Bilinear transformation method in Signal Processing - Mini Project: Build & Apply

Bilinear Transformation Method

📖 Scenario: You are working as a signal processing engineer. You need to convert an analog filter into a digital filter using the bilinear transformation method. This method helps to map the analog filter's frequency response into the digital domain.

🎯 Goal: Build a Python program that applies the bilinear transformation method to convert an analog filter's transfer function coefficients into digital filter coefficients.

📋 What You'll Learn

Create variables for analog filter numerator and denominator coefficients

Define the sampling frequency

Apply the bilinear transformation using the scipy.signal.bilinear function

Print the resulting digital filter numerator and denominator coefficients

💡 Why This Matters

🌍 Real World

Bilinear transformation is used in designing digital filters from analog prototypes, which is common in audio processing, communications, and control systems.

💼 Career

Understanding bilinear transformation helps engineers convert analog designs to digital implementations, a key skill in signal processing and embedded systems jobs.

1

Create analog filter coefficients

Create two lists called b_analog and a_analog with these exact values: b_analog = [1, 0] and a_analog = [1, 1].

Signal Processing

# Create the analog filter numerator coefficients b_analog and denominator coefficients a_analog
# Your code here

Hint

The numerator and denominator coefficients represent the analog filter's transfer function.

2

Define the sampling frequency

Create a variable called fs and set it to 1000 to represent the sampling frequency in Hertz.

Signal Processing

b_analog = [1, 0]
a_analog = [1, 1]
# Define the sampling frequency fs
# Your code here

Hint

The sampling frequency is how often the digital signal is sampled per second.

3

Apply the bilinear transformation

Import bilinear from scipy.signal. Use bilinear(b_analog, a_analog, fs) to compute the digital filter coefficients. Store the results in variables b_digital and a_digital.

Signal Processing

b_analog = [1, 0]
a_analog = [1, 1]
fs = 1000
# Import bilinear and apply the bilinear transformation
# Your code here

Hint

The bilinear function converts analog coefficients to digital coefficients using the sampling frequency.

4

Print the digital filter coefficients

Print the variables b_digital and a_digital to display the digital filter numerator and denominator coefficients.

Signal Processing

from scipy.signal import bilinear

b_analog = [1, 0]
a_analog = [1, 1]
fs = 1000

b_digital, a_digital = bilinear(b_analog, a_analog, fs)
# Print the digital filter coefficients
# Your code here

Hint

The printed arrays show the digital filter coefficients after transformation.

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