Common Z-transform pairs in Signal Processing - Mini Project: Build & Apply

Common Z-transform Pairs

📖 Scenario: You are working with digital signals and want to understand how common time-domain sequences relate to their Z-transforms. This helps in analyzing and designing digital filters.

🎯 Goal: Build a Python dictionary of common discrete-time signals and their Z-transform pairs, then filter and display specific pairs based on a condition.

📋 What You'll Learn

Create a dictionary called z_pairs with given time-domain sequences as keys and their Z-transform expressions as values.

Create a variable called filter_key to select a specific time-domain sequence.

Use a dictionary comprehension to create a new dictionary filtered_pairs that contains only the pair matching filter_key.

Print the filtered_pairs dictionary.

💡 Why This Matters

🌍 Real World

Digital signal processing engineers use Z-transform pairs to analyze and design filters and systems that work with discrete signals.

💼 Career

Understanding Z-transform pairs is essential for roles in signal processing, communications, and control systems engineering.

1

Create the Z-transform pairs dictionary

Create a dictionary called z_pairs with these exact entries: 'delta[n]': '1', 'u[n]': '1 / (1 - z^-1)', 'a^n u[n]': '1 / (1 - a z^-1)', 'n u[n]': 'z^-1 / (1 - z^-1)^2'.

Signal Processing

# Create the dictionary z_pairs with the given time-domain sequences and their Z-transforms
# Your code here

Hint

Use curly braces {} to create the dictionary and separate each key-value pair with a comma.

2

Set the filter key

Create a variable called filter_key and set it to the string 'a^n u[n]'.

Signal Processing

z_pairs = {
    'delta[n]': '1',
    'u[n]': '1 / (1 - z^-1)',
    'a^n u[n]': '1 / (1 - a z^-1)',
    'n u[n]': 'z^-1 / (1 - z^-1)^2'
}
# Create the variable filter_key with the value 'a^n u[n]'
# Your code here

Hint

Assign the string 'a^n u[n]' to the variable filter_key.

3

Filter the dictionary using comprehension

Use a dictionary comprehension to create a new dictionary called filtered_pairs that contains only the key-value pair from z_pairs where the key matches filter_key.

Signal Processing

z_pairs = {
    'delta[n]': '1',
    'u[n]': '1 / (1 - z^-1)',
    'a^n u[n]': '1 / (1 - a z^-1)',
    'n u[n]': 'z^-1 / (1 - z^-1)^2'
}
filter_key = 'a^n u[n]'
# Create filtered_pairs using dictionary comprehension
# Your code here

Hint

Use {k: v for k, v in z_pairs.items() if k == filter_key} to filter the dictionary.

4

Print the filtered dictionary

Print the variable filtered_pairs to display the filtered Z-transform pair.

Signal Processing

z_pairs = {
    'delta[n]': '1',
    'u[n]': '1 / (1 - z^-1)',
    'a^n u[n]': '1 / (1 - a z^-1)',
    'n u[n]': 'z^-1 / (1 - z^-1)^2'
}
filter_key = 'a^n u[n]'
filtered_pairs = {k: v for k, v in z_pairs.items() if k == filter_key}
# Print the filtered_pairs dictionary
# Your code here

Hint

Use print(filtered_pairs) to display the result.

Practice

1. Which of the following is the Z-transform of the sequence x[n] = a^n u[n], where u[n] is the unit step?

easy

A. X(z) = z / (z - a), |z| < |a|

B. X(z) = 1 / (1 - a z^{-1}), |z| > |a|

C. X(z) = (1 - a z^{-1}) / z, |z| > |a|

D. X(z) = a / (1 - z^{-1}), |z| > 1

Solution

1. Step 1: Recall the definition of the Z-transform for geometric sequences

   The Z-transform of a^n u[n] is a standard pair given by the sum of a geometric series: \( \sum_{n=0}^\infty a^n z^{-n} = \frac{1}{1 - a z^{-1}} \) for \( |z| > |a| \).

2. Step 2: Match the formula with the options

   X(z) = 1 / (1 - a z^{-1}), |z| > |a| matches the formula exactly with the correct region of convergence.

3. Final Answer:

   X(z) = 1 / (1 - a z^{-1}), |z| > |a| -> Option B

4. Quick Check:

   Geometric sequence Z-transform = 1/(1 - a z^{-1}) [OK]

Hint: Geometric sequences transform to 1/(1 - a z^{-1}) [OK]

Common Mistakes:

• Confusing region of convergence
• Mixing numerator and denominator terms
• Using wrong inequality for |z|

2. Which of the following is the correct Z-transform expression for the unit impulse sequence δ[n]?

easy

A. X(z) = z / (z - 1)

B. X(z) = z^{-1}

C. X(z) = 1 / (1 - z^{-1})

D. X(z) = 1

Solution

1. Step 1: Understand the unit impulse sequence

   The unit impulse δ[n] is 1 at n=0 and 0 elsewhere.

2. Step 2: Calculate the Z-transform

   By definition, \( X(z) = \sum_{n=-\infty}^\infty δ[n] z^{-n} = z^{0} = 1 \).

3. Final Answer:

   X(z) = 1 -> Option D

4. Quick Check:

   Impulse Z-transform = 1 [OK]

Hint: Impulse transforms to 1 always [OK]

Common Mistakes:

• Confusing impulse with step function
• Adding z terms unnecessarily
• Forgetting impulse is zero except at n=0

3. What is the Z-transform of the sequence x[n] = n u[n]?

medium

A. X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1

B. X(z) = 1 / (1 - z^{-1})^2, |z| > 1

C. X(z) = z / (z - 1), |z| > 1

D. X(z) = 1 / (1 - z)^2, |z| > 1

Solution

1. Step 1: Recall the Z-transform of the sequence n u[n]

   The sequence n u[n] has Z-transform \( X(z) = \frac{z^{-1}}{(1 - z^{-1})^2} \) with region of convergence \( |z| > 1 \).

2. Step 2: Match the formula with the options

   X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1 matches the formula exactly including the region of convergence.

3. Final Answer:

   X(z) = z^{-1} / (1 - z^{-1})^2, |z| > 1 -> Option A

4. Quick Check:

   Linear sequence Z-transform = z^{-1}/(1 - z^{-1})^2 [OK]

Hint: Linear sequences transform to z^{-1}/(1 - z^{-1})^2 [OK]

Common Mistakes:

• Confusing powers of z
• Using (z - 1) instead of (1 - z^{-1})
• Ignoring region of convergence

4. Identify the error in the following Z-transform expression for x[n] = a^n u[n]: X(z) = 1 / (1 - z a^{-1})

medium

A. The variable z and a are swapped in the denominator

B. The region of convergence is missing

C. The exponent on a should be negative

D. The formula is correct as is

Solution

1. Step 1: Recall the correct Z-transform formula for a^n u[n]

   The correct formula is \( X(z) = \frac{1}{1 - a z^{-1}} \), not \( 1/(1 - z a^{-1}) \).

2. Step 2: Analyze the given expression

   The given expression swaps z and a in the denominator, which changes the meaning and is incorrect.

3. Final Answer:

   The variable z and a are swapped in the denominator -> Option A

4. Quick Check:

   Correct formula uses a z^{-1}, not z a^{-1} [OK]

Hint: Check order of variables in denominator carefully [OK]

Common Mistakes:

• Swapping z and a in denominator
• Ignoring negative powers of z
• Missing region of convergence

5. Given the sequence x[n] = (1/2)^n u[n] + 3 δ[n], what is the combined Z-transform X(z)?

hard

A. X(z) = 3 * z / (z - 1/2), |z| > 1/2

B. X(z) = 3 z^{-1} + 1 / (1 - (1/2) z^{-1}), |z| > 1/2

C. X(z) = 3 + 1 / (1 - (1/2) z^{-1}), |z| > 1/2

D. X(z) = 3 + 1 / (1 - 2 z^{-1}), |z| > 2

Solution

1. Step 1: Find Z-transform of each sequence part

   The Z-transform of (1/2)^n u[n] is \( \frac{1}{1 - (1/2) z^{-1}} \) with \( |z| > 1/2 \). The Z-transform of 3 δ[n] is 3.

2. Step 2: Add the transforms for combined sequence

   Since Z-transform is linear, add both: \( X(z) = 3 + \frac{1}{1 - (1/2) z^{-1}} \).

3. Final Answer:

   X(z) = 3 + 1 / (1 - (1/2) z^{-1}), |z| > 1/2 -> Option C

4. Quick Check:

   Sum of transforms = 3 + 1/(1 - 0.5 z^{-1}) [OK]

Hint: Sum transforms for combined sequences linearly [OK]

Common Mistakes:

• Multiplying instead of adding transforms
• Incorrect region of convergence
• Wrong coefficient placement