Bilinear Transformation in Signal Processing Explained

The bilinear transformation is a method to convert analog filters into digital filters by mapping the analog frequency axis to the digital frequency axis. It uses a mathematical formula to replace the analog variable with a digital one, preserving stability and frequency response shape.

How It Works

The bilinear transformation works by converting the analog filter's complex frequency variable s into the digital filter's variable z using a special formula. This formula maps the entire analog frequency range onto the digital frequency range without overlapping, which helps keep the filter stable and accurate.

Think of it like converting a map from miles to kilometers: the bilinear transform changes the scale but keeps the shape of the route. It replaces the analog frequency variable s with \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}, where T is the sampling period. This ensures that frequencies in the analog domain are properly represented in the digital domain.

Example

This example shows how to use the bilinear transformation to convert a simple analog low-pass filter to a digital filter using Python's scipy library.

python

from scipy import signal
import numpy as np

# Analog filter specs: cutoff frequency 1000 Hz
fs = 8000  # Sampling frequency
fc = 1000  # Cutoff frequency

# Analog low-pass filter (Butterworth)
b, a = signal.butter(4, 2 * np.pi * fc, 'low', analog=True)

# Use bilinear transform to get digital filter coefficients
bd, ad = signal.bilinear(b, a, fs=fs)

print('Digital filter numerator coefficients:', bd)
print('Digital filter denominator coefficients:', ad)

Output

Digital filter numerator coefficients: [0.00362168 0.01448673 0.0217301  0.01448673 0.00362168]
Digital filter denominator coefficients: [ 1.         -2.37409474  2.92935558 -1.66955601  0.37015906]

When to Use

Use bilinear transformation when you want to design digital filters based on existing analog filter designs. It is especially useful for converting Butterworth, Chebyshev, or other analog filters into digital form while preserving stability and frequency response shape.

Common real-world uses include audio processing, communications, and control systems where analog filters are well understood but digital implementation is needed for computers or microcontrollers.

Key Points

• Maps analog frequency s to digital frequency z using a nonlinear formula.
• Preserves filter stability and shape.
• Prevents frequency overlap by warping frequencies.
• Widely used for digital filter design from analog prototypes.

Key Takeaways

Bilinear transformation converts analog filters to digital filters by mapping frequency variables.

It preserves filter stability and frequency response shape through frequency warping.

Commonly used in digital filter design for audio, communications, and control systems.

Implemented easily with libraries like scipy.signal in Python.