circular_convolution.py

# https://en.wikipedia.org/wiki/Circular_convolution

"""
Circular convolution, also known as cyclic convolution,
is a special case of periodic convolution, which is the convolution of two
periodic functions that have the same period. Periodic convolution arises,
for example, in the context of the discrete-time Fourier transform (DTFT).
In particular, the DTFT of the product of two discrete sequences is the periodic
convolution of the DTFTs of the individual sequences. And each DTFT is a periodic
summation of a continuous Fourier transform function.

Source: https://en.wikipedia.org/wiki/Circular_convolution
"""

import doctest
from collections import deque

import numpy as np


class CircularConvolution:
 """
 This class stores the first and second signal and performs the circular convolution
 """

 def __init__(self) -> None:
 """
 First signal and second signal are stored as 1-D array
 """

 self.first_signal = [2, 1, 2, -1]
 self.second_signal = [1, 2, 3, 4]

 def circular_convolution(self) -> list[float]:
 """
 This function performs the circular convolution of the first and second signal
 using matrix method

 Usage:
 >>> convolution = CircularConvolution()
 >>> convolution.circular_convolution()
 [10.0, 10.0, 6.0, 14.0]

 >>> convolution.first_signal = [0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6]
 >>> convolution.second_signal = [0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5]
 >>> convolution.circular_convolution()
 [5.2, 6.0, 6.48, 6.64, 6.48, 6.0, 5.2, 4.08]

 >>> convolution.first_signal = [-1, 1, 2, -2]
 >>> convolution.second_signal = [0.5, 1, -1, 2, 0.75]
 >>> convolution.circular_convolution()
 [6.25, -3.0, 1.5, -2.0, -2.75]

 >>> convolution.first_signal = [1, -1, 2, 3, -1]
 >>> convolution.second_signal = [1, 2, 3]
 >>> convolution.circular_convolution()
 [8.0, -2.0, 3.0, 4.0, 11.0]

 """

 length_first_signal = len(self.first_signal)
 length_second_signal = len(self.second_signal)

 max_length = max(length_first_signal, length_second_signal)

 # create a zero matrix of max_length x max_length
 matrix = [[0] * max_length for i in range(max_length)]

 # fills the smaller signal with zeros to make both signals of same length
 if length_first_signal < length_second_signal:
 self.first_signal += [0] * (max_length - length_first_signal)
 elif length_first_signal > length_second_signal:
 self.second_signal += [0] * (max_length - length_second_signal)

 """
 Fills the matrix in the following way assuming 'x' is the signal of length 4
 [
 [x[0], x[3], x[2], x[1]],
 [x[1], x[0], x[3], x[2]],
 [x[2], x[1], x[0], x[3]],
 [x[3], x[2], x[1], x[0]]
 ]
 """
 for i in range(max_length):
 rotated_signal = deque(self.second_signal)
 rotated_signal.rotate(i)
 for j, item in enumerate(rotated_signal):
 matrix[i][j] += item

 # multiply the matrix with the first signal
 final_signal = np.matmul(np.transpose(matrix), np.transpose(self.first_signal))

 # rounding-off to two decimal places
 return [float(round(i, 2)) for i in final_signal]


if __name__ == "__main__":
 doctest.testmod()