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

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

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)

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()

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()

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()

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()

Output: