Transfer function H(z) in Signal Processing

Introduction

The transfer function H(z) helps us understand how a digital filter changes an input signal to produce an output signal.

When designing a digital filter to remove noise from audio recordings.

When analyzing how a system responds to different frequencies in a signal.

When simulating the behavior of digital control systems.

When studying the stability and performance of digital signal processing algorithms.

Syntax

Signal Processing

H(z) = Y(z) / X(z)

where:
- H(z) is the transfer function
- Y(z) is the output signal in z-domain
- X(z) is the input signal in z-domain

The transfer function is a ratio of output to input in the z-domain (frequency domain for discrete signals).

It is often represented as a ratio of polynomials in z or z^-1.

Examples

This is a simple first-order digital filter transfer function.

Signal Processing

H(z) = (1 - 0.5z^{-1}) / (1 - 0.8z^{-1})

A second-order filter with numerator and denominator polynomials.

Signal Processing

H(z) = (1 + 0.3z^{-1} + 0.2z^{-2}) / (1 - 0.4z^{-1} + 0.12z^{-2})

Sample Program

This code plots the frequency response of a simple digital filter defined by its transfer function H(z).

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import freqz

# Define filter coefficients for H(z) = (1 - 0.5z^{-1}) / (1 - 0.8z^{-1})
b = [1, -0.5]  # numerator coefficients
 a = [1, -0.8]  # denominator coefficients

# Calculate frequency response
w, h = freqz(b, a, worN=8000)

# Plot magnitude response
plt.plot(w / np.pi, 20 * np.log10(abs(h)))
plt.title('Frequency Response of H(z)')
plt.xlabel('Normalized Frequency (×π rad/sample)')
plt.ylabel('Magnitude (dB)')
plt.grid(True)
plt.show()

Important Notes

The transfer function H(z) is key to analyzing digital filters and systems.

Using tools like Python's scipy.signal.freqz helps visualize how filters affect signals.

Summary

H(z) shows how input signals are changed by digital filters.

It is a ratio of output to input in the z-domain.

Plotting frequency response helps understand filter behavior.