Bilinear transformation method in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the purpose of the bilinear transformation method

What is the main purpose of using the bilinear transformation method in signal processing?

A To perform Fourier transform on a signal for frequency analysis

B To convert a discrete-time filter into a continuous-time filter without changing frequency response

C To increase the sampling rate of a digital signal without distortion

D To convert a continuous-time filter design into a discrete-time filter design while preserving stability

💡 Hint

Think about how analog filters are adapted for digital use.

The bilinear transformation method is used to convert analog filter designs into digital filter designs. It maps the s-plane to the z-plane, preserving the stability of the filter and avoiding aliasing.

❓ Predict Output

intermediate

Output of bilinear transform frequency mapping

What is the output of the following Python code that uses the bilinear transform formula to map an analog frequency to digital frequency?

Signal Processing

import numpy as np

# Analog frequency in radians per second
omega_analog = np.pi / 2

# Sampling frequency
fs = 2 * np.pi

# Bilinear transform frequency mapping formula
omega_digital = 2 * fs * np.tan(omega_analog / 2)

print(round(omega_digital, 3))

A 3.142

B 4.712

C 6.283

D 1.571

💡 Hint

Recall that tan(π/4) = 1 and fs = 2π.

The code calculates omega_digital = 2 * 2π * tan(π/4) = 4π ≈ 12.566, but since fs is 2π, the calculation is 2 * 2π * 1 = 4π ≈ 12.566. The printed rounded value is 12.566, which is not among the options. However, the formula for frequency warping is usually omega_digital = 2 * fs * tan(omega_analog / (2 * fs)), so the code or options need correction.

❓ data_output

advanced

Resulting poles after bilinear transformation

Given an analog filter with poles at s = -1 and s = -2, what are the poles of the digital filter after applying the bilinear transformation with sampling frequency fs = 10 rad/s?

Signal Processing

import numpy as np

fs = 10
s_poles = np.array([-1, -2])
z_poles = (1 + s_poles / (2 * fs)) / (1 - s_poles / (2 * fs))
print(np.round(z_poles, 3))

A [0.9+0j 0.8+0j]

B [0.818+0j 0.667+0j]

C [1.222+0j 1.333+0j]

D [-0.818+0j -0.667+0j]

💡 Hint

Use the bilinear transform formula: z = (1 + s/(2fs)) / (1 - s/(2fs))

Applying the formula for each pole: For s = -1, z = (1 - 1/(20)) / (1 + 1/(20)) = 0.818; for s = -2, z = (1 - 2/(20)) / (1 + 2/(20)) = 0.667.

❓ visualization

advanced

Frequency warping visualization by bilinear transform

Which plot correctly shows the frequency warping effect of the bilinear transformation mapping analog frequencies (0 to π) to digital frequencies (0 to π)?

A A curve starting at 0 and bending downwards, reaching π at analog frequency π

B A straight line from 0 to π with slope 1

C A curve starting at 0 and bending upwards, reaching π at analog frequency π

D A horizontal line at π for all analog frequencies

💡 Hint

Remember that bilinear transform compresses high frequencies.

The bilinear transform causes frequency warping where low frequencies map almost linearly, but higher frequencies are compressed, so the curve bends downward approaching π.

🔧 Debug

expert

Identify the error in bilinear transform implementation

What error does the following Python code raise when applying the bilinear transform to poles?

Signal Processing

import numpy as np

fs = 5
s_poles = np.array([-10, -20])
z_poles = (1 + s_poles / (2 * fs)) / (1 - s_poles / (2 * fs))
print(z_poles)

A ZeroDivisionError

B TypeError

C No error, outputs poles correctly

D ValueError

💡 Hint

Check the denominator values carefully.

For s = -10 and fs = 5, denominator 1 - s/(2*fs) becomes zero for s = -10, causing division by zero and raising ZeroDivisionError.