Pole-zero analysis for stability in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the function that finds poles and zeros of a system.

Signal Processing

from scipy.signal import [1]

Drag options to blanks, or click blank then click option'

Atf2zpk

Blfilter

Cfreqz

Dbutter

Attempts:

3 left

💡 Hint

Common Mistakes

Using filter design functions like butter instead of tf2zpk.

Importing freqz which computes frequency response, not poles and zeros.

The tf2zpk function returns zeros, poles, and gain of a transfer function.

2fill in blank

medium

Complete the code to find poles of the system given numerator and denominator coefficients.

Signal Processing

zeros, poles, gain = tf2zpk(num, [1])

Drag options to blanks, or click blank then click option'

Aden

Bnum

Cgain

Dpoles

Attempts:

3 left

💡 Hint

Common Mistakes

Passing numerator coefficients twice.

Confusing gain or poles variable as input.

The denominator coefficients den define the poles of the system.

3fill in blank

hard

Fix the error in the code to check if all poles are inside the unit circle for stability.

Signal Processing

is_stable = all(abs(p) [1] 1 for p in poles)

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A>

B<

C>=

D<=

Attempts:

3 left

💡 Hint

Common Mistakes

Using > instead of < which checks for instability.

Using >= or <= which includes poles on the unit circle (marginally stable).

For discrete systems, stability requires all poles to have magnitude less than 1.

4fill in blank

hard

Fill both blanks to create a dictionary of poles and zeros magnitudes.

Signal Processing

magnitudes = {'poles': [abs(p) for p in [1]], 'zeros': [abs(z) for z in [2]]}

Drag options to blanks, or click blank then click option'

Apoles

Bzeros

Cnum

Dden

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing numerator or denominator coefficients instead of poles or zeros.

Swapping poles and zeros variables.

We calculate magnitudes of poles and zeros from their respective lists.

5fill in blank

hard

Fill all three blanks to filter stable poles and zeros with magnitude less than 1.

Signal Processing

stable_poles = [p for p in [1] if abs(p) [2] 1]
stable_zeros = [z for z in [3] if abs(z) [2] 1]

Drag options to blanks, or click blank then click option'

Apoles

B<

Czeros

Dden

Attempts:

3 left

💡 Hint

Common Mistakes

Using denominator coefficients instead of poles or zeros.

Using greater than sign which selects unstable poles or zeros.

We select poles and zeros with magnitude less than 1 to identify stable ones.

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