Discrete Time Stable LTI Systems

**************************************
%Linearity Property of two sequences
**************************************
n=0:40; a=2; b=-3;
x1=cos(2*pi*0.1*n);
x2=cos(2*pi*0.4*n);
x=a*x1+b*x2;
ic=[0 0];
num=[2.2403 2.4908 2.2403];
den=[1 -0.4 0.75];
y1=filter(num,den,x1,ic);
y2=filter(num,den,x2,ic);
y=filter(num,den,x,ic);
yt=a*y1+b*y2;
d=y-yt;
subplot(3,1,1), stem(n,y); grid
subplot(3,1,2), stem(n,yt); grid
subplot(3,1,3), stem(n,d); grid

**************************************
%shift Invariance property
**************************************
n=0:40;D=10;
x=3*cos(2*pi*0.1*n)-2*cos(2*pi*0.4*n);
xd=[zeros(1,D) x];
num=[2.2403 2.4908 2.2403];
den=[1 -0.4 0.75];
ic=[0 0];
y=filter(num,den,x,ic)
yd=filter(num,den,xd,ic)
d=y-yd(1+D:41+D);
subplot(3,1,1),stem(y),grid;
subplot(3,1,2),stem(yd),grid;
subplot(3,1,3),stem(d),grid;

**************************************
%To check stability of a system
**************************************
num=[1 0.8];
den=[1 1.5 .9];
N=200;
h=impz(num,den,N+1);
sum=0;
n=0:N;
for k=1:N+1
if abs(h(k))<10^(-6);
break
end
sum=sum+h(k);
end
stem(n,h); grid;
disp('Value='),
disp(sum)