IIR stands for Infinite Impulse Response, It is one of the striking characteristics of many linear-time invariant systems that are characterized from having an impulse response h(t)/h(n) that does not reach 0 at any stage but instead persists indefinitely.
What is IIR Bandpass Elliptic Filter ?
Elliptical Filter is a special type of Filter that is used in digital signal processing when there is a need for a fast transition from pass to stop band.
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 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(). The mfreqz is a function for magnitude and phase plot and the impz is a function for impulse and step response].
Python3
# Function to depict magnitude # and phase plotdef 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 specification# Sampling frequency in HzFs = 7000# Pass band frequency in Hzfp = np.array([1400, 2100])# Stop band frequency in Hzfs = np.array([1050, 2450])# Pass band ripple in dBAp = 0.4# Stop band attenuation in dBAs = 50 |
Step 4: Compute the cut-off frequency
Python3
# Compute pass band and stop band edge frequencies# Normalized passband edge# frequencies w.r.t. Nyquist ratewp = fp/(Fs/2)# Normalized stopband# edge frequenciesws = fs/(Fs/2) |
Step 5: Compute order of the Elliptic Bandpass digital filter.
Python3
# Compute order of the elliptic filter # using signal.ellipordN, wc = signal.ellipord(wp, ws, Ap, As)# Print the order of the filter and # cutoff frequenciesprint('Order of the filter=', N)print('Cut-off frequency=', wc) |
Step 6: Design digital Elliptical bandpass filter.
Python3
# Design digital elliptic bandpass filter # using signal.ellip functionz, p = signal.ellip(N, Ap, As, wc, 'bandpass')# Print numerator and denomerator # coefficients of the filterprint('Numerator Coefficients:', z)print('Denominator Coefficients:', p) |
Step 7: Plot magnitude and phase response.
Python3
# Depicting visualizations# Call mfreqz to plot the magnitude and phase responsemfreqz(z, p, Fs) |
Step 8: Plot impulse and step response of the filter.
Python3
# Call impz function to plot impulse # and step response of the filterimpz(z, p) |
Below is the complete implementation of the above stepwise approach:
Python3
# Import required libraryimport numpy as npimport scipy.signal as signalimport matplotlib.pyplot as plt# Function to depict magnitude # and phase plotdef 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 specification# Sampling frequency in HzFs = 7000# Pass band frequency in Hzfp = np.array([1400, 2100])# Stop band frequency in Hzfs = np.array([1050, 2450])# Pass band ripple in dBAp = 0.4# Stop band attenuation in dBAs = 50# Compute pass band and# stop band edge frequencies# Normalized passband edge frequencies # w.r.t. Nyquist ratewp = fp/(Fs/2)# Normalized stopband edge frequenciesws = fs/(Fs/2)# Compute order of the elliptic filter # using signal.ellipordN, wc = signal.ellipord(wp, ws, Ap, As)# Print the order of the filter and cutoff frequenciesprint('Order of the filter=', N)print('Cut-off frequency=', wc)# Design digital elliptic bandpass filter # using signal.ellip() functionz, p = signal.ellip(N, Ap, As, wc, 'bandpass')# Print numerator and denomerator coefficients of the filterprint('Numerator Coefficients:', z)print('Denominator Coefficients:', p)# Depicting visualizations# 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:
