Digital Band Reject Butterworth Filter in Python

In this article, we are going to discuss how to design a Digital Band Reject 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.

What is Digital Bandreject Filter?

A band-pass filter is a filter that passes frequencies within a range and rejects frequencies outside that range.

How it’s different from Highpass, Lowpass & Bandpass:

The main difference can be spotted by observing the magnitude response of the Band Pass Filter. In the Band-Reject filter, all the signals between the specified frequency range get rejected by the filter. 

The specifications are as follows:  

• The sampling rate of 12 kHz.
• Pass band edge frequencies are 2100 Hz & 4500 Hz.
• Stop band edge frequencies are 2700 Hz & 3900 Hz.
• Pass band ripple of 0.6 dB.
• Minimum stop band attenuation of 45 dB.

We will plot the magnitude, phase, and impulse response of the filter.

Step-by-step Approach:

Before starting, first, we will create a user-defined function to convert the edge frequencies, we are defining it as the convert() method.

Python3

# explicit function to convert
# edge frequencies


def convertX(f_sample, f):
    w = []

    for i in range(len(f)):
        b = 2*((f[i]/2) / (f_sample/2))
        w.append(b)

    omega_mine = []

    for i in range(len(w)):
        c = (2/Td)*np.tan(w[i]/2)
        omega_mine.append(c)

    return omega_mine

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 = 12000

# pass band frequency
f_pass = [2100, 4500]

# stop band frequency
f_stop = [2700, 3900]

# pass band ripple
fs = 0.5

# Sampling Time
Td = 1

# pass band ripple
g_pass = 0.6

# stop band attenuation
g_stop = 45

Step 3: Building the filter using signal.buttord() function.

Python3

# Conversion to prewrapped analog
# frequency
omega_p = convertX(f_sample, f_pass)
omega_s = convertX(f_sample, f_stop)

# 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
# N is the order
print("Order of the Filter=", N)

# Wn is the cut-off freq of the filter
print("Cut-off frequency= {:} 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, 'bandpass', 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, markerfmt='D', 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

# Frequency 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:

Python3

# import required modules
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
import math

# explicit function to convert
# edge frequencies
def convertX(f_sample, f):
    w = []

    for i in range(len(f)):
        b = 2*((f[i]/2)/(f_sample/2))
        w.append(b)

    omega_mine = []

    for i in range(len(w)):
        c = (2/Td)*np.tan(w[i]/2)
        omega_mine.append(c)

    return omega_mine

# Specifications of Filter
# sampling frequency
f_sample = 12000

# pass band frequency
f_pass = [2100, 4500]

# stop band frequency
f_stop = [2700, 3900]

# pass band ripple
fs = 0.5

# Sampling Time
Td = 1

# pass band ripple
g_pass = 0.6

# stop band attenuation
g_stop = 45

# Conversion to prewrapped analog
# frequency
omega_p = convertX(f_sample, f_pass)
omega_s = convertX(f_sample, f_stop)

# 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
# N is the order
print("Order of the Filter=", N)

# Wn is the cut-off freq of the filter
print("Cut-off frequency= {:} 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, 'bandpass', 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,
         markerfmt='D',
         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()

# Frequency 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: