 Hello everyone. Welcome to this session. Myself, K. R. Biradhar, Assistant Professor, Department of Electronics and Communication, Valchin Institute of Technology, Swallapur. Today, I am going to discuss the topic called circular convolution of sequence. Let us start with the learning outcomes first. At the end of this session, students will be able to find the circular convolution of two sequences, compare different methods of finding the circular convolution. These are the contents. Start with the definition of the circular convolution, different methods of finding the circular convolution, matrix multiplication method, concentric circle method and finally, references, circular convolution. Circular convolution for the input x of n and its impulse response h of n is defined as y of n is equal to x of n circularly convolved with h of n. There are different methods to find the circular conditions are. There are two types. First one is matrix multiplication method, second one is concentric circle method. Let us see the first method, matrix multiplication method. In matrix multiplication method, circular convolution of two sequences x 1 of n and x 2 of n can be calculated by representing a sequences in matrix form as follows. See x 1 of 0, x 1 of 1, x 1 of 2, x 1 of n minus 1 first sequence will be written in the first column. Second column will be formed by shifting the one element down each of the first elements forms the second element. For example, x 1 of 0 which is occupied the position 1 in the first column moves to second position in the second column. Similarly, third position in the third column at the last it moves to last position of the last column. Similarly, x 1 of 1 which is present in the second position of the first column moves to third position of the second column. Keep on shifting this one. This forms the x 1 of n that is first matrix. Similarly, x 2 of n will be written in the second matrix as x 2 of 0, x 2 of 1, x 2 of 2, x 2 of n minus 1. So, by multiplication of these two sequences we are going to get y of 0, y of 1, y of 2, up to y of n minus 1. Let us recall what is the length of the output sequence of the circular convolution. We have seen already that is linear convolution of two sequences the length of the linear convolution output is equal to length of the first input sequence plus length of the second input sequence minus 1. Whereas, circular convolution will have the output length which is equal to length of the input sequence. One more condition is the input shall have both same length then only you can perform the circular convolution. Let us take an example for finding the circular convolution of two sequences using matrix multiplication method. This example will be shown here. See here the first sequence is x 1 of n which is equal to 2, 1, 2 and the second sequence is x 2 of n which is equal to 1, 2, 3. You write the first sequence in the first column that is 2, 1, 2 as usual and now you shift the 2 which is present in the first column moves to second portion in the second column. Similarly, this one moves to that is third portion which is occupied second portion in the first column. Now whatever the last element present in the first column reaches to that is this position 2. Similarly, when you want to form the third column whatever the 2 present in the second column occupied at first position moves to second portion. Similarly, this 2 moves to here and this one last one goes at this place. So, whatever the started with the two element in the first column reaches to the last corner of the right hand side just below last element of the this column. This is first sequence forming the matrix. Similarly, second sequence as usual we need to write that is 1, 2, 3 you write this 1, 1, 2, 3 serially and multiply these things. Whatever the 2 which is there will be multiplied with 1 that is 2 into 1 is 2. Similarly, 2 into 2 which is equal to 4 and 1 into 3 that is 3. Summit of these things we are going to get 2 plus 4, 6 plus 3, 9. Similarly, second row element that is 1 will be multiplied with 1 that is 1, 2 will be multiplied with 2, 4, 2 will be multiplied with 6 then sum it up you are going to get the 11. Similarly, that is 2 will be multiplied with 1, 2, 1 will be multiplied with 2, 2, 2 into multiplied with 3, 6 then total answer is equal to 10. This is how we are going to find the circular convolution using matrix method. Let us explain now x1 of n is equal to 2, 1, 2 and x2 of n is equal to 1, 2, 3. What the procedure says is you draw the that is outer circle and represent 2, 1, 2 in anticlockwise direction. For example, outer circle consists of x1 of 0, x1 of 1, x1 of 2 then first x1 of 0 is value is 2 and x1 of 1 value is 1 and x1 of 2 value is 2. You need to draw that is nothing but the outer circle and mark that is the given first sequences in anticlockwise direction. Similarly, you draw inner circle and represent those values which are present in the x2 of n that is 1, 2, 3 will be represented in clockwise direction that is x2 of 0, x2 of 1 and x2 of 2. So, x2 of 0 is 1 which has already represented inside then 2 and 3. Now, we need to multiply those values that is 2 into 1 this is 2 into 1 and similarly 1 into 3, 1 into 3 and 2 into 2. Now, 2 into 1 is 2, 3 this is 4, total is equal to 9. This is about the first step. What the second step says is in the second step we need to rotate the inner circle such that we need to rotate the inner circle in anticlockwise direction that means x2 whatever x2 of 0 which is present in this position moves to x2 of 0 in this position. Similarly, x2 of 1 which is just going towards that is nothing but a anticlockwise direction and x2 of 2 which is moving this place. Now, the corresponding values we need to multiply whatever 2 which is there here will be multiplied with 2 and that you can get it here and similarly 1 will be multiplied with 1, 1 will be multiplied with 1 that is 1 into 1 next 2 will be multiplied with 3 that is 2 into 3. So, 4 plus 1, 5 plus 6 total 11. Similarly, in the third step what you are going to do is we need to again rotate the inner circle anticlockwise direction whatever the x2 of 0 which is there here is moving to next position that is in this position. So, similarly x2 of 0 x2 of 1 and x2 of 2 their values are 1, 2, 3. So, initially we have started with 1, 2, 3 here. So, 1 we have started 1, 2, 3. Now, 1 moves at this position because it is moving towards anticlockwise direction. Now, 1 is now here 1, 2 and 3. Now, we are going to multiply the corresponding elements that is 3 or 2 into 3 is multiplied and similarly 1 into 2 will be multiplied and 2 into 1 will be multiplied. 2 into 3, 6 plus 2 into 1 is 2, 6 plus 2, 8 plus 2 into 1 is 2 raise total 10. This is how we are going to find using concentric circle method. This is the answer 9, 11 and 12 for forming the circular convolution using concentric circle method. Now, let us move to the next slide. This is the references which are going to consider. These are some of the references. Thank you.