Transfer function H(z) in Signal Processing - Mini Project: Build & Apply

Calculate Transfer Function H(z) from Filter Coefficients

📖 Scenario: You are working with digital filters in signal processing. Filters are described by their coefficients, and the transfer function H(z) shows how the filter changes signals in the frequency domain.Understanding H(z) helps you analyze and design filters for audio, communications, and control systems.

🎯 Goal: Build a Python program that calculates the transfer function H(z) for a digital filter given its numerator and denominator coefficients.You will create the coefficient lists, set a frequency point, compute H(z) at that point, and print the result.

📋 What You'll Learn

Create two lists called numerator and denominator with exact coefficients

Create a variable z representing a complex number on the unit circle

Calculate the transfer function H_z as the ratio of numerator and denominator polynomials evaluated at z

Print the value of H_z

💡 Why This Matters

🌍 Real World

Digital filters are used in audio processing, communications, and control systems to shape signals. Calculating H(z) helps engineers understand how filters affect different frequencies.

💼 Career

Signal processing engineers and data scientists working with time series or sensor data often analyze transfer functions to design and evaluate filters.

1

Create filter coefficient lists

Create a list called numerator with values [0.0675, 0.1349, 0.0675] and a list called denominator with values [1.0, -1.1430, 0.4128].

Signal Processing

# Your code here

Hint

Use square brackets to create lists with the exact numbers given.

2

Set the complex variable z on the unit circle

Create a variable called z and set it to the complex number exp(-1j * pi / 4) using the cmath module.

Signal Processing

numerator = [0.0675, 0.1349, 0.0675]
denominator = [1.0, -1.1430, 0.4128]
import cmath
# Your code here

Hint

Use cmath.exp() and cmath.pi to create the complex number.

3

Calculate the transfer function H(z)

Calculate the numerator polynomial value at z and store it in num_val. Calculate the denominator polynomial value at z and store it in den_val. Then calculate H_z as num_val / den_val.

Signal Processing

numerator = [0.0675, 0.1349, 0.0675]
denominator = [1.0, -1.1430, 0.4128]
import cmath
z = cmath.exp(-1j * cmath.pi / 4)
# Your code here

Hint

Use enumerate() to get index and coefficient, then sum coef * z**(-i) for numerator and denominator.

4

Print the transfer function value

Print the value of H_z.

Signal Processing

numerator = [0.0675, 0.1349, 0.0675]
denominator = [1.0, -1.1430, 0.4128]
import cmath
z = cmath.exp(-1j * cmath.pi / 4)
num_val = sum(coef * z**(-i) for i, coef in enumerate(numerator))
den_val = sum(coef * z**(-i) for i, coef in enumerate(denominator))
H_z = num_val / den_val
# Your code here

Hint

Use print(H_z) to display the transfer function value.