How to design and implement a digital low-pass filter on an Arduino

How to Design and Implement a Low-Pass Filter for Arduino Projects

Introduction to Signal Processing

• In Arduino projects, sensor measurements often contain noise. This video demonstrates how to design and implement a low-pass filter suitable for such applications.

• An artificial signal is created for testing the filter, consisting of a 2 Hz sine wave as the fundamental component and a 50 Hz sine wave representing unwanted noise.

Understanding Fourier Transform

• The Discrete Fourier Transform (DFT) provides insight into the signal by displaying each sine curve as a peak in the Fourier domain.

• A first-order low-pass filter is introduced, represented by the transfer function omega_0/s + omega_0 , which will be used to eliminate high-frequency noise while preserving lower frequencies.

Designing the Low-Pass Filter

• The cutoff frequency ( omega_0 ) is set at 2pi times 5 radians per second (5 Hz), aimed at preserving the 2 Hz signal while attenuating the 50 Hz noise.

• Testing shows that when passing through the filter, the 2 Hz signal remains largely unaffected, whereas the magnitude of the 50 Hz signal is significantly reduced.

Bode Plot Analysis

• The Bode plot illustrates how different frequencies are affected by the filter; at 1 Hz, signals remain largely unchanged, while at 5 Hz they are attenuated to about half their original magnitude.

• At higher frequencies like 50 Hz and beyond, only a small fraction of those signals remains after filtering—10% at 50 Hz and just 1% at 500 Hz.

Phase Delay Considerations

• The phase plot indicates minimal delay at lower frequencies (1 Hz), but significant delays occur as frequency increases—up to about -90 degrees at higher frequencies.

Transitioning to Arduino Implementation

• The continuous transfer function isn't suitable for real-time processing on Arduino; thus, conversion into discrete form is necessary for practical implementation.

• A Python script has been prepared to handle complex calculations required for creating this digital filter.

Sampling Frequency and Discrete Transfer Function

• To simplify processing on Arduino, sampling frequency is throttled down to one kilohertz. This affects how we derive our discrete transfer function using bilinear transformation methods.

Constructing Difference Equations

• Coefficients from discrete transfer functions inform us about constructing difference equations needed for filtering in real-time applications.

Coding and Testing on Arduino

• The code begins with defining test signals followed by implementing difference equations where new filtered values depend on previous raw values and filtered outputs.

Results of Filtering Process

• Initial results show that high-frequency components have been effectively removed from test signals; however, some attenuation occurs along with slight delays in output response.

Evaluating Filter Performance

Low-Pass Filter Design and Implementation

Adjusting Cutoff Frequency and Test Signal

• The cutoff frequency in the script is set to 30 Hz, preserving signals from 0 to 20 Hz. The main test signal is adjusted to 20 Hz.

• The power spectrum of the test signal is displayed, followed by generating the continuous transfer function of the filter and its bode diagram.

Understanding Filter Performance

• A low-pass filter has a broad transition band where frequencies near the cutoff are attenuated rather than completely removed.

• The pass band (0-2 Hz for a 5 Hz cutoff) preserves signals, while the stop band (above 30 Hz) removes most content; between these ranges, attenuation occurs without complete removal.

Ideal vs. Real Filters

• Ideally, a low-pass filter would perfectly pass all signals below the cutoff frequency with no transition band; however, real-world filters have limitations.

• Higher-order filters like Butterworth provide better performance: a second-order offers greater attenuation and smaller transition bands compared to first-order filters.

Implementing Butterworth Filters

• While higher-order Butterworth filters approach ideal performance, practical considerations limit their use due to complexity and potential drawbacks.

• The Butterworth filter's transfer function resembles that of first-order filters but uses polynomial denominators based on order n.

Coding and Coefficient Calculation

• Python code can compute coefficients for any order Butterworth filter using recursion formulas established since 1930.

• After obtaining the continuous transfer function, discrete transfer functions can be computed for implementation on Arduino.

Comparing Filter Orders

• Testing shows that while a second-order Butterworth filter retains lower frequencies better than first-order ones, it introduces more delay in filtered signals.

• Bode plots reveal that higher orders increase delay significantly; for instance, a fourth-order may introduce up to 45 degrees of phase shift within its pass band.

Balancing Attenuation and Delay

• When designing low-pass filters, one must balance attenuation characteristics against delays introduced by higher orders as shown in Bode plots.

• Alternative high-order filters exist beyond Butterworth; if their characteristics do not suit your project needs, consider exploring other standard options.

Practical Application Insights

• A comparison between first and second order low-pass filters applied to raw accelerometer readings indicates both effectively remove high-frequency components but differ in smoothness and delay effects.