

   One pole is at 0.9 (inside), the other at 1.1 (outside) the unit circle.

2. Step 2: Determine stability from pole locations

   Since a pole lies outside the unit circle, the system is unstable regardless of zeros.

3. Final Answer:

   Unstable because one pole is outside the unit circle -> Option D

4. Quick Check:

   Any pole outside unit circle = unstable [OK]

Hint: Any pole outside unit circle means unstable system [OK]

Common Mistakes:

• Ignoring poles outside the unit circle
• Confusing zeros with poles for stability
• Assuming zeros affect stability directly

4. The following Python code attempts to plot poles and zeros but produces an error. What is the error and how to fix it?

import matplotlib.pyplot as plt
poles = [0.7+0.7j, 0.7-0.7j]
zeros = [0.9]
plt.scatter(poles.real, poles.imag, marker='x', color='r')
plt.scatter(zeros.real, zeros.imag, marker='o', color='b')
plt.show()

medium

A. Error because marker symbols are incorrect

B. Error because poles and zeros are lists, need numpy arrays for .real and .imag

C. Error because plt.show() is missing

D. No error, code runs fine

Solution

1. Step 1: Identify data types of poles and zeros

   Poles and zeros are Python lists of complex numbers, which do not support .real or .imag attributes directly.

2. Step 2: Fix by converting lists to numpy arrays

   Convert poles and zeros to numpy arrays to access .real and .imag properties without error.

3. Final Answer:

   Error because poles and zeros are lists, need numpy arrays for .real and .imag -> Option B

4. Quick Check:

   Lists lack .real/.imag; use numpy arrays [OK]

Hint: Use numpy arrays for complex .real and .imag access [OK]

Common Mistakes:

• Trying to access .real on Python lists
• Ignoring missing plt.show()
• Using wrong marker symbols

5. You have a discrete-time system with poles at [0.6, 0.8*1j, -0.8*1j] and zeros at [0.9]. You want to improve stability by moving poles closer to the origin without changing zeros. Which pole movement will make the system more stable?

hard

A. Move poles to [0.6, 0.8j, -0.8j] (no change)

B. Move poles to [1.1, 0.9j, -0.9j]

C. Move poles to [0.4, 0.5j, -0.5j]

D. Move poles to [0.7, 0.85j, -0.85j]

Solution

1. Step 1: Understand stability and pole radius

   Poles closer to origin (smaller magnitude) mean faster decay and better stability.

2. Step 2: Compare given pole movements

   Move poles to [0.4, 0.5j, -0.5j] moves poles closer to origin (0.4 and 0.5j), reducing magnitude from original values.

3. Step 3: Confirm other options increase or keep magnitude same

   Options B and C move poles farther or slightly farther from origin, reducing stability; A is no change.

4. Final Answer:

   Move poles to [0.4, 0.5j, -0.5j] -> Option C

5. Quick Check:

   Smaller pole radius = better stability [OK]

Hint: Move poles closer to origin to improve stability [OK]

Common Mistakes:

• Moving poles outside unit circle
• Changing zeros instead of poles
• Not reducing pole magnitude