# Digital Low Pass Butterworth Filter in Python

In this article, we are going to discuss how to design a Digital Low Pass Butterworth Filter using Python. The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the pass band. Let us take the below specifications to design the filter and observe the Magnitude, Phase & Impulse Response of the Digital Butterworth Filter.

The specifications are as follows:

• Sampling rate of 40 kHz
• Pass band edge frequency of 4 kHz
• Stop band edge frequency of 8kHz
• Pass band ripple of 0.5 dB
• Minimum stop band attenuation of40 dB

We will plot the magnitude, phase, and impulse response of the filter.

### Step-by-step Approach:

Step 1: Importing all the necessary libraries.

## Python3

 `# import required modules ` `import` `numpy as np ` `import` `matplotlib.pyplot as plt ` `from` `scipy ``import` `signal ` `import` `math`

Step 2: Define variables with the given specifications of the filter.

## Python3

 `# Specifications of Filter ` ` `  ` ``# sampling frequency ` `f_sample ``=` `40000`  ` `  `# pass band frequency ` `f_pass ``=` `4000`   ` `  `# stop band frequency ` `f_stop ``=` `8000`   ` `  `# pass band ripple ` `fs ``=` `0.5` ` `  `# pass band freq in radian ` `wp ``=` `f_pass``/``(f_sample``/``2``)   ` ` `  `# stop band freq in radian ` `ws ``=` `f_stop``/``(f_sample``/``2``)  ` ` `  `# Sampling Time ` `Td ``=` `1`   ` `  ` ``# pass band ripple ` `g_pass ``=` `0.5`  ` `  `# stop band attenuation ` `g_stop ``=` `40`

Step3: Building the filter using signal.buttord function.

## Python3

 `# Conversion to prewrapped analog frequency ` `omega_p ``=` `(``2``/``Td)``*``np.tan(wp``/``2``) ` `omega_s ``=` `(``2``/``Td)``*``np.tan(ws``/``2``) ` ` `  ` `  `# Design of Filter using signal.buttord function ` `N, Wn ``=` `signal.buttord(omega_p, omega_s, g_pass, g_stop, analog``=``True``) ` ` `  ` `  `# Printing the values of order & cut-off frequency! ` `print``(``"Order of the Filter="``, N)  ``# N is the order ` `# Wn is the cut-off freq of the filter ` `print``(``"Cut-off frequency= {:.3f} rad/s "``.``format``(Wn)) ` ` `  ` `  `# Conversion in Z-domain ` ` `  `# b is the numerator of the filter & a is the denominator ` `b, a ``=` `signal.butter(N, Wn, ``'low'``, ``True``) ` `z, p ``=` `signal.bilinear(b, a, fs) ` `# w is the freq in z-domain & h is the magnitude in z-domain ` `w, h ``=` `signal.freqz(z, p, ``512``)`

Output:

Step 4: Plotting the Magnitude Response.

## Python3

 `# Magnitude Response ` `plt.semilogx(w, ``20``*``np.log10(``abs``(h))) ` `plt.xscale(``'log'``) ` `plt.title(``'Butterworth filter frequency response'``) ` `plt.xlabel(``'Frequency [Hz]'``) ` `plt.ylabel(``'Amplitude [dB]'``) ` `plt.margins(``0``, ``0.1``) ` `plt.grid(which``=``'both'``, axis``=``'both'``) ` `plt.axvline(``100``, color``=``'green'``) ` `plt.show()`

Output:

Step 5: Plotting the Impulse Response.

## Python3

 `# Impulse Response ` `imp ``=` `signal.unit_impulse(``40``) ` `c, d ``=` `signal.butter(N, ``0.5``) ` `response ``=` `signal.lfilter(c, d, imp) ` ` `  `plt.stem(np.arange(``0``, ``40``), imp, use_line_collection``=``True``) ` `plt.stem(np.arange(``0``, ``40``), response, use_line_collection``=``True``) ` `plt.margins(``0``, ``0.1``) ` ` `  `plt.xlabel(``'Time [samples]'``) ` `plt.ylabel(``'Amplitude'``) ` `plt.grid(``True``) ` `plt.show()`

Output:

Step 6: Plotting the Phase Response.

## Python3

 `# Phase Response ` `fig, ax1 ``=` `plt.subplots() ` ` `  `ax1.set_title(``'Digital filter frequency response'``) ` `ax1.set_ylabel(``'Angle(radians)'``, color``=``'g'``) ` `ax1.set_xlabel(``'Frequency [Hz]'``) ` ` `  `angles ``=` `np.unwrap(np.angle(h)) ` ` `  `ax1.plot(w``/``2``*``np.pi, angles, ``'g'``) ` `ax1.grid() ` `ax1.axis(``'tight'``) ` `plt.show() `

Output:

Below is the complete program based on the above approach:

## Python

 `# import required modules ` `import` `numpy as np ` `import` `matplotlib.pyplot as plt ` `from` `scipy ``import` `signal ` `import` `math ` ` `  ` `  `# Specifications of Filter ` ` `  ` ``# sampling frequency ` `f_sample ``=` `40000`  ` `  `# pass band frequency ` `f_pass ``=` `4000`   ` `  `# stop band frequency ` `f_stop ``=` `8000`   ` `  `# pass band ripple ` `fs ``=` `0.5` ` `  `# pass band freq in radian ` `wp ``=` `f_pass``/``(f_sample``/``2``)   ` ` `  `# stop band freq in radian ` `ws ``=` `f_stop``/``(f_sample``/``2``)  ` ` `  `# Sampling Time ` `Td ``=` `1`   ` `  ` ``# pass band ripple ` `g_pass ``=` `0.5`  ` `  `# stop band attenuation ` `g_stop ``=` `40`   ` `  ` `  `# Conversion to prewrapped analog frequency ` `omega_p ``=` `(``2``/``Td)``*``np.tan(wp``/``2``) ` `omega_s ``=` `(``2``/``Td)``*``np.tan(ws``/``2``) ` ` `  ` `  `# Design of Filter using signal.buttord function ` `N, Wn ``=` `signal.buttord(omega_p, omega_s, g_pass, g_stop, analog``=``True``) ` ` `  ` `  `# Printing the values of order & cut-off frequency! ` `print``(``"Order of the Filter="``, N)  ``# N is the order ` `# Wn is the cut-off freq of the filter ` `print``(``"Cut-off frequency= {:.3f} rad/s "``.``format``(Wn)) ` ` `  ` `  `# Conversion in Z-domain ` ` `  `# b is the numerator of the filter & a is the denominator ` `b, a ``=` `signal.butter(N, Wn, ``'low'``, ``True``) ` `z, p ``=` `signal.bilinear(b, a, fs) ` `# w is the freq in z-domain & h is the magnitude in z-domain ` `w, h ``=` `signal.freqz(z, p, ``512``) ` ` `  ` `  `# Magnitude Response ` `plt.semilogx(w, ``20``*``np.log10(``abs``(h))) ` `plt.xscale(``'log'``) ` `plt.title(``'Butterworth filter frequency response'``) ` `plt.xlabel(``'Frequency [Hz]'``) ` `plt.ylabel(``'Amplitude [dB]'``) ` `plt.margins(``0``, ``0.1``) ` `plt.grid(which``=``'both'``, axis``=``'both'``) ` `plt.axvline(``100``, color``=``'green'``) ` `plt.show() ` ` `  ` `  `# Impulse Response ` `imp ``=` `signal.unit_impulse(``40``) ` `c, d ``=` `signal.butter(N, ``0.5``) ` `response ``=` `signal.lfilter(c, d, imp) ` `plt.stem(np.arange(``0``, ``40``), imp, use_line_collection``=``True``) ` `plt.stem(np.arange(``0``, ``40``), response, use_line_collection``=``True``) ` `plt.margins(``0``, ``0.1``) ` `plt.xlabel(``'Time [samples]'``) ` `plt.ylabel(``'Amplitude'``) ` `plt.grid(``True``) ` `plt.show() ` ` `  ` `  `# Phase Response ` `fig, ax1 ``=` `plt.subplots() ` `ax1.set_title(``'Digital filter frequency response'``) ` `ax1.set_ylabel(``'Angle(radians)'``, color``=``'g'``) ` `ax1.set_xlabel(``'Frequency [Hz]'``) ` `angles ``=` `np.unwrap(np.angle(h)) ` `ax1.plot(w``/``2``*``np.pi, angles, ``'g'``) ` `ax1.grid() ` `ax1.axis(``'tight'``) ` `plt.show() `

Output:

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