Bilinear transformation method in Signal Processing

Introduction

The bilinear transformation method helps convert analog filters into digital filters. It keeps the filter's important features while changing from continuous to discrete time.

When designing digital filters based on existing analog filter designs.

To convert a continuous-time filter into a discrete-time filter for digital signal processing.

When you want to avoid aliasing effects during filter conversion.

To maintain the stability of the filter after conversion.

When implementing filters in software or digital hardware that work with sampled data.

Syntax

Signal Processing

digital_filter = bilinear(analog_numerator, analog_denominator, sampling_frequency)

analog_numerator and analog_denominator are the coefficients of the analog filter's transfer function.

sampling_frequency is the rate at which the digital system samples the signal.

Examples

This converts a simple analog filter with numerator 1 and denominator s+1 into a digital filter with a sampling frequency of 1000 Hz.

Signal Processing

b, a = bilinear([1], [1, 1], 1000)

Converts a second-order analog filter to digital with a 2000 Hz sampling rate.

Signal Processing

b, a = bilinear([0, 0, 1], [1, 0.5, 1], 2000)

Sample Program

This program converts a simple analog filter with transfer function 1/(s+1) into a digital filter using the bilinear transformation method with a sampling frequency of 1000 Hz. It prints the digital filter coefficients.

Signal Processing

from scipy.signal import bilinear

# Analog filter coefficients for H(s) = 1 / (s + 1)
analog_num = [1]
analog_den = [1, 1]

# Sampling frequency in Hz
fs = 1000

# Convert analog filter to digital filter
b, a = bilinear(analog_num, analog_den, fs)

print('Digital filter numerator coefficients:', b)
print('Digital filter denominator coefficients:', a)

Important Notes

The bilinear transform maps the entire analog frequency axis to the digital frequency axis without aliasing.

Frequency warping occurs, so pre-warping frequencies before transformation can improve accuracy.

The method always produces stable digital filters if the analog filter is stable.

Summary

The bilinear transformation converts analog filters to digital filters safely.

It keeps filter stability and avoids aliasing.

Use it when you want to implement analog filter designs in digital systems.