Common Z-transform pairs in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Z-transform for a unit step sequence

What is the Z-transform of the unit step sequence u[n] = 1 for n >= 0 and 0 otherwise?

Signal Processing

import sympy as sp
z = sp.symbols('z')
X = 1/(1 - 1/z)
X_simplified = sp.simplify(X)
print(X_simplified)

A (z - 1) / z

B 1 / (1 - z)

C z / (z - 1)

D 1 / (z - 1)

💡 Hint

Recall the Z-transform formula for the unit step sequence is a geometric series sum.

The Z-transform of u[n] is the sum from n=0 to infinity of z^(-n), which equals 1/(1 - z^(-1)) = z/(z-1).

❓ data_output

intermediate

Number of terms in Z-transform of finite sequence

Given the finite sequence x[n] = {1, 2, 3} for n=0,1,2 and zero elsewhere, how many non-zero terms does its Z-transform sum contain?

Signal Processing

x = [1, 2, 3]
non_zero_terms = len([val for val in x if val != 0])
print(non_zero_terms)

A 2

B 3

C 1

D 0

💡 Hint

Count the number of non-zero elements in the sequence.

The sequence has three non-zero values at n=0,1,2, so the Z-transform sum has three terms.

🧠 Conceptual

advanced

Region of convergence for causal exponential sequence

For the causal sequence x[n] = a^n u[n], where |a| < 1, what is the region of convergence (ROC) of its Z-transform?

A |z| < |a|

B |z| = |a|

C All complex z except z=0

D |z| > |a|

💡 Hint

ROC depends on the convergence of the infinite sum for n >= 0.

For the sum to converge, |z| must be greater than |a|, so ROC is |z| > |a|.

🔧 Debug

advanced

Identify error in Z-transform calculation code

What error does the following code produce when calculating the Z-transform of x[n] = 2^n u[n]? import sympy as sp z = sp.symbols('z') X = sum(2**n * z**n for n in range(5)) print(X)

Signal Processing

import sympy as sp
z = sp.symbols('z')
X = sum(2**n * z**(-n) for n in range(5))
print(X)

A No error, outputs a polynomial

B TypeError due to unsupported operand types

C SyntaxError due to missing colon

D NameError because z is not defined

💡 Hint

Check if the code runs and what it outputs.

The code sums finite terms and outputs a polynomial expression without error.

🚀 Application

expert

Z-transform of a shifted sequence

Given the sequence x[n] = u[n-2], where u[n] is the unit step, what is the Z-transform X(z)?

A z^(-2) / (1 - z^(-1))

B 1 / (1 - z^(-1))

C z^2 / (1 - z)

D z / (z - 1)

💡 Hint

Use the time-shifting property of the Z-transform.

The Z-transform of u[n] is z/(z-1). Shifting by 2 delays multiplies by z^(-2). So X(z) = z^(-2) * z/(z-1) = z^(-2)/(1 - z^(-1)).

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