Bilinear transformation method in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the function used for bilinear transformation from scipy.signal.

Signal Processing

from scipy.signal import [1]

Drag options to blanks, or click blank then click option'

Afft

Bbilinear

Cconvolve

Dlfilter

Attempts:

3 left

💡 Hint

Common Mistakes

Importing unrelated functions like fft or convolve.

Using lfilter which is for filtering signals, not transformation.

The bilinear function from scipy.signal is used to perform the bilinear transformation.

2fill in blank

medium

Complete the code to define the analog filter numerator coefficients as a list.

Signal Processing

b = [1]

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A1, 0

B(1, 0)

C[1, 0]

D{1, 0}

Attempts:

3 left

💡 Hint

Common Mistakes

Using tuples or sets instead of lists.

Not enclosing coefficients in any brackets.

Filter coefficients should be defined as a list of numbers, e.g., [1, 0].

3fill in blank

hard

Fix the error in the code to correctly perform the bilinear transformation.

Signal Processing

b_z, a_z = bilinear(b, [1], fs)

Drag options to blanks, or click blank then click option'

A1

Bb

Cfs

Da

Attempts:

3 left

💡 Hint

Common Mistakes

Passing numerator coefficients twice.

Passing sampling frequency as the second argument.

The second argument to bilinear is the denominator coefficients a of the analog filter.

4fill in blank

hard

Fill both blanks to define the sampling frequency and apply bilinear transform correctly.

Signal Processing

fs = [1]
b_z, a_z = bilinear(b, a, [2])

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A1000

B500

Cfs

D1

Attempts:

3 left

💡 Hint

Common Mistakes

Using incorrect sampling frequency values.

Passing a literal number instead of the variable fs.

The sampling frequency fs is set to 1000 Hz, and then passed as the third argument to bilinear.

5fill in blank

hard

Fill all three blanks to create a digital filter using bilinear transform and print the numerator coefficients.

Signal Processing

b = [1]
a = [2]
fs = 2000
b_z, a_z = bilinear(b, a, [3])
print(b_z)

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A[0.0675, 0.1349, 0.0675]

B[1, -1.143, 0.4128]

Cfs

D[1, 0.5, 0.25]

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up numerator and denominator coefficients.

Passing sampling frequency as a list or wrong variable.

Not printing the correct variable.

The analog filter numerator b and denominator a coefficients are defined as lists. The sampling frequency fs is passed to bilinear. The output b_z contains the digital filter numerator coefficients.

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