IR 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 Chebyshev Filter?
IIR Chebyshev is a filter that is linear-time invariant filter just like the Butterworth however, it has a steeper roll-off compared to the Butterworth Filter. Chebyshev Filter is further classified as Chebyshev Type-I and Chebyshev Type-II according to the parameters such as pass band ripple and stop ripple.
How is Chebyshev Filter different from Butterworth?
Chebyshev Filter has a steeper roll-off compared to the Butterworth Filter.
What is Chebyshev Type-2 Filter?
Chebyshev Type-2 minimizes the absolute difference between the ideal and actual frequency response over the entire stopband by incorporating an equal ripple in the stopband.
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.
# import required library import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt
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Step 2: Defining user-defined functions mfreqz() and impz(). The mfreqz is a function for magnitude and phase plotand the impz is a function for impulse and step response]
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)
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()
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Step 3:Define variables with the given specifications of the filter.
# Given specification # Sampling frequency in Hz Fs = 7000 # Pass band frequency in Hz fp = np.array([ 1400 , 2100 ])
# Stop band frequency in Hz fs = np.array([ 1050 , 2450 ])
# Pass band ripple in dB Ap = 0.4
# Stop band attenuation in dB As = 50 |
Step 4: Compute the cut-off frequency
# Compute pass band and stop band edge frequencies # Normalized passband edge # frequencies w.r.t. Nyquist rate wp = fp / (Fs / 2 )
# Normalized stopband # edge frequencies ws = fs / (Fs / 2 )
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Step 5: Compute order of the Chebyshev type-2 digital filter.
# Compute order of the Chebyshev type-2 # digital filter using signal.cheb2ord N, wc = signal.cheb2ord(wp, ws, Ap, As)
# Print the order of the filter # and cutoff frequencies print ( 'Order of the filter=' , N)
print ( 'Cut-off frequency=' , wc)
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Output:
Step 6: Design digital Chebyshev type-2 bandpass filter.
# Design digital Chebyshev type-2 bandpass # filter using signal.cheby2 function z, p = signal.cheby2(N, As, wc, 'bandpass' )
# Print numerator and denomerator # coefficients of the filter print ( 'Numerator Coefficients:' , z)
print ( 'Denominator Coefficients:' , p)
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Output:
Step 7: Plot magnitude and phase response.
# Call mfreqz to plot the # magnitude and phase response mfreqz(z, p, Fs) |
Output:
Step 8: Plot impulse and step response of the filter.
# Call impz function to plot impulse # and step response of the filter impz(z, p) |
Output:
Below is the complete implementation of the above stepwise approach:
# import required library import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt
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 # Sampling frequency in Hz Fs = 7000 # Pass band frequency in Hz fp = np.array([ 1400 , 2100 ])
# Stop band frequency in Hz fs = np.array([ 1050 , 2450 ])
# Pass band ripple in dB Ap = 0.4
# Stop band attenuation in dB As = 50 # Compute pass band and # stop band edge frequencies # Normalized passband edge frequencies w.r.t. Nyquist rate wp = fp / (Fs / 2 )
# Normalized stopband edge frequencies ws = fs / (Fs / 2 )
# Compute order of the Chebyshev type-2 # digital filter using signal.cheb2ord N, wc = signal.cheb2ord(wp, ws, Ap, As)
# Print the order of the filter and cutoff frequencies print ( 'Order of the filter=' , N)
print ( 'Cut-off frequency=' , wc)
# Design digital Chebyshev type-2 bandpass # filter using signal.cheby2 function z, p = signal.cheby2(N, As, wc, 'bandpass' )
# Print numerator and denomerator coefficients of the filter print ( 'Numerator Coefficients:' , z)
print ( 'Denominator Coefficients:' , 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) |