How to implement IIR Bandpass Butterworth Filter using Scipy – Python?

IIR stands for Infinite Impulse Response, It is one of the striking features of many linear-time invariant systems that are distinguished by having an impulse response h(t)/h(n) which does not become zero after some point but instead continues infinitely.

What is IIR Bandpass Butterworth ?

It basically behaves just like an ordinary digital Bandpass Butterworth Filter with an infinite impulse response.

The specifications are as follows: 

• Pass band frequency: 1400-2100 Hz
• Stop band frequency: 1050-24500 Hz
• Pass band ripple: 0.4dB
• Stop band attenuation: 50 dB
• Sampling frequency: 7 kHz

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

Step-by-step Approach:

Step 1: Importing all the necessary libraries.

Python3

# import required library
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt

Step 2: Defining user defined functions mfreqz() and impz(). [mfreqz is a function for magnitude and phase plot & impz is function for impulse and step response]

Python3

def mfreqz(b, a, Fs):

    # Compute frequency response of the filter
    # using signal.freqz function
    wz, hz = signal.freqz(b, a)

    # Calculate Magnitude from hz in dB
    Mag = 20*np.log10(abs(hz))

    # Calculate phase angle in degree from hz
    Phase = np.unwrap(np.arctan2(np.imag(hz), np.real(hz)))*(180/np.pi)

    # Calculate frequency in Hz from wz
    Freq = wz*Fs/(2*np.pi)

    # Plot filter magnitude and phase responses using subplot.
    fig = plt.figure(figsize=(10, 6))

    # Plot Magnitude response
    sub1 = plt.subplot(2, 1, 1)
    sub1.plot(Freq, Mag, 'r', linewidth=2)
    sub1.axis([1, Fs/2, -100, 5])
    sub1.set_title('Magnitude Response', fontsize=20)
    sub1.set_xlabel('Frequency [Hz]', fontsize=20)
    sub1.set_ylabel('Magnitude [dB]', fontsize=20)
    sub1.grid()

    # Plot phase angle
    sub2 = plt.subplot(2, 1, 2)
    sub2.plot(Freq, Phase, 'g', linewidth=2)
    sub2.set_ylabel('Phase (degree)', fontsize=20)
    sub2.set_xlabel(r'Frequency (Hz)', fontsize=20)
    sub2.set_title(r'Phase response', fontsize=20)
    sub2.grid()

    plt.subplots_adjust(hspace=0.5)
    fig.tight_layout()
    plt.show()

# Define impz(b,a) to calculate impulse response
# and step response of a system
# input: b= an array containing numerator coefficients,
# a= an array containing denominator coefficients
def impz(b, a):

    # Define the impulse sequence of length 60
    impulse = np.repeat(0., 60)
    impulse[0] = 1.
    x = np.arange(0, 60)

    # Compute the impulse response
    response = signal.lfilter(b, a, impulse)

    # Plot filter impulse and step response:
    fig = plt.figure(figsize=(10, 6))
    plt.subplot(211)
    plt.stem(x, response, 'm', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Impulse response', fontsize=15)

    plt.subplot(212)
    step = np.cumsum(response)  # Compute step response of the system

    plt.stem(x, step, 'g', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Step response', fontsize=15)
    plt.subplots_adjust(hspace=0.5)

    fig.tight_layout()
    plt.show()

Step 3:Define variables with the given specifications of the filter.

Python3

# Given specification
Fs = 7000  # Sampling frequency in Hz
fp = np.array([1400, 2100])  # Pass band frequency in Hz
fs = np.array([1050, 2450])  # Stop band frequency in Hz
Ap = 0.4  # Pass band ripple in dB
As = 50  # stop band attenuation in dB

Step 4: Computing the cut-off frequency

Python3

# Compute pass band and stop band edge frequencies
wp = fp/(Fs/2)  # Normalized passband edge frequencies w.r.t. Nyquist rate
ws = fs/(Fs/2)  # Normalized stopband edge frequencies

Step 5: Compute cut-off frequency & order

Python3

# Compute order of the digital Butterworth filter using signal.buttord
N, wc = signal.buttord(wp, ws, Ap, As, analog=True)
# Print the order of the filter and cutoff frequencies
print('Order of the filter=', N)
print('Cut-off frequency=', wc)

Output:

Step 6: Compute the filter co-efficient 

Python3

# Design digital Butterworth band pass
# filter using signal.butter function
z, p = signal.butter(N, wc, 'bandpass')

# Print numerator and denomerator
# coefficients of the filter
print('Numerator Coefficients:', z)
print('Denominator Coefficients:', p)

Output:

Step 7: Compute frequency response using signal.freqz() function

Python3

# Compute frequency response of the filter using signal.freqz function
wz, hz = signal.freqz(z, p)

Step 8: Plotting the Magnitude & Phase Response

Python3

# Call mfreqz to plot the magnitude and phase response
mfreqz(z, p, Fs)

Output:

Step 9: Plotting the Impulse and Step Response

Python3

# Call impz function to plot impulse
# and step response of the filter
impz(z, p)

Output:

Below is the implementation:

Python3

# import required library
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt

# Compute magnitude and phase response
# using mfreqz function
def mfreqz(b, a, Fs):

    # Compute frequency response of the filter
    # using signal.freqz function
    wz, hz = signal.freqz(b, a)

    # Calculate Magnitude from hz in dB
    Mag = 20*np.log10(abs(hz))

    # Calculate phase angle in degree from hz
    Phase = np.unwrap(np.arctan2(np.imag(hz), np.real(hz)))*(180/np.pi)

    # Calculate frequency in Hz from wz
    Freq = wz*Fs/(2*np.pi)

    # Plot filter magnitude and phase responses using subplot.
    fig = plt.figure(figsize=(10, 6))

    # Plot Magnitude response
    sub1 = plt.subplot(2, 1, 1)
    sub1.plot(Freq, Mag, 'r', linewidth=2)
    sub1.axis([1, Fs/2, -100, 5])
    sub1.set_title('Magnitude Response', fontsize=20)
    sub1.set_xlabel('Frequency [Hz]', fontsize=20)
    sub1.set_ylabel('Magnitude [dB]', fontsize=20)
    sub1.grid()

    # Plot phase angle
    sub2 = plt.subplot(2, 1, 2)
    sub2.plot(Freq, Phase, 'g', linewidth=2)
    sub2.set_ylabel('Phase (degree)', fontsize=20)
    sub2.set_xlabel(r'Frequency (Hz)', fontsize=20)
    sub2.set_title(r'Phase response', fontsize=20)
    sub2.grid()

    plt.subplots_adjust(hspace=0.5)
    fig.tight_layout()
    plt.show()

# Define impz(b,a) to calculate impulse response
# and step response of a system
# input: b= an array containing numerator coefficients,
# a= an array containing denominator coefficients
def impz(b, a):

    # Define the impulse sequence of length 60
    impulse = np.repeat(0., 60)
    impulse[0] = 1.
    x = np.arange(0, 60)

    # Compute the impulse response
    response = signal.lfilter(b, a, impulse)

    # Plot filter impulse and step response:
    fig = plt.figure(figsize=(10, 6))
    plt.subplot(211)
    plt.stem(x, response, 'm', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Impulse response', fontsize=15)

    plt.subplot(212)
    step = np.cumsum(response)  # Compute step response of the system

    plt.stem(x, step, 'g', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Step response', fontsize=15)
    plt.subplots_adjust(hspace=0.5)

    fig.tight_layout()
    plt.show()


# Given specification
Fs = 7000  # Sampling frequency in Hz
fp = np.array([1400, 2100])  # Pass band frequency in Hz
fs = np.array([1050, 2450])  # Stop band frequency in Hz
Ap = 0.4  # Pass band ripple in dB
As = 50  # stop band attenuation in dB


# Compute pass band and stop band edge frequencies
wp = fp/(Fs/2)  # Normalized passband edge frequencies w.r.t. Nyquist rate
ws = fs/(Fs/2)  # Normalized stopband edge frequencies

# Compute order of the digital Butterworth filter using signal.buttord
N, wc = signal.buttord(wp, ws, Ap, As, analog=True)

# Print the order of the filter and cutoff frequencies
print('Order of the filter=', N)
print('Cut-off frequency=', wc)

# Design digital Butterworth band pass
# filter using signal.butter function
z, p = signal.butter(N, wc, 'bandpass')


# Print numerator and denomerator
# coefficients of the filter
print('Numerator Coefficients:', z)
print('Denominator Coefficients:', p)

# Compute frequency response of the filter
# using signal.freqz function
wz, hz = signal.freqz(z, p)

# Call mfreqz to plot the magnitude and phase response
mfreqz(z, p, Fs)

# Call impz function to plot impulse
# and step response of the filter
impz(z, p)

Output: