How to filter a signal with a Butterworth filter in SciPy

A sampled signal can carry energy from the phenomenon of interest and from faster variation that obscures it. A low-pass Butterworth filter preserves slower content with a smooth passband while reducing frequencies above a chosen cutoff.

The scipy.signal.butter() function turns the filter order, cutoff, and sampling rate into coefficients. Requesting second-order sections with output=“sos” avoids the numerical sensitivity of transfer-function coefficients and produces the representation expected by sosfiltfilt().

The sosfiltfilt() function processes an array forward and backward, which cancels phase delay for offline analysis but uses future samples. A two-tone signal at 5 Hz and 40 Hz makes the attenuation measurable in the FFT; live or causal pipelines need sosfilt() instead.

1. Create butterworth_filter.py with the imports and two-tone sampled signal.

   butterworth_filter.py

   import numpy as np
   from scipy.signal import butter, sosfiltfilt
    
   sample_rate = 200.0
   time = np.arange(0.0, 2.0, 1.0 / sample_rate)
   raw = np.sin(2 * np.pi * 5 * time) + 0.5 * np.sin(2 * np.pi * 40 * time)

   The sample_rate value must match measured data. Its 100 Hz Nyquist frequency is above the 12 Hz cutoff used here.

2. Append the Butterworth design and forward-backward filter below the signal definition.

   sos = butter(4, 12.0, btype="lowpass", fs=sample_rate, output="sos")
   filtered = sosfiltfilt(sos, raw)

   Supplying fs expresses the cutoff in hertz. For a Butterworth filter, the cutoff is the point where the passband gain falls by about 3 dB.

3. Append the FFT amplitude comparison below the filtering call.

   frequencies = np.fft.rfftfreq(raw.size, d=1 / sample_rate)
   raw_amplitudes = np.abs(np.fft.rfft(raw)) * 2 / raw.size
   filtered_amplitudes = np.abs(np.fft.rfft(filtered)) * 2 / filtered.size
    
   for target in (5.0, 40.0):
       index = np.argmin(np.abs(frequencies - target))
       print(
           f"{target:>4.0f} Hz: raw={raw_amplitudes[index]:.3f}, "
           f"filtered={filtered_amplitudes[index]:.3f}"
       )

4. Run butterworth_filter.py to confirm the retained and attenuated frequency components.

   $ python3 butterworth_filter.py
      5 Hz: raw=1.000, filtered=1.000
     40 Hz: raw=0.500, filtered=0.002

   The 5 Hz component keeps its input amplitude, while the 40 Hz component falls from 0.500 to 0.002 after filtering.