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

10 Nov, 2021
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.
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 libraryimport numpy as npimport scipy.signal as signalimport 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 coefficientsdef 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 specificationFs = 7000  # Sampling frequency in Hzfp = np.array([1400, 2100])  # Pass band frequency in Hzfs = np.array([1050, 2450])  # Stop band frequency in HzAp = 0.4  # Pass band ripple in dBAs = 50  # stop band attenuation in dB
Step 4: Computing the cut-off frequency
Python3
# Compute pass band and stop band edge frequencieswp = fp/(Fs/2)  # Normalized passband edge frequencies w.r.t. Nyquist ratews = fs/(Fs/2)  # Normalized stopband edge frequencies
Step 5: Compute cut-off frequency & order
Python3
# Compute order of the digital Butterworth filter using signal.buttordN, wc = signal.buttord(wp, ws, Ap, As, analog=True)# Print the order of the filter and cutoff frequenciesprint('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 functionz, p = signal.butter(N, wc, 'bandpass') # Print numerator and denomerator# coefficients of the filterprint('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 functionwz, hz = signal.freqz(z, p)
Step 8: Plotting the Magnitude & Phase Response
Python3
# Call mfreqz to plot the magnitude and phase responsemfreqz(z, p, Fs)
Output:
Step 9: Plotting the Impulse and Step Response
Python3
# Call impz function to plot impulse# and step response of the filterimpz(z, p)
Output:
Below is the implementation:
Python3
# import required libraryimport numpy as npimport scipy.signal as signalimport matplotlib.pyplot as plt # Compute magnitude and phase response# using mfreqz functiondef 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 coefficientsdef 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 specificationFs = 7000  # Sampling frequency in Hzfp = np.array([1400, 2100])  # Pass band frequency in Hzfs = np.array([1050, 2450])  # Stop band frequency in HzAp = 0.4  # Pass band ripple in dBAs = 50  # stop band attenuation in dB  # Compute pass band and stop band edge frequencieswp = fp/(Fs/2)  # Normalized passband edge frequencies w.r.t. Nyquist ratews = fs/(Fs/2)  # Normalized stopband edge frequencies # Compute order of the digital Butterworth filter using signal.buttordN, wc = signal.buttord(wp, ws, Ap, As, analog=True) # Print the order of the filter and cutoff frequenciesprint('Order of the filter=', N)print('Cut-off frequency=', wc) # Design digital Butterworth band pass# filter using signal.butter functionz, p = signal.butter(N, wc, 'bandpass')  # Print numerator and denomerator# coefficients of the filterprint('Numerator Coefficients:', z)print('Denominator Coefficients:', p) # Compute frequency response of the filter# using signal.freqz functionwz, hz = signal.freqz(z, p) # Call mfreqz to plot the magnitude and phase responsemfreqz(z, p, Fs) # Call impz function to plot impulse# and step response of the filterimpz(z, p)
Output:
gulshankumarar231
Data Visualization
Python-scipy
Python
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