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 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('Magnitute 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('Magnitute 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:
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