Stability analysis (pole-zero plot) in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

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

Signal Processing

from scipy.signal import [1]

zeros, poles, _ = tf2zpk([1, 0.5], [1, -0.8, 0.15])

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Afreqz

Btf2zpk

Clfilter

Dbutter

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'freqz' which computes frequency response, not poles and zeros.

Using 'lfilter' which filters signals, not related to poles and zeros.

Using 'butter' which designs Butterworth filters, not for poles and zeros extraction.

The tf2zpk function computes zeros, poles, and gain from numerator and denominator coefficients.

2fill in blank

medium

Complete the code to plot the poles and zeros on the complex plane.

Signal Processing

import matplotlib.pyplot as plt

plt.scatter(zeros.real, zeros.imag, marker='o', label='Zeros')
plt.scatter(poles.real, poles.[1], marker='x', label='Poles')
plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.legend()
plt.grid(True)
plt.title('Pole-Zero Plot')
plt.show()

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Aimag

Breal

Cangle

Dconj

Attempts:

3 left

💡 Hint

Common Mistakes

Using '.real' for the imaginary axis.

Using '.angle' which gives the phase angle, not the imaginary part.

Using '.conj' which gives the complex conjugate, not the imaginary part.

To plot poles on the complex plane, use their imaginary parts with poles.imag.

3fill in blank

hard

Fix the error in the code that checks system stability by verifying if all poles are inside the unit circle.

Signal Processing

stable = all(abs(p) [1] 1 for p in poles)
print('System is stable:', stable)

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A<

B>

C<=

D>=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '>' which checks poles outside the unit circle.

Using '>=' or '<=' which includes poles on the unit circle, not strictly inside.

A discrete-time system is stable if all poles lie strictly inside the unit circle, so their magnitude is less than 1.

4fill in blank

hard

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

Signal Processing

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

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Azeros

Bpoles

Croots

Dvalues

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'roots' or 'values' which are not defined variables here.

Mixing zeros and poles in the wrong blanks.

We use the variables zeros and poles to get their magnitudes with abs().

5fill in blank

hard

Fill all three blanks to filter poles with magnitude greater than 1 and create a list of their angles in radians.

Signal Processing

import cmath

unstable_poles = [p for p in [1] if abs(p) [2] 1]
angles = [[3](p) for p in unstable_poles]

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Apoles

B>

Ccmath.phase

Dzeros

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'zeros' instead of 'poles' for filtering.

Using '<' instead of '>' for magnitude comparison.

Using 'abs' instead of 'cmath.phase' for angle calculation.

We filter poles with magnitude greater than 1 and use cmath.phase to get their angles.