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)).