IIR filter design (butter, cheby1) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Butterworth filter coefficients

What is the output of the following code that designs a 3rd order Butterworth lowpass filter with cutoff frequency 0.4 (normalized) and prints the numerator coefficients?

SciPy

from scipy.signal import butter
b, a = butter(3, 0.4, btype='low')
print(b)

A [0.01809893 0.05429679 0.05429679 0.01809893]

B [ 0.09763107 0.29289322 0.29289322 0.09763107]

C [0.00482434 0.01447302 0.01447302 0.00482434]

D [0.25 0.25 0.25 0.25]

💡 Hint

Recall that butter() returns numerator and denominator coefficients of the filter.

The butter function with order 3 and cutoff 0.4 returns numerator coefficients approximately equal to [0.0181, 0.0543, 0.0543, 0.0181]. Other options are incorrect values.

❓ data_output

intermediate

Number of zeros in Chebyshev Type I filter numerator

After designing a 4th order Chebyshev Type I highpass filter with 1 dB ripple and cutoff frequency 0.3 (normalized), how many zeros does the numerator polynomial have?

SciPy

from scipy.signal import cheby1
b, a = cheby1(4, 1, 0.3, btype='high')
print(len(b) - 1)

A 3

B 4

C 5

D 1

💡 Hint

The numerator polynomial degree equals the filter order for IIR filters.

The numerator coefficients array length is order + 1, so zeros count is 4 for a 4th order filter.

🔧 Debug

advanced

Identify the error in filter design code

What error does the following code raise when trying to design a Butterworth bandpass filter with cutoff frequencies 0.2 and 0.5?

SciPy

from scipy.signal import butter
b, a = butter(3, [0.2, 0.5], btype='bandpass')

A No error, returns filter coefficients

B TypeError: 'bandpass' is not a valid filter type

C ValueError: Digital filter critical frequencies must be strictly increasing and 0 < Wn < 1

D SyntaxError: invalid syntax

💡 Hint

Check the valid filter types and cutoff frequency ranges for butter().

No error occurs because 'bandpass' is a valid btype for designing bandpass filters, and the cutoff frequencies [0.2, 0.5] satisfy 0 < 0.2 < 0.5 < 1 and are strictly increasing.

❓ visualization

advanced

Frequency response plot of Chebyshev Type I filter

Which option correctly plots the frequency response magnitude (in dB) of a 5th order Chebyshev Type I lowpass filter with 0.5 normalized cutoff and 0.5 dB ripple?

SciPy

import matplotlib.pyplot as plt
import numpy as np
from scipy.signal import cheby1, freqz
b, a = cheby1(5, 0.5, 0.5, btype='low')
w, h = freqz(b, a)
plt.plot(w / 3.1415, 20 * np.log10(abs(h)))
plt.xlabel('Normalized Frequency')
plt.ylabel('Magnitude (dB)')
plt.title('Frequency Response')
plt.grid(True)
plt.show()

A Plots frequency response with x-axis from 0 to 1 and y-axis showing magnitude in dB

B Plots frequency response with x-axis in Hz (0 to 5000) and y-axis linear magnitude

C Plots phase response instead of magnitude

D Raises NameError due to missing numpy import

💡 Hint

Check if all required libraries are imported before plotting.

The code uses np.log10 and np without importing numpy as np, causing NameError.

🚀 Application

expert

Choosing filter type for sharp cutoff with minimal ripple

You need to design a digital IIR filter with a very sharp cutoff and minimal passband ripple for a biomedical signal sampled at 1000 Hz. Which filter design and parameters below best meet this requirement?

A Use a 6th order Chebyshev Type I lowpass filter with 0.5 dB ripple and cutoff 100 Hz

B Use a 6th order Butterworth lowpass filter with cutoff 100 Hz

C Use a 6th order Chebyshev Type II lowpass filter with 40 dB stopband attenuation and cutoff 100 Hz

D Use a 6th order Elliptic lowpass filter with 0.5 dB ripple and 40 dB stopband attenuation

💡 Hint

Chebyshev Type II filters have flat passband and sharp cutoff with stopband ripple.

Chebyshev Type II filters have no ripple in passband and sharp cutoff, suitable for minimal passband ripple and sharp transition.