fft bandpass filter in python

Question

What I try is to filter my data with fft. I have a noisy signal recorded with 500Hz as a 1d- array. My high-frequency should cut off with 20Hz and my low-frequency with 10Hz

Answer 1

There's a fundamental flaw in what you are trying to do here - you're applying a rectangular window in the frequency domain which will result in a time domain signal which has been convolved with a sinc function. In other words there will be a large amount of "ringing" in the time domain signal due to the step changes you have introduced in the frequency domain. The proper way to do this kind of frequency domain filtering is to apply a suitable window function in the frequency domain. Any good introductory DSP book should cover this.

Answer 2

It's worth noting that the magnitude of the units of your bp are not necessarily going to be in Hz, but are dependent on the sampling frequency of signal, you should use scipy.fftpack.fftfreq for the conversion. Also if your signal is real you should be using scipy.fftpack.rfft. Here is a minimal working example that filters out all frequencies less than a specified amount:

import numpy as np
from scipy.fftpack import rfft, irfft, fftfreq

time   = np.linspace(0,10,2000)
signal = np.cos(5*np.pi*time) + np.cos(7*np.pi*time)

W = fftfreq(signal.size, d=time[1]-time[0])
f_signal = rfft(signal)

# If our original signal time was in seconds, this is now in Hz    
cut_f_signal = f_signal.copy()
cut_f_signal[(W<6)] = 0

cut_signal = irfft(cut_f_signal)

We can plot the evolution of the signal in real and fourier space:

import pylab as plt
plt.subplot(221)
plt.plot(time,signal)
plt.subplot(222)
plt.plot(W,f_signal)
plt.xlim(0,10)
plt.subplot(223)
plt.plot(W,cut_f_signal)
plt.xlim(0,10)
plt.subplot(224)
plt.plot(time,cut_signal)
plt.show()