Deep Dive into Butterworth Low Pass Filter

This is the 7th article in the AI Chat series, where we’ll explore the principles and implementation of the Butterworth low-pass filter.

Table of Contents

1. Introduction
2. Mathematical Foundation
3. Filter Design Process
4. Implementation Methods
5. Experimental Validation
6. Conclusion

1. Introduction

In signal processing, particularly for positioning and sensor data applications, filter selection is crucial. The Butterworth low-pass filter has become one of the most widely used filters due to its:

• Maximally flat frequency response in the passband
• Simple mathematical structure
• Straightforward implementation
• Predictable behavior

This article will guide you through the complete design process, from mathematical principles to practical implementation, using a real-world example from the VQF IMU attitude fusion algorithm.

2. Mathematical Foundation

2.1 Butterworth Filter Characteristics

The Butterworth filter is designed to have a maximally flat magnitude response in the passband. For a second-order filter, this means:

• DC gain: 1 (0 dB)
• Cutoff frequency: -3 dB point
• Roll-off rate: -40 dB/decade after cutoff

2.2 Normalized Prototype Filter

The standard approach starts with a normalized prototype filter with cutoff frequency at 1 rad/s:

Transfer Function:

H(s) = 1 / (s² + √2s + 1)

Why 1 rad/s?

• This is a convention in filter design
• Provides a standardized starting point
• Makes calculations and comparisons easier
• All Butterworth filters begin with this normalized form

Mathematical Verification: At ω = 1 rad/s:

H(j·1) = 1 / ((j·1)² + √2·j·1 + 1)
        = 1 / (-1 + j√2 + 1)
        = 1 / (j√2)

The magnitude is exactly 1/√2 ≈ 0.707, which corresponds to the -3 dB cutoff point.

3. Filter Design Process

3.1 Traditional Design Steps

The classical approach involves four main steps:

1. Normalized Prototype: Start with H(s) = 1/(s² + √2s + 1)
2. Frequency Scaling: Scale to desired cutoff frequency ωc
3. Bilinear Transform: Convert from s-domain to z-domain
4. Coefficient Extraction: Derive the difference equation coefficients

3.2 Frequency Scaling

To scale the cutoff frequency from 1 rad/s to ωc:

Substitute s → s/ωc:

H(s) = 1 / ((s/ωc)² + √2(s/ωc) + 1)
     = ωc² / (s² + √2ωc·s + ωc²)

3.3 Bilinear Transform

Convert from continuous-time to discrete-time:

Transform Formula:

s = (2/T) × (z-1)/(z+1)

Where T is the sampling period.

4. Implementation Methods

4.1 VQF Implementation Approach

The VQF algorithm uses a more sophisticated approach that implicitly handles frequency pre-warping:

void filterCoeffs(vqf_real_t tau, vqf_real_t Ts, double outB[], double outA[])
{
    assert(tau > 0);
    assert(Ts > 0);
    
    // Calculate pre-warped cutoff frequency
    double fc = (M_SQRT2 / (2.0*M_PI))/double(tau);
    double C = tan(M_PI*fc*double(Ts));
    double D = C*C + sqrt(2)*C + 1;
    
    // Filter coefficients
    double b0 = C*C/D;
    outB[0] = b0;      // b0
    outB[1] = 2*b0;    // b1
    outB[2] = b0;      // b2
    
    outA[0] = 2*(C*C-1)/D;           // a1
    outA[1] = (1-sqrt(2)*C+C*C)/D;   // a2
}

4.2 Why VQF Method Works

The key insight is frequency pre-warping compensation:

Pre-warping Formula:

ωc = (2/T) × tan(ωd×T/2)

Where:

• ωc = pre-warped cutoff frequency
• ωd = desired digital cutoff frequency
• T = sampling period

Why This Matters:

• The bilinear transform introduces frequency distortion
• Low frequencies are compressed, high frequencies are expanded
• Without pre-warping, the digital filter would have the wrong cutoff frequency
• The VQF method automatically compensates for this distortion

4.3 Traditional vs. VQF Approach

5. Experimental Validation

5.1 Test Setup

Let’s validate our understanding by comparing the VQF implementation with SciPy’s industry-standard implementation:

Parameters:

• Sampling Frequency: 1000 Hz
• Cutoff Frequency: 100 Hz
• Filter Order: 2
• Nyquist Frequency: 500 Hz

5.2 Coefficient Comparison

Method 1: SciPy Implementation

b coefficients: [0.06745527 0.13491055 0.06745527]
a coefficients: [ 1.        -1.1429805  0.4128016]

Method 2: VQF Implementation

b coefficients: [0.06745527 0.13491055 0.06745527]
a coefficients: [ 1.        -1.1429805  0.4128016]

Result: Perfect match! Both methods produce identical coefficients.

5.3 Frequency Response Analysis

The frequency response confirms:

• -3 dB cutoff exactly at 100 Hz
• Passband flatness as expected
• Roll-off rate of -40 dB/decade
• No ripples in passband or stopband

6. Conclusion

6.1 Key Takeaways

1. Mathematical Foundation: The Butterworth filter’s normalized prototype provides a solid starting point
2. Pre-warping Importance: Frequency compensation is crucial for accurate digital filter design
3. VQF Method Superiority: The VQF approach automatically handles pre-warping, making it more robust
4. Validation: Both theoretical and experimental results confirm the correctness of our implementation

6.2 Practical Applications

This understanding is essential for:

• Sensor data processing in robotics and IoT
• Audio signal filtering in consumer electronics
• Control system design in automotive and aerospace
• Biomedical signal processing in healthcare devices

6.3 Further Reading

For deeper exploration, consider:

• VQF IMU Attitude Fusion Algorithm - The source of our implementation
• Stack Overflow Discussion - Detailed mathematical derivation
• Signal Processing textbooks on digital filter design

The Butterworth filter remains a cornerstone of signal processing, and understanding its implementation details empowers engineers to create robust, efficient filtering solutions for real-world applications.