Time series data may contain signals at many different frequencies. Sharp increases or decreases have a high frequency. Slow increases or decreases have a low frequency. Filtering allows us to take different frequency components out of the data.
Signal filtering is a science on its own and I’ll focus on the practical aspects here and stick to two filter types: butterworth and Chebyshev type I. Each of those filters can be used for different purposes. We can use them as low pass, high pass, band pass or notch filters. Low pass filters leave low frequencies alone but attack high frequencies. High pass filters leave high frequencies alone but attach low frequencies. The title image shows an example of low and high pass filters used on the same data. Band pass filters leave a specific frequency band alone and attack all other frequencies. Notch filters attack a specific frequency band, leaving the rest alone. Let’s look at an example.
import numpy as np import matplotlib.pyplot as plt import seaborn as sns from scipy.signal import butter, cheby1, filtfilt data = np.load("example_data.npy") order = 3 Wn = 4000 # in Hz btype = 'lowpass' fs = 50000 # in Hz b, a = butter(order, Wn, btype, fs = fs) data_butter = filtfilt(b, a, data)
This is a butterworth lowpass filter with a cutoff frequency of 4000Hz (
Wn). That means, signals below 4000Hz are is the pass band. They are largely left alone. Signals above 4000Hz are in the stop band, they are diminished.
fs is the sampling frequency of the data. If the units are Hz, it tells us how many data points are recorded during one second.
filtfilt is the function that does the actual filtering on the data, based on the filter (
b, a) that was designed previously. Filtering is not a perfect process. Filters have what is called roll-off at the critical 4000Hz frequency.
Ideally, we would like a filter response that falls down straight. Anything in the pass band is untouched, anything in the stop band is shutdown the same way. As you can see, our actual filter does no live up to the ideal. It already slightly attenuates signal that is part of the pass band and it falls much slower in the stop band. If we need a steeper roll off, we can increase the order of our filter.
Some filter types have steeper roll off than others. For example, the Chebyshev type I filter achieves steeper roll off by tolerating some ripple in the pass band.
This can lead to distortions in the data depending on the size of the ripple. The Chebyshev type I filter takes an argument
rp that defines the amount of pass band ripple that is tolerated in units of dB. The more ripple we tolerate, the steeper the roll off will be. Here you can see how large ripple causes oscillations in the data.
Generally, the butterworth filter is sufficient for most situations and is safer because it does not ripple. I hope this post helped you filtering your own data. If you want to learn more, check out the SciPy signal docs. Both the
cheby1 filter are there with many, many more.