Circular Convolution using Matrix Method

Given two array X[] and H[] of length N and M respectively, the task is to find the circular convolution of the given arrays using Matrix method. Multiplication of the Circularly Shifted Matrix and the column-vector is the Circular-Convolution of the arrays.
Examples:

Input: X[] = {1, 2, 4, 2}, H[] = {1, 1, 1} 
Output: 7 5 7 8
Input: X[] = {5, 7, 3, 2}, H[] = {1, 5} 
Output: 15 32 38 17 

Explanation: 

• Create a circularly shifted Matrix circular_shift_mat of K * K using the elements of array whose length is maximum(X_n in this case) where K is MAX(N, M). 
  

Circularly shifted matrix of the array X_n.

• Create a column-vector col_vec of length K
• Insert the elements of the array H_m into the col_vec in positions [0, m).
• As K = max(N, M), here N; M < K. Therefore fill the rest of the positions of col_vec [m, K) with 0.Therefore the col_vec will be

col_vec = { 1, 1, 1, 0 }

• Multiply the circular_shift_mat and the col_vec
• Multiplication of the Circularly Shifted Matrix (circular_shift_mat) and the column-vector (col_vec) is the Circular-Convolution of the arrays.

Approach: 

• Create a Circularly shifted Matrix of N * N using the elements of array of the maximum length.
• Create a column-vector of length N using elements of another array and fill up rest of the positions by 0.
• Multiplication of Matrix and the column-vector is the Circular-Convolution of arrays.

Below is the implementation of the above approach. 

C++

// C++ program to compute circular
// convolution of two arrays
#include <bits/stdc++.h>
using namespace std;

#define MAX_SIZE 10

// Function to find circular convolution
void convolution(vector<int>& x, vector<int>& h, int n, int m)
{
    int row_vec[MAX_SIZE], col_vec[MAX_SIZE];
    int out[MAX_SIZE] = { 0 };
    int circular_shift_mat[MAX_SIZE][MAX_SIZE];

    // Finding the maximum size between the
    // two input sequence sizes
    int maxSize = n > m ? n : m;

    // Copying elements of x to row_vec and padding
    // zeros if size of x < maxSize
    for (int i = 0; i < maxSize; i++) {
        if (i >= n) {
            row_vec[i] = 0;
        }
        else {
            row_vec[i] = x[i];
        }
    }

    // Copying elements of h to col_vec and padding
    // zeros if size of h is less than maxSize
    for (int i = 0; i < maxSize; i++) {
        if (i >= m) {
            col_vec[i] = 0;
        }
        else {
            col_vec[i] = h[i];
        }
    }

    // Generating 2D matrix of
    // circularly shifted elements
    int k = 0, d = 0;

    for (int i = 0; i < maxSize; i++) {
        int curIndex = k - d;
        for (int j = 0; j < maxSize; j++) {
            circular_shift_mat[j][i] = row_vec
                [curIndex % maxSize];
            curIndex++;
        }
        k = maxSize;
        d++;
    }

    // Computing result by matrix
    // multiplication and printing results
    for (int i = 0; i < maxSize; i++) {
        for (int j = 0; j < maxSize; j++) {

            out[i] += circular_shift_mat[i][j]
                      * col_vec[j];
        }
        cout << out[i] << " ";
    }
}

// Driver program
int main()
{
    vector<int> x = { 5, 7, 3, 2 };
    int n = x.size();
    vector<int> h = { 1, 5 };
    int m = h.size();

    convolution(x, h, n, m);

    return 0;
}

Java

// Java program to compute circular
// convolution of two arrays
class GFG
{
    final static int MAX_SIZE = 10 ;

    // Function to find circular convolution
    static void convolution(int []x, int []h, int n, int m)
    {
        int row_vec[] = new int[MAX_SIZE];
        int col_vec[] = new int[MAX_SIZE];
        int out[] = new int [MAX_SIZE];
        int circular_shift_mat[][] = new int[MAX_SIZE][MAX_SIZE];

        // Finding the maximum size between the
        // two input sequence sizes
        int maxSize = n > m ? n : m;

        // Copying elements of x to row_vec and padding
        // zeros if size of x < maxSize
        for (int i = 0; i < maxSize; i++)
        {
            if (i >= n)
            {
                row_vec[i] = 0;
            }
            else
            {
                row_vec[i] = x[i];
            }
        }

        // Copying elements of h to col_vec and padding
        // zeros if size of h is less than maxSize
        for (int i = 0; i < maxSize; i++)
        {
            if (i >= m)
            {
                col_vec[i] = 0;
            }
            else
            {
                col_vec[i] = h[i];
            }
        }

        // Generating 2D matrix of
        // circularly shifted elements
        int k = 0, d = 0;

        for (int i = 0; i < maxSize; i++)
        {
            int curIndex = k - d;
            for (int j = 0; j < maxSize; j++)
            {
                circular_shift_mat[j][i] =
                row_vec[curIndex % maxSize];
                curIndex++;
            }
            k = maxSize;
            d++;
        }

        // Computing result by matrix
        // multiplication and printing results
        for (int i = 0; i < maxSize; i++)
        {
            for (int j = 0; j < maxSize; j++)
            {

                out[i] += circular_shift_mat[i][j] * col_vec[j];
            }
            System.out.print(out[i] + " ");
        }
    }

    // Driver program
    public static void main (String[] args)
    {
        int x[] = { 5, 7, 3, 2 };
        int n = x.length;
        int h[] = { 1, 5 };
        int m = h.length;

        convolution(x, h, n, m);
    }
}

// This code is contributed by AnkitRai01

Python3

# Python program to compute circular
# convolution of two arrays
MAX_SIZE = 10;

# Function to find circular convolution
def convolution(x, h, n, m):
    row_vec = [0] * MAX_SIZE;
    col_vec = [0] * MAX_SIZE;
    out = [0] * MAX_SIZE;
    circular_shift_mat = [[0 for i in range(MAX_SIZE)]
                            for j in range(MAX_SIZE)] ;

    # Finding the maximum size between the
    # two input sequence sizes
    if(n > m ):
        maxSize = n;
    else:
        maxSize = m;

    # Copying elements of x to row_vec and padding
    # zeros if size of x < maxSize
    for i in range(maxSize):
        if (i >= n):
            row_vec[i] = 0;
        else:
            row_vec[i] = x[i];

    # Copying elements of h to col_vec and padding
    # zeros if size of h is less than maxSize
    for i in range(maxSize):
        if (i >= m):
            col_vec[i] = 0;
        else:
            col_vec[i] = h[i];

    # Generating 2D matrix of
    # circularly shifted elements
    k = 0;
    d = 0;

    for i in range(maxSize):
        curIndex = k - d;
        for j in range(maxSize):
            circular_shift_mat[j][i] = \
            row_vec[curIndex % maxSize];
            curIndex += 1;

        k = maxSize;
        d += 1;

    # Computing result by matrix
    # multiplication and printing results
    for i in range(maxSize):
        for j in range(maxSize):
            out[i] += circular_shift_mat[i][j] * \
                                    col_vec[j];

        print(out[i], end = " ");

# Driver program
if __name__ == '__main__':
    x = [ 5, 7, 3, 2 ];
    n = len(x);
    h = [ 1, 5 ];
    m = len(h);

    convolution(x, h, n, m);

# This code is contributed by 29AjayKumar

C#

// C# program to compute circular
// convolution of two arrays
using System;

class GFG
{
    readonly static int MAX_SIZE = 10 ;

    // Function to find circular convolution
    static void convolution(int []x, int []h,
                            int n, int m)
    {
        int []row_vec = new int[MAX_SIZE];
        int []col_vec = new int[MAX_SIZE];
        int []out_ = new int [MAX_SIZE];
        int [,]circular_shift_mat =
            new int[MAX_SIZE,MAX_SIZE];

        // Finding the maximum size between the
        // two input sequence sizes
        int maxSize = n > m ? n : m;

        // Copying elements of x to row_vec and padding
        // zeros if size of x < maxSize
        for (int i = 0; i < maxSize; i++)
        {
            if (i >= n)
            {
                row_vec[i] = 0;
            }
            else
            {
                row_vec[i] = x[i];
            }
        }

        // Copying elements of h to col_vec and padding
        // zeros if size of h is less than maxSize
        for (int i = 0; i < maxSize; i++)
        {
            if (i >= m)
            {
                col_vec[i] = 0;
            }
            else
            {
                col_vec[i] = h[i];
            }
        }

        // Generating 2D matrix of
        // circularly shifted elements
        int k = 0, d = 0;

        for (int i = 0; i < maxSize; i++)
        {
            int curIndex = k - d;
            for (int j = 0; j < maxSize; j++)
            {
                circular_shift_mat[j, i] =
                row_vec[curIndex % maxSize];
                curIndex++;
            }
            k = maxSize;
            d++;
        }

        // Computing result by matrix
        // multiplication and printing results
        for (int i = 0; i < maxSize; i++)
        {
            for (int j = 0; j < maxSize; j++)
            {

                out_[i] += circular_shift_mat[i, j] *
                            col_vec[j];
            }
            Console.Write(out_[i] + " ");
        }
    }

    // Driver program
    public static void Main(String[] args)
    {
        int []x = {5, 7, 3, 2};
        int n = x.Length;
        int []h = {1, 5};
        int m = h.Length;

        convolution(x, h, n, m);
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

    // JavaScript program to compute circular
    // convolution of two arrays

    let MAX_SIZE = 10 ;

    // Function to find circular convolution
    function convolution(x, h, n, m)
    {
        let row_vec = new Array(MAX_SIZE);
        row_vec.fill(0);
        let col_vec = new Array(MAX_SIZE);
        col_vec.fill(0);
        let out = new Array(MAX_SIZE);
        out.fill(0);
        let circular_shift_mat = new Array(MAX_SIZE);
        circular_shift_mat.fill(0);
        for (let i = 0; i < MAX_SIZE; i++)
        {
            circular_shift_mat[i] = new Array(MAX_SIZE);
            for (let j = 0; j < MAX_SIZE; j++)
            {
                circular_shift_mat[i][j] = 0;
            }
        }

        // Finding the maximum size between the
        // two input sequence sizes
        let maxSize = n > m ? n : m;

        // Copying elements of x to row_vec and padding
        // zeros if size of x < maxSize
        for (let i = 0; i < maxSize; i++)
        {
            if (i >= n)
            {
                row_vec[i] = 0;
            }
            else
            {
                row_vec[i] = x[i];
            }
        }

        // Copying elements of h to col_vec and padding
        // zeros if size of h is less than maxSize
        for (let i = 0; i < maxSize; i++)
        {
            if (i >= m)
            {
                col_vec[i] = 0;
            }
            else
            {
                col_vec[i] = h[i];
            }
        }

        // Generating 2D matrix of
        // circularly shifted elements
        let k = 0, d = 0;

        for (let i = 0; i < maxSize; i++)
        {
            let curIndex = k - d;
            for (let j = 0; j < maxSize; j++)
            {
                circular_shift_mat[j][i] =
                row_vec[curIndex % maxSize];
                curIndex++;
            }
            k = maxSize;
            d++;
        }

        // Computing result by matrix
        // multiplication and printing results
        for (let i = 0; i < maxSize; i++)
        {
            for (let j = 0; j < maxSize; j++)
            {

                out[i] += circular_shift_mat[i][j] * col_vec[j];
            }
            document.write(out[i] + " ");
        }
    }

    let x = [ 5, 7, 3, 2 ];
    let n = x.length;
    let h = [ 1, 5 ];
    let m = h.length;

    convolution(x, h, n, m);

</script>

Output:

15 32 38 17

Time Complexity: O(MAX_SIZE * MAX_SIZE)
Auxiliary Space: O(MAX_SIZE * MAX_SIZE)