MATLAB: How can I find y[n]

dİscrete signal outputdiscrete time signal

I want to find output of system y[n] for n = 1,2,3,4 , y[n] is equal to y[n]=0.5*y[n-1] + x[n] + 0.5*x[n-3] .I have a knowledge about this system , system is LTI .I don't know right the code that below.

n = [1:4];
x = cos(pi.*n).*[0 ones(1,3)];
x_shifted=[0 x(1:end-1)];
x_scaled=2.*x;
% stem(n,x)
y(3) = 0;
y2(3) = 0;
for k=4:length(x)
        y(k) = 0.5*y(k-1)  + x_shifted(k) +0.5*x_shifted(k-1)+0.5*x_shifted(k-3);
        y2(k)=0.5*y(k-1)  + x(k) + 0.5*x(k-3);
end
stem(n,y,'rx')
hold on 
stem(n,y2,'b')

Best Answer

% write your equation as : y(n)-0.5y(n-1) = x(n)+0x(n-1)+0x(n-2)+0.5x(n-3)
% and define the coefficient vectors a and b as shown. 
n = 1:4;
a = [1 -0.5];
b = [1 0 0 0.5];
x = cos(pi.*n).*[0 ones(1,3)];
y = filter(b,a,x);  % output of the LTI system
stem(n,y,'linewidth',3,'color','m')
grid;
a = title('Output of an LTI System y(n)');
set(a,'fontsize',14);
a = ylabel('y(n)');
set(a,'Fontsize',14);
a = xlabel('n [1 4]');
set(a,'Fontsize',14);
% figure
% stem(n,x,'linewidth',3,'color','g')

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MATLAB: Vectorization with a loop and repmat

That sounds like a nice cody task ;-) I'm sure there are dozens of solution, so my suggestion is:

m=k+mod(bsxfun(@(x,y) y+x,(0:n-k)',0:n-k),n-k+1);

MATLAB: How to write a sigma calculation

In part two of the problem, you will have to use the definition of the DTFT to compute X(omega). The resultant function is just a periodic version of X(jw) from part one of the problem. See the following code below.

%%Part 1
clear variables
close all
N= 200;
T = 2;
t  = -2:4/(N-1):2;
x = rectpuls(t,T);
f= -2:4/(N-1):2;
X = T*sin(pi*f*T)./(pi*f*T);
figure
subplot(121)
plot(t,x,'linewidth',2,'color','b')
grid;
a = title('x(t)');
set(a,'fontsize',14);
a = ylabel('x');
set(a,'Fontsize',14);
a = xlabel('t');
set(a,'Fontsize',14);
subplot(122)
plot(f,abs(X)/(max(X)),'linewidth',2,'color','m')
a = title('|X(jw)|');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid
%%Part 2
M = 1000;
N = 7;
n = 0:N-1;
xn = ones(1,N);
w = 8*pi;
omega = -w:2*w/(M-1):w; 
Xn= exp(-1i.*omega.*(N-1)./2).*(sin(omega*N/2)./sin(omega/2)); %define DTFT function
Mag = abs(Xn)/max(Xn);   %compute magnitude
Phase = angle(Xn);       %compute phase
figure
subplot(3,1,1)
stem(n,xn,'linewidth',2,'color','b'); 
a = title('x(n)');
set(a,'fontsize',14);
a = ylabel('xn');
set(a,'Fontsize',14);
a = xlabel('n');
set(a,'Fontsize',14);
grid
subplot(3,1,2)
plot(omega/pi,real(Mag),'linewidth',2,'color','k'); 
a = title('|X(\omega)|');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid
subplot(3,1,3)
plot(omega./pi,Phase,'linewidth',2,'color','k'); 
a = title('<X(\omega)');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid

Best Answer

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MATLAB: Vectorization with a loop and repmat

MATLAB: How to write a sigma calculation

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