A common problem in signal processing is to automatically determine the optimal cutoff frequency that should be employed in a low-pass filter to attenuate as much as possible the noise without compromising the signal content of the data.
Before we continue, see this Jupyter notebook for an overview about data filtering if needed.
Unfortunately, there is no definite solution for this problem, but there are some techniques, with different degrees of success, to try to determine the optimal cutoff frequency.
David Winter, in his classic book Biomechanics and motor control of human movement, proposed a method to find the optimal cutoff frequency based on residual analysis of the difference between filtered and unfiltered signals over a range of cutoff frequencies. The optimal cutoff frequency is the one where the residual starts to change very little because it is considered that from this point, it's being filtered mostly noise and minimally signal, ideally. This concept is straightforward to implement.
optcutfreq.py from Python module
optcutfreq is an implementation of this method and it is divided in three parts (after the help section): first, the residuals over a range of cutoff frequencies are calculated; second, an algorithm tries to find the noisy region (with a supposed linear behavior in the frequency domain) of the residuals versus cutoff frequencies plot and finds the optimal cutoff frequency; and third, the results are plotted. The code is lengthy relatively to the simplicity of the idea because of the long help section, the implementation of the automatic search and a rich plot. Here is the function signature:
fc_opt = optcutfreq(y, freq=1, fclim=, show=False, ax=None):
Let's test this function with benchmark data.
In 1977, Pezzack, Norman and Winter published a paper where they investigated the effects of differentiation and filtering processes on experimental data (the angle of a bar manipulated in space). Since then, these data have became a benchmark to test new algorithms. Let's work with these data (available at http://isbweb.org/data/pezzack/index.html). The data have the angular displacement measured by video and the angular acceleration directly measured by an accelerometer, which we will consider as the true acceleration.
Part of these data are showing next:
# Import the necessary libraries import numpy as np %matplotlib inline import matplotlib.pyplot as plt
# load data file time, disp, disp2, aacc = np.loadtxt('./../data/Pezzack.txt', skiprows=6, unpack=True) dt = np.mean(np.diff(time)) # plot data fig, (ax1,ax2) = plt.subplots(1, 2, sharex = True, figsize=(11, 4)) plt.suptitle("Pezzack's benchmark data", fontsize=20) ax1.plot(time, disp, 'b') ax1.set_xlabel('Time [s]'); ax1.set_ylabel('Angular displacement [rad]') ax2.plot(time, aacc, 'g') ax2.set_xlabel('Time [s]'); ax2.set_ylabel('Angular acceleration [rad/s$^2$]') plt.subplots_adjust(wspace=0.3)
And using the residual analsysis code:
from optcutfreq import optcutfreq freq = np.mean(1/np.diff(time)) fc_opt = optcutfreq(disp, freq=freq, show=True)
The optimal cutoff frequency found is 5.6 Hz. Note that the filtering process is relevant only for the derivative of the data; we cannot distinguish the unfiltered and unfiltered displacements (see that the RMSE residual is very small).
Let's employ this filter, differentiate the data twice and compare with the true acceleration as we did before:
from scipy.signal import butter, filtfilt # Butterworth filter # Correct the cutoff frequency for the number of passes in the filter C = 0.802 # for dual pass; C = (2**(1/npasses) - 1)**0.25 b, a = butter(2, (fc_opt/C)/(freq/2)) dispf = filtfilt(b, a, disp) aaccBW = np.diff(dispf, 2)*freq*freq # RMSE: rmseBW = np.sqrt(np.mean((aaccBW-aacc[1:-1])**2)) # plot data fig, ax1 = plt.subplots(1, 1, figsize=(11, 4)) plt.suptitle("Pezzack's benchmark data", fontsize=20) ax1.plot(time[1:-1], aacc[1:-1], 'g', label='Analog acceleration: (True value)') ax1.plot(time[1:-1], aaccBW, 'r', label='Butterworth %.3g Hz: RMSE = %0.2f' %(fc_opt,rmseBW)) ax1.set_xlabel('Time [s]'); ax1.set_ylabel('Angular acceleration [rad/s$^2$]'); plt.legend(frameon=False, fontsize=12, loc='upper left');
The performance seems satisfactory (see this Jupyter notebook for a comparison using other filters), but it is known that this residual analysis algorithm results in oversmoothing the kinematic data (see http://www.clinicalgaitanalysis.com/faq/cutoff.html).
To read more about the determination of the optimal cutoff frequency, see the following papers: