06-05-2011, 03:35 PM
Procedure:-
1. Enter the sequence x[n]
2. Enter the sequence y[n]
3. Find the lengths of x[n] and y[n],ie;Nx and Ny respectively
4. Check Nx=Ny:proceed if equal
5. Initialize a loop variable number of output points
6. For each out sample , access the samples of y[n] in the cyclic order
7. Find the sum of products of x[n] and cyclically folded and shifted y[n]
Description:-
Steps for Cyclic Convolution
Steps for cyclic convolution are the same as the usual convolution, except all index calculations are done "mod N" = "on the wheel"
Step1: “Plot f[m] and h[−m]
Subfigure 1.1 Subfigure 1.2
Step 2: "Spin" h[−m] n times Anti Clock Wise (counter-clockwise) to get h[n-m]
(i.e. Simply rotate the sequence, h[n], clockwise by n steps)
Figure 2: Step 2
Step 3: Point wise multiply the f[m] wheel and the h[n−m] wheel. Sum=y[n]
Step 4: Repeat for all 0≤n≤N−1
Example 1: Convolve (n = 4)
Subfigure 3.1 Subfigure 3.2
Figure 3: Two discrete-time signals to be convolved.
• h[−m] =
Figure 4
Multiply f[m] and sum to yield: y[0] =3
• h[1−m]
Figure 5
Multiply f[m] and sum to yield: y[1] =5
• h[2−m]
Figure 6
Multiply f[m] and sum to yield: y[2] =3
• h[3−m]
Figure 7
Multiply f[m] and sum to yield: y[3] =1
MATLAB program for circular convolution
Code:
clear
clc
close all
% FIRST SEQUENCE X1
x1 = [2 1 2 1];
nx1 = 0:length(x1)-1;
subplot(3,1,1);
stem(nx1,x1);
xlabel('n-->');
ylabel('x1-->');
title('First Sequence');
grid on;
% SECOND SEQUENCE X2
x2 = [1 2 3 4];
nx2 = 0:length(x2)-1;
subplot(3,1,2);
stem(nx2,x2);
xlabel('n-->');
ylabel('x2-->');
title('Second Sequence');
grid on;
% CIRCULAR CONVOLUTION ORDER IS N
N = 4;
% CHECK FOR LENGTH OF X1 AND X2 WITH RESPECT TO N
if length(x1) > N
error(' N must be >= the length of x1');
end;
if length(x2) > N
error(' N must be >= the length of x2');
end;
% if N > length(x1) or length(x2) zero pad
x1 = [x1,zeros(1,N-length(x1))];
x2 = [x2,zeros(1,N-length(x2))];
% CIRCULARLY FOLD X2,X2(-M,MOD N), FOR M = 0,1,...,N-1
% X2 = X2(MOD(-M,N)+1);
% CONVOLVE THE CIRCULAR SHIFTED FOLDED SEQUENCE WITH X1
% Y(N) = SUM(X1(M) * X2((N-M)MOD N))
m = [0:N-1];
for n = 0:N-1,
y(n+1) = sum(x2(mod(n-m,N)+1).*x1);
end;
ny = 0:3;
subplot(3,1,3);
stem(ny,y);
xlabel('n-->');
ylabel('y-->');
title('First Sequence');
grid on;
disp(‘First Sequence is:’);
x1
disp(‘Second Sequence is:’)
x2
disp(‘Circular Convolution is:’);
y
