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# Digital Band Reject Butterworth Filter in Python

In this article, we are going to discuss how to design a Digital Band Reject 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.

## What is Digital Bandreject Filter?

A band-pass filter is a filter that passes frequencies within a range and rejects frequencies outside that range.

## How it’s different from Highpass, Lowpass & Bandpass:

The main difference can be spotted by observing the magnitude response of the Band Pass Filter. In the Band-Reject filter, all the signals between the specified frequency range get rejected by the filter.

The specifications are as follows:

• The sampling rate of 12 kHz.
• Pass band edge frequencies are 2100 Hz & 4500 Hz.
• Stop band edge frequencies are 2700 Hz & 3900 Hz.
• Pass band ripple of 0.6 dB.
• Minimum stop band attenuation of 45 dB.

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

### Step-by-step Approach:

Before starting, first, we will create a user-defined function to convert the edge frequencies, we are defining it as the convert() method.

## Python3

 `# explicit function to convert``# edge frequencies`` ` ` ` `def` `convertX(f_sample, f):``    ``w ``=` `[]`` ` `    ``for` `i ``in` `range``(``len``(f)):``        ``b ``=` `2``*``((f[i]``/``2``) ``/` `(f_sample``/``2``))``        ``w.append(b)`` ` `    ``omega_mine ``=` `[]`` ` `    ``for` `i ``in` `range``(``len``(w)):``        ``c ``=` `(``2``/``Td)``*``np.tan(w[i]``/``2``)``        ``omega_mine.append(c)`` ` `    ``return` `omega_mine`

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 ``=` `12000`` ` `# pass band frequency``f_pass ``=` `[``2100``, ``4500``]`` ` `# stop band frequency``f_stop ``=` `[``2700``, ``3900``]`` ` `# pass band ripple``fs ``=` `0.5`` ` `# Sampling Time``Td ``=` `1`` ` `# pass band ripple``g_pass ``=` `0.6`` ` `# stop band attenuation``g_stop ``=` `45`

Step 3: Building the filter using signal.buttord() function.

## Python3

 `# Conversion to prewrapped analog``# frequency``omega_p ``=` `convertX(f_sample, f_pass)``omega_s ``=` `convertX(f_sample, f_stop)`` ` `# 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``# N is the order``print``(``"Order of the Filter="``, N)`` ` `# Wn is the cut-off freq of the filter``print``(``"Cut-off frequency= {:} 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, ``'bandpass'``, ``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, markerfmt``=``'D'``, 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

 `# Frequency 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:

## Python3

 `# import required modules``import` `numpy as np``import` `matplotlib.pyplot as plt``from` `scipy ``import` `signal``import` `math`` ` `# explicit function to convert``# edge frequencies``def` `convertX(f_sample, f):``    ``w ``=` `[]`` ` `    ``for` `i ``in` `range``(``len``(f)):``        ``b ``=` `2``*``((f[i]``/``2``)``/``(f_sample``/``2``))``        ``w.append(b)`` ` `    ``omega_mine ``=` `[]`` ` `    ``for` `i ``in` `range``(``len``(w)):``        ``c ``=` `(``2``/``Td)``*``np.tan(w[i]``/``2``)``        ``omega_mine.append(c)`` ` `    ``return` `omega_mine`` ` `# Specifications of Filter``# sampling frequency``f_sample ``=` `12000`` ` `# pass band frequency``f_pass ``=` `[``2100``, ``4500``]`` ` `# stop band frequency``f_stop ``=` `[``2700``, ``3900``]`` ` `# pass band ripple``fs ``=` `0.5`` ` `# Sampling Time``Td ``=` `1`` ` `# pass band ripple``g_pass ``=` `0.6`` ` `# stop band attenuation``g_stop ``=` `45`` ` `# Conversion to prewrapped analog``# frequency``omega_p ``=` `convertX(f_sample, f_pass)``omega_s ``=` `convertX(f_sample, f_stop)`` ` `# 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``# N is the order``print``(``"Order of the Filter="``, N)`` ` `# Wn is the cut-off freq of the filter``print``(``"Cut-off frequency= {:} 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, ``'bandpass'``, ``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,``         ``markerfmt``=``'D'``,``         ``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()`` ` `# Frequency 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: