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

Inverse Z-transform

📖 Scenario: You are working with digital signals and need to find the original time-domain sequence from its Z-transform representation. This is common in signal processing when analyzing discrete-time systems.

🎯 Goal: Learn how to compute the inverse Z-transform of a simple rational function to get the original discrete-time signal sequence.

📋 What You'll Learn

Create a dictionary representing the Z-transform coefficients.

Set a maximum time index for the inverse transform calculation.

Calculate the inverse Z-transform sequence using a loop.

Print the resulting time-domain sequence.

💡 Why This Matters

🌍 Real World

Inverse Z-transform is used in digital signal processing to find the original signal from its Z-transform, which helps in analyzing and designing filters and systems.

💼 Career

Understanding inverse Z-transform is important for roles in signal processing, communications engineering, and control systems where discrete-time system analysis is required.

1

Create the Z-transform coefficients dictionary

Create a dictionary called z_transform with these exact entries: 0: 1, 1: -0.5, 2: 0.25 representing the coefficients of the Z-transform polynomial.

Signal Processing

# Create the dictionary z_transform with keys 0, 1, 2 and values 1, -0.5, 0.25
# Your code here

Hint

Think of the Z-transform as a polynomial with powers of z. The dictionary keys are powers, and values are coefficients.

2

Set the maximum time index for the inverse transform

Create a variable called max_time and set it to 5 to define how many time points to calculate in the inverse Z-transform.

Signal Processing

z_transform = {0: 1, 1: -0.5, 2: 0.25}
# Set max_time to 5
# Your code here

Hint

This variable controls how many terms of the time sequence you want to find.

3

Calculate the inverse Z-transform sequence

Create a list called time_sequence and use a for loop with variable n from 0 to max_time (inclusive) to calculate each term of the inverse Z-transform. For each n, sum the products of z_transform[k] and (-1)**k times n choose k for all k in z_transform keys where k <= n. Use the math.comb function for combinations.

Signal Processing

z_transform = {0: 1, 1: -0.5, 2: 0.25}
max_time = 5
# Calculate time_sequence list using a for loop and math.comb
# Your code here

Hint

Use the binomial coefficient and alternating signs to compute each term of the inverse sequence.

4

Print the inverse Z-transform time sequence

Write a print statement to display the time_sequence list.

Signal Processing

import math

z_transform = {0: 1, 1: -0.5, 2: 0.25}
max_time = 5

time_sequence = []
for n in range(max_time + 1):
    value = 0
    for k in z_transform:
        if k <= n:
            value += z_transform[k] * ((-1) ** k) * math.comb(n, k)
    time_sequence.append(value)

# Print the time_sequence list
# Your code here

Hint

The printed list shows the original discrete-time signal values from n=0 to n=5.