2. To find linear and circular convolution.

 Code:

clc;

clear all;

close;

//linear convolution

x=[1,2,3,4,5]; // First Signal

h=[1,2,1,2]; // Second Signal

nx=length(x);

nh=length(h);

N=nx+nh-1;

n1=0:nx-1;

n2=0:nh-1;

n=0:nx+nh-2;

X_lin=[x,zeros(1,nh-1)];

//append zeros to make length equal to nx+nh-1

H_lin=[h,zeros(1,(nx-1))];

//append zeros to make length equal to nx+nh-1

// Formation of Toeplitz Matrix

for i=1:N

for j=1:N

if (i<j) then

X2_lin(i,j)=0;

else

X2_lin(i,j)=X_lin(i-j+1);

end

end

end

Y_lin=X2_lin*H_lin';// matrix multiplication to obtain linear convolution

disp('The first signal x(n) is ');

disp(x);

disp('The second signal h(n) is ');

disp(h);

disp('The toeplitz matrix of x(n) to calculate linear convolution is ');

disp(X2_lin);

disp('The linear convolution for x(n)and h(n) is ');

disp(Y_lin');

//Plotting of the signals

figure(1);

subplot(2,2,1);

a=gca();

a.x_location="origin";

a.y_location="origin";

plot2d3(n1,x);

a.children.children.foreground=5;

xtitle('input signal x(n)','n','input');

subplot(2,2,2);

a=gca();

a.x_location="origin";

a.y_location="origin";

plot2d3(n2,h);

a.children.children.foreground=5;

xtitle('input signal h(n)','n','input');

subplot(2,2,3);

a=gca();

a.x_location="origin";

a.y_location="origin";

plot2d3(n,Y_lin);

a.children.children.foreground=5;

xtitle('Linearly convolved signal','n','Y(n)');

//Circular Convolution

M=max(nx,nh);

X_circ=[x,zeros(1,M-nx)];

H_circ=[h,zeros(1,M-nh)];

//Formation of toeplitz matrix

for i=1:M

for j=1:M

if (i-j+1)>0 &(i-j+1)<M+1 then

X2_circ(i,j)=X_circ(i-j+1);

elseif (i<j) then

X2_circ(i,j)=X_circ(i-j+1+M)

end

end

end

Y_circ=X2_circ*H_circ';

// Matrix multiplication to obtain circular convolution

disp('The toeplitz matrix of x(n) to calculate Circular convolution is

');

disp(X2_circ);

disp('The Circular convolution for x(n)and h(n) is ');

disp(Y_circ');

//Plotting of the signals

m=0:M-1;

subplot(2,2,4);

a=gca();

a.x_location="origin";

a.y_location="origin";

plot2d3(m,Y_circ);

a.children.children.foreground=5;

xtitle('Circilarly convolved signal','n','Y(n)');