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})