# Design an IIR Highpass Butterworth Filter using Bilinear Transformation Method in Scipy – Python

• Last Updated : 07 Jan, 2022

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 Highpass Butterworth ?

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

The specifications are as follows:

• Pass band frequency: 2-4 kHz
• Stop band frequency: 0-500 Hz
• Pass band ripple: 3dB
• Stop band attenuation: 20 dB
• Sampling frequency: 8 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 a 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)  ``# START CODE HERE ### (≈ 1 line of code)`` ` `    ``# 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 ``=` `8000`  `# Sampling frequency in Hz``fp ``=` `2000`  `# Pass band frequency in Hz``fs ``=` `500`  `# Stop Band frequency in Hz``Ap ``=` `3`  `# Pass band ripple in dB``As ``=` `20`  `# Stop band attenuation in dB`` ` `# Compute Sampling parameter``Td ``=` `1``/``Fs`

Step 4:Computing the cut-off frequency

## Python3

 `# Compute cut-off frequency in radian/sec``wp ``=` `2``*``np.pi``*``fp  ``# pass band frequency in radian/sec``ws ``=` `2``*``np.pi``*``fs  ``# stop band frequency in radian/sec`

Step 5: Pre-wrapping the cut-off frequency

## Python3

 `# Prewarp the analog frequency``Omega_p ``=` `(``2``/``Td)``*``np.tan(wp``*``Td``/``2``)  ``# Prewarped analog passband frequency``Omega_s ``=` `(``2``/``Td)``*``np.tan(ws``*``Td``/``2``)  ``# Prewarped analog stopband frequency`

Step 6: Computing the Butterworth Filter

## Python3

 `# Compute Butterworth filter order and cutoff frequency``N, wc ``=` `signal.buttord(Omega_p, Omega_s, Ap, As, analog``=``True``)`` ` `# Print the values of order and cut-off frequency``print``(``'Order of the filter='``, N)``print``(``'Cut-off frequency='``, wc)`

Output:

Step 7: Design analog Butterworth filter using N and wc by signal.butter() function.

## Python3

 `# Design analog Butterworth filter using N and``# wc by signal.butter function``b, a ``=` `signal.butter(N, wc, ``'high'``, analog``=``True``)`` ` `# Perform bilinear Transformation``z, p ``=` `signal.bilinear(b, a, fs``=``Fs)`` ` `# Print numerator and denomerator coefficients ``# of the filter``print``(``'Numerator Coefficients:'``, z)``print``(``'Denominator Coefficients:'``, p)`

Output:

Step 8: Plotting the Magnitude & Phase Response

## Python3

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

Output:

Step 9: Plotting the impulse & 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`` ` `# User defined functions mfreqz for ``# Magnitude & Phase 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)  ``# START CODE HERE ### (≈ 1 line of code)`` ` `    ``# 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 ``=` `8000`  `# Sampling frequency in Hz``fp ``=` `2000`  `# Pass band frequency in Hz``fs ``=` `500`  `# Stop Band frequency in Hz``Ap ``=` `3`  `# Pass band ripple in dB``As ``=` `20`  `# Stop band attenuation in dB`` ` `# Compute Sampling parameter``Td ``=` `1``/``Fs`` ` `# Compute cut-off frequency in radian/sec``wp ``=` `2``*``np.pi``*``fp  ``# pass band frequency in radian/sec``ws ``=` `2``*``np.pi``*``fs  ``# stop band frequency in radian/sec`` ` `# Prewarp the analog frequency``Omega_p ``=` `(``2``/``Td)``*``np.tan(wp``*``Td``/``2``)  ``# Prewarped analog passband frequency``Omega_s ``=` `(``2``/``Td)``*``np.tan(ws``*``Td``/``2``)  ``# Prewarped analog stopband frequency`` ` `# Compute Butterworth filter order and cutoff frequency``N, wc ``=` `signal.buttord(Omega_p, Omega_s, Ap, As, analog``=``True``)`` ` `# Print the values of order and cut-off frequency``print``(``'Order of the filter='``, N)``print``(``'Cut-off frequency='``, wc)`` ` `# Design analog Butterworth filter using N and``# wc by signal.butter function``b, a ``=` `signal.butter(N, wc, ``'high'``, analog``=``True``)`` ` `# Perform bilinear Transformation``z, p ``=` `signal.bilinear(b, a, fs``=``Fs)`` ` `# Print numerator and denomerator coefficients of the filter``print``(``'Numerator Coefficients:'``, z)``print``(``'Denominator Coefficients:'``, p)`` ` `# Call mfreqz function 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:

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