Z-transform definition in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the Z-transform formula

Which of the following expressions correctly defines the Z-transform of a discrete-time signal x[n]?

A X(z) = \int_{-\infty}^{\infty} x(t) e^{-zt} dt

B X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{n}

C X(z) = \sum_{n=0}^{\infty} x[n] z^{n}

D X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}

💡 Hint

Recall that the Z-transform uses a sum over all integer n with z raised to the negative power of n.

The Z-transform of a discrete-time signal x[n] is defined as the sum over all n from minus infinity to infinity of x[n] times z to the power of negative n.

❓ Predict Output

intermediate

Output of Z-transform calculation for a finite signal

What is the output of the following Python code that computes the Z-transform for x[n] = {1, 2, 3} at z=2?

Signal Processing

import numpy as np
x = [1, 2, 3]
z = 2
X_z = sum(x[n] * z**(-n) for n in range(len(x)))
print(X_z)

A 2.75

B 2.25

C 4.0

D 3.5

💡 Hint

Calculate each term: x[0]*z^0 + x[1]*z^-1 + x[2]*z^-2

The calculation is 1*2^0 + 2*2^-1 + 3*2^-2 = 1 + 1 + 0.75 = 2.75.

❓ data_output

advanced

Z-transform sequence output

Given the signal x[n] = {1, -1, 2} for n=0,1,2, what is the sequence of Z-transform values X(z) for z = 1, 2, 3?

Signal Processing

x = [1, -1, 2]
z_values = [1, 2, 3]
X_z = [sum(x[n] * z**(-n) for n in range(len(x))) for z in z_values]
print(X_z)

A [2.0, 1.25, 1.1111111111111112]

B [2.0, 1.0, 0.8888888888888888]

C [0.0, 0.5, 0.6666666666666666]

D [1.0, 0.75, 0.5555555555555556]

💡 Hint

Calculate X(z) for each z by summing x[n]*z^(-n).

For z=1: 1 -1 + 2 = 2.0; for z=2: 1 -0.5 + 0.5 = 1.0; for z=3: 1 - 1/3 + 2/9 ≈ 0.8889.

🔧 Debug

advanced

Identify the error in Z-transform code

What error does the following Python code produce when computing the Z-transform?

Signal Processing

x = [1, 2, 3]
z = 2
X_z = sum(x[n] * z**(-n) for n in range(len(x)))
print(X_z)

A IndexError because n is out of range

B TypeError because z is not defined

C SyntaxError due to incorrect exponentiation syntax

D No error, outputs 2.75

💡 Hint

Check the use of curly braces in the exponent.

The code uses curly braces { } instead of parentheses ( ) for exponentiation, which is invalid syntax in Python.

🚀 Application

expert

Applying Z-transform to solve difference equation

Given the difference equation y[n] - 0.5 y[n-1] = x[n], and the input x[n] = (0.5)^n u[n] (u[n] is the step function), what is the Z-transform Y(z) of the output y[n]?

A Y(z) = X(z) / (1 - 0.5 z^{-1})

B Y(z) = X(z) * (1 - 0.5 z^{-1})

C Y(z) = X(z) / (1 + 0.5 z^{-1})

D Y(z) = X(z) * (1 + 0.5 z^{-1})

💡 Hint

Take the Z-transform of both sides and solve for Y(z).

Taking Z-transform: Y(z) - 0.5 z^{-1} Y(z) = X(z) => Y(z)(1 - 0.5 z^{-1}) = X(z) => Y(z) = X(z) / (1 - 0.5 z^{-1})

Practice

1. What is the primary purpose of the Z-transform in signal processing?

easy

A. To convert discrete-time signals into a complex frequency domain for easier analysis

B. To convert continuous signals into discrete signals

C. To amplify the signal strength

D. To filter out noise from signals

Solution

1. Step 1: Understand the role of Z-transform

   The Z-transform changes discrete-time signals into a complex frequency domain representation.

2. Step 2: Compare options with definition

   Only To convert discrete-time signals into a complex frequency domain for easier analysis correctly describes this transformation purpose.

3. Final Answer:

   To convert discrete-time signals into a complex frequency domain for easier analysis -> Option A

4. Quick Check:

   Z-transform = frequency domain conversion [OK]

Hint: Z-transform moves signals to complex frequency domain [OK]

Common Mistakes:

• Confusing Z-transform with filtering
• Thinking it converts continuous to discrete signals
• Assuming it amplifies signals

2. Which of the following is the correct formula for the Z-transform of a discrete sequence x[n]?

easy

A. X(z) = \sum_{n=-\infty}^{\infty} x[n] z^n

B. X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}

C. X(z) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt

D. X(z) = \sum_{n=0}^{N} x[n] z^n

Solution

1. Step 1: Recall the Z-transform formula

   The Z-transform is defined as X(z) = sum from n=0 to infinity of x[n] times z to the power -n.

2. Step 2: Match formula with options

   X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} matches the standard Z-transform formula exactly.

3. Final Answer:

   X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} -> Option B

4. Quick Check:

   Z-transform formula = sum x[n] z^-n [OK]

Hint: Look for sum from 0 to infinity with z to negative n [OK]

Common Mistakes:

• Using positive powers of z instead of negative
• Confusing Z-transform with Fourier transform integral
• Summing over all integers instead of starting at 0

3. Given the sequence x[n] = {1, 2, 3} for n=0,1,2 and zero elsewhere, what is the Z-transform X(z)?

medium

A. X(z) = 1z^{-1} + 2z^{-2} + 3z^{-3}

B. X(z) = 1 + 2z + 3z^{2}

C. X(z) = 1 + 2z^{-1} + 3z^{-2}

D. X(z) = 1z + 2z^{2} + 3z^{3}

Solution

1. Step 1: Apply Z-transform formula to given sequence

   Use X(z) = sum x[n] z^{-n} from n=0 to 2: 1*z^0 + 2*z^{-1} + 3*z^{-2}.

2. Step 2: Write the expression clearly

   This equals 1 + 2z^{-1} + 3z^{-2}, matching X(z) = 1 + 2z^{-1} + 3z^{-2}.

3. Final Answer:

   1 + 2z^{-1} + 3z^{-2} -> Option C

4. Quick Check:

   Sum x[n] z^-n = 1+2z^-1+3z^-2 [OK]

Hint: Use powers of z as negative indices matching n [OK]

Common Mistakes:

• Using positive powers of z
• Starting sum at n=1 instead of 0
• Ignoring zero values outside given n

4. Identify the error in this Z-transform expression: X(z) = \sum_{n=0}^{\infty} x[n] z^{n}

medium

A. The variable z should be real, not complex

B. The summation limits are incorrect

C. The sequence x[n] should be continuous, not discrete

D. The power of z should be negative, not positive

Solution

1. Step 1: Review the Z-transform formula

   The Z-transform uses z raised to the negative power of n, i.e., z^{-n}.

2. Step 2: Compare with given expression

   The expression uses z^{n} (positive power), which is incorrect.

3. Final Answer:

   The power of z should be negative, not positive -> Option D

4. Quick Check:

   Z-transform uses z^-n, not z^n [OK]

Hint: Check sign of exponent on z carefully [OK]

Common Mistakes:

• Using positive powers of z
• Misunderstanding summation limits
• Confusing discrete and continuous signals

5. You have a signal x[n] defined as 1 for n=0 to 3 and 0 elsewhere. How would you express its Z-transform X(z) and what does it represent?

hard

A. X(z) = 1 + z^{-1} + z^{-2} + z^{-3}, representing the signal's frequency components

B. X(z) = z + z^{2} + z^{3} + z^{4}, representing the time-domain signal

C. X(z) = \sum_{n=0}^{\infty} x[n] z^{n}, representing the signal's energy

D. X(z) = 1 + 2z^{-1} + 3z^{-2} + 4z^{-3}, representing the signal's amplitude

Solution

1. Step 1: Write the Z-transform for given x[n]

   Since x[n] = 1 for n=0 to 3, X(z) = sum from 0 to 3 of 1*z^{-n} = 1 + z^{-1} + z^{-2} + z^{-3}.

2. Step 2: Understand what X(z) represents

   X(z) shows the signal's frequency components in the complex z-plane.

3. Final Answer:

   X(z) = 1 + z^{-1} + z^{-2} + z^{-3}, representing the signal's frequency components -> Option A

4. Quick Check:

   Sum of z^-n terms = frequency components [OK]

Hint: Sum z^-n terms for signal duration [OK]

Common Mistakes:

• Using positive powers of z
• Incorrect coefficients in sum
• Misinterpreting what X(z) represents