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

• Last Updated : 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.

## 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 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 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 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 ``=` `7000`  `# Sampling frequency in Hz` `fp ``=` `np.array([``1400``, ``2100``])  ``# Pass band frequency in Hz` `fs ``=` `np.array([``1050``, ``2450``])  ``# Stop band frequency in Hz` `Ap ``=` `0.4`  `# Pass band ripple in dB` `As ``=` `50`  `# stop band attenuation in dB`

Step 4: Computing the cut-off frequency

## Python3

 `# Compute pass band and stop band edge frequencies` `wp ``=` `fp``/``(Fs``/``2``)  ``# Normalized passband edge frequencies w.r.t. Nyquist rate` `ws ``=` `fs``/``(Fs``/``2``)  ``# Normalized stopband edge frequencies`

Step 5: Compute cut-off frequency & order

## Python3

 `# Compute order of the digital Butterworth filter using signal.buttord` `N, wc ``=` `signal.buttord(wp, ws, Ap, As, analog``=``True``)` `# Print the order of the filter and cutoff frequencies` `print``(``'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 function` `z, p ``=` `signal.butter(N, wc, ``'bandpass'``)`   `# Print numerator and denomerator ` `# coefficients of the filter` `print``(``'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 function` `wz, hz ``=` `signal.freqz(z, p)`

Step 8: Plotting the Magnitude & Phase Response

## Python3

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

Output:

Step 9: Plotting the Impulse and 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`   `# Compute magnitude and phase response` `# using mfreqz function` `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` `Fs ``=` `7000`  `# Sampling frequency in Hz` `fp ``=` `np.array([``1400``, ``2100``])  ``# Pass band frequency in Hz` `fs ``=` `np.array([``1050``, ``2450``])  ``# Stop band frequency in Hz` `Ap ``=` `0.4`  `# Pass band ripple in dB` `As ``=` `50`  `# stop band attenuation in dB`     `# Compute pass band and stop band edge frequencies` `wp ``=` `fp``/``(Fs``/``2``)  ``# Normalized passband edge frequencies w.r.t. Nyquist rate` `ws ``=` `fs``/``(Fs``/``2``)  ``# Normalized stopband edge frequencies`   `# Compute order of the digital Butterworth filter using signal.buttord` `N, wc ``=` `signal.buttord(wp, ws, Ap, As, analog``=``True``)`   `# Print the order of the filter and cutoff frequencies` `print``(``'Order of the filter='``, N)` `print``(``'Cut-off frequency='``, wc)`   `# Design digital Butterworth band pass ` `# filter using signal.butter function` `z, p ``=` `signal.butter(N, wc, ``'bandpass'``)`     `# Print numerator and denomerator ` `# coefficients of the filter` `print``(``'Numerator Coefficients:'``, z)` `print``(``'Denominator Coefficients:'``, p)`   `# Compute frequency response of the filter ` `# using signal.freqz function` `wz, hz ``=` `signal.freqz(z, 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)`

Output:

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