Digital Low Pass Butterworth Filter in Python
In this article, we are going to discuss how to design a Digital Low Pass Butterworth Filter using Python. The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the pass band. Let us take the below specifications to design the filter and observe the Magnitude, Phase & Impulse Response of the Digital Butterworth Filter.
The specifications are as follows:
- Sampling rate of 40 kHz
- Pass band edge frequency of 4 kHz
- Stop band edge frequency of 8kHz
- Pass band ripple of 0.5 dB
- Minimum stop band attenuation of40 dB
We will plot the magnitude, phase, and impulse response of the filter.
Step-by-step Approach:
Step 1: Importing all the necessary libraries.
Python3
# import required modules import numpy as np import matplotlib.pyplot as plt from scipy import signal import math |
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Step 2: Define variables with the given specifications of the filter.
Python3
# Specifications of Filter # sampling frequency f_sample = 40000 # pass band frequency f_pass = 4000 # stop band frequency f_stop = 8000 # pass band ripple fs = 0.5 # pass band freq in radian wp = f_pass/(f_sample/2) # stop band freq in radian ws = f_stop/(f_sample/2) # Sampling Time Td = 1 # pass band ripple g_pass = 0.5 # stop band attenuation g_stop = 40 |
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Step3: Building the filter using signal.buttord function.
Python3
# Conversion to prewrapped analog frequency omega_p = (2/Td)*np.tan(wp/2) omega_s = (2/Td)*np.tan(ws/2) # Design of Filter using signal.buttord function N, Wn = signal.buttord(omega_p, omega_s, g_pass, g_stop, analog=True) # Printing the values of order & cut-off frequency! print("Order of the Filter=", N) # N is the order # Wn is the cut-off freq of the filter print("Cut-off frequency= {:.3f} rad/s ".format(Wn)) # Conversion in Z-domain # b is the numerator of the filter & a is the denominator b, a = signal.butter(N, Wn, 'low', True) z, p = signal.bilinear(b, a, fs) # w is the freq in z-domain & h is the magnitude in z-domain w, h = signal.freqz(z, p, 512) |
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Output:
Step 4: Plotting the Magnitude Response.
Python3
# Magnitude Response plt.semilogx(w, 20*np.log10(abs(h))) plt.xscale('log') plt.title('Butterworth filter frequency response') plt.xlabel('Frequency [Hz]') plt.ylabel('Amplitude [dB]') plt.margins(0, 0.1) plt.grid(which='both', axis='both') plt.axvline(100, color='green') plt.show() |
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Output:
Step 5: Plotting the Impulse Response.
Python3
# Impulse Response imp = signal.unit_impulse(40) c, d = signal.butter(N, 0.5) response = signal.lfilter(c, d, imp) plt.stem(np.arange(0, 40), imp, use_line_collection=True) plt.stem(np.arange(0, 40), response, use_line_collection=True) plt.margins(0, 0.1) plt.xlabel('Time [samples]') plt.ylabel('Amplitude') plt.grid(True) plt.show() |
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Output:
Step 6: Plotting the Phase Response.
Python3
# Phase Response fig, ax1 = plt.subplots() ax1.set_title('Digital filter frequency response') ax1.set_ylabel('Angle(radians)', color='g') ax1.set_xlabel('Frequency [Hz]') angles = np.unwrap(np.angle(h)) ax1.plot(w/2*np.pi, angles, 'g') ax1.grid() ax1.axis('tight') plt.show() |
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Output:
Below is the complete program based on the above approach:
Python
# import required modules import numpy as np import matplotlib.pyplot as plt from scipy import signal import math # Specifications of Filter # sampling frequency f_sample = 40000 # pass band frequency f_pass = 4000 # stop band frequency f_stop = 8000 # pass band ripple fs = 0.5 # pass band freq in radian wp = f_pass/(f_sample/2) # stop band freq in radian ws = f_stop/(f_sample/2) # Sampling Time Td = 1 # pass band ripple g_pass = 0.5 # stop band attenuation g_stop = 40 # Conversion to prewrapped analog frequency omega_p = (2/Td)*np.tan(wp/2) omega_s = (2/Td)*np.tan(ws/2) # Design of Filter using signal.buttord function N, Wn = signal.buttord(omega_p, omega_s, g_pass, g_stop, analog=True) # Printing the values of order & cut-off frequency! print("Order of the Filter=", N) # N is the order # Wn is the cut-off freq of the filter print("Cut-off frequency= {:.3f} rad/s ".format(Wn)) # Conversion in Z-domain # b is the numerator of the filter & a is the denominator b, a = signal.butter(N, Wn, 'low', True) z, p = signal.bilinear(b, a, fs) # w is the freq in z-domain & h is the magnitude in z-domain w, h = signal.freqz(z, p, 512) # Magnitude Response plt.semilogx(w, 20*np.log10(abs(h))) plt.xscale('log') plt.title('Butterworth filter frequency response') plt.xlabel('Frequency [Hz]') plt.ylabel('Amplitude [dB]') plt.margins(0, 0.1) plt.grid(which='both', axis='both') plt.axvline(100, color='green') plt.show() # Impulse Response imp = signal.unit_impulse(40) c, d = signal.butter(N, 0.5) response = signal.lfilter(c, d, imp) plt.stem(np.arange(0, 40), imp, use_line_collection=True) plt.stem(np.arange(0, 40), response, use_line_collection=True) plt.margins(0, 0.1) plt.xlabel('Time [samples]') plt.ylabel('Amplitude') plt.grid(True) plt.show() # Phase Response fig, ax1 = plt.subplots() ax1.set_title('Digital filter frequency response') ax1.set_ylabel('Angle(radians)', color='g') ax1.set_xlabel('Frequency [Hz]') angles = np.unwrap(np.angle(h)) ax1.plot(w/2*np.pi, angles, 'g') ax1.grid() ax1.axis('tight') plt.show() |
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Output:
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