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.

Practice

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

easy

A. To convert analog filter designs into digital filter designs

B. To convert digital signals into analog signals

C. To increase the sampling rate of a digital signal

D. To reduce noise in digital signals

Solution

1. Step 1: Understand the bilinear transformation concept

   The bilinear transformation substitutes the analog frequency variable s with a function of the digital variable z, enabling analog filter designs to be converted to digital filters.

2. Step 2: Identify the main use

   This method is primarily used to convert analog filter designs into digital filter designs for digital signal processing.

3. Final Answer:

   To convert analog filter designs into digital filter designs -> Option A

4. Quick Check:

   Bilinear transformation = analog to digital filter conversion [OK]

Hint: Bilinear transforms analog filters to digital filters [OK]

Common Mistakes:

• Confusing bilinear transform with sampling rate conversion
• Thinking it converts digital signals back to analog
• Assuming it reduces noise directly

2. Which of the following is the correct substitution for s in the bilinear transformation?

easy

A. s = (1 - z^{-1}) / (1 + z^{-1})

B. s = (1 + z^{-1}) / (1 - z^{-1})

C. s = (2/T) * (1 - z^{-1}) / (1 + z^{-1})

D. s = (z - 1) / (z + 1)

Solution

1. Step 1: Recall the bilinear transformation formula

   The bilinear transformation substitutes s with (2/T) * (1 - z^{-1}) / (1 + z^{-1}), where T is the sampling period.

2. Step 2: Match the correct formula

   Among the options, only s = (2/T) * (1 - z^{-1}) / (1 + z^{-1}) correctly includes the factor 2/T and the fraction form.

3. Final Answer:

   s = (2/T) * (1 - z^{-1}) / (1 + z^{-1}) -> Option C

4. Quick Check:

   Bilinear substitution = (2/T)(1 - z^{-1})/(1 + z^{-1}) [OK]

Hint: Remember the 2/T factor in bilinear substitution [OK]

Common Mistakes:

• Omitting the 2/T factor
• Swapping numerator and denominator
• Using incorrect signs in numerator or denominator

3. Given an analog low-pass filter transfer function H(s) = 1 / (s + 1) and sampling period T = 1, what is the digital filter transfer function H(z) after applying the bilinear transformation?

medium

A. H(z) = (1 + z^{-1}) / (3 + z^{-1})

B. H(z) = (1 + z^{-1}) / (3 - z^{-1})

C. H(z) = (1 - z^{-1}) / (3 + z^{-1})

D. H(z) = (1 - z^{-1}) / (3 - z^{-1})

Solution

1. Step 1: Substitute s using bilinear transform

   With T=1, substitute s = 2 * (1 - z^{-1}) / (1 + z^{-1}) into H(s) = 1 / (s + 1).
   So, H(z) = 1 / (2 * (1 - z^{-1}) / (1 + z^{-1}) + 1).

2. Step 2: Simplify the expression

   Rewrite denominator:
   2 * (1 - z^{-1}) / (1 + z^{-1}) + 1 = (2(1 - z^{-1}) + (1 + z^{-1})) / (1 + z^{-1}) = (2 - 2z^{-1} + 1 + z^{-1}) / (1 + z^{-1}) = (3 - z^{-1}) / (1 + z^{-1}).
   Therefore, H(z) = 1 / ((3 - z^{-1}) / (1 + z^{-1})) = (1 + z^{-1}) / (3 - z^{-1}).

3. Final Answer:

   H(z) = (1 + z^{-1}) / (3 - z^{-1}) -> Option B

4. Quick Check:

   Substitution and simplification give (1 + z^{-1})/(3 - z^{-1}) [OK]

Hint: Substitute s, then simplify fractions carefully [OK]

Common Mistakes:

• Forgetting to add 1 in denominator
• Incorrect algebraic simplification
• Mixing signs in numerator or denominator

4. A student tries to apply the bilinear transformation but writes s = (1 - z) / (1 + z) without the 2/T factor. What is the main issue with this approach?

medium

A. The digital filter will be unstable

B. The transformation will cause a syntax error

C. The analog filter cannot be converted

D. The frequency scaling is incorrect, causing distortion

Solution

1. Step 1: Identify missing factor impact

   The bilinear transformation requires the factor 2/T to correctly scale frequencies from analog to digital domain.

2. Step 2: Consequence of missing factor

   Without 2/T, the frequency mapping is incorrect, leading to frequency distortion or warping in the digital filter.

3. Final Answer:

   The frequency scaling is incorrect, causing distortion -> Option D

4. Quick Check:

   Missing 2/T causes frequency distortion [OK]

Hint: Always include 2/T factor to avoid frequency errors [OK]

Common Mistakes:

• Ignoring the 2/T factor
• Assuming syntax error instead of scaling error
• Thinking filter stability is directly affected

5. You want to design a digital Butterworth low-pass filter with cutoff frequency 1000 Hz using bilinear transformation. The sampling frequency is 8000 Hz. Which step is essential to correctly map the analog cutoff frequency to the digital domain?

hard

A. Pre-warp the analog cutoff frequency using Ω = 2 * fs * tan(π * f_c / fs)

B. Use the cutoff frequency directly without modification

C. Divide the cutoff frequency by 2 before applying bilinear transform

D. Multiply the cutoff frequency by the sampling frequency

Solution

1. Step 1: Understand frequency warping in bilinear transform

   The bilinear transformation causes frequency warping, so the analog cutoff frequency must be pre-warped to match the desired digital cutoff.

2. Step 2: Apply pre-warping formula

   The correct pre-warping formula is Ω = 2 * fs * tan(π * f_c / fs), where fs is sampling frequency and f_c is desired cutoff frequency.

3. Final Answer:

   Pre-warp the analog cutoff frequency using Ω = 2 * fs * tan(π * f_c / fs) -> Option A

4. Quick Check:

   Pre-warping corrects frequency warping [OK]

Hint: Pre-warp cutoff frequency to fix bilinear warping [OK]

Common Mistakes:

• Skipping pre-warping step
• Using cutoff frequency directly
• Incorrectly scaling cutoff frequency