sampling and similarities

we all know cos x is periodic with a period of 2*n*pi.so if two signals are sampled at a rate less than the Nyquist rate then we can have same samples from two different signals.The reason is as follows:

lets say sampling frequency is Fs.Hence Ts=1/Fs.
if a signal say cos(2*pi*f*t) is sampled then we get the sampled version as cos(2*pi*F*n*Ts)

now lets take another signal cos(2*pi*(f+Fs)*t).If this is sampled at the same rate then we get:
cos(2*pi*(f+Fs)*n*Ts)=cos(2*pi*f*n*Ts+2*pi*Fs*n*Ts) now the term 2*pi*n*Fs*Ts yields nothing but 2*pi*n.Since cos x is periodic with period 2*pi*n hence the second sampled signal reduces to cos(2*pi*f*n*Ts) which is the same sampled signal as the previous one.

PROBLEM :
 Four sinusoidal signals are sampled and they are found to have same sampled version in pairs.Now you know why.


function sinseq1()

A=2;
phi=0;
n=0:10;

%case 1
x=A*cos(2*pi*1.1*n+phi);
subplot(2,2,1);
stem(n,x);

%case 2
x=A*cos(2*pi*0.9*n+phi);
subplot(2,2,2);
stem(n,x);

%case 3
x=A*cos(2*pi*0.8*n+phi);
subplot(2,2,3);
stem(n,x);

%case 4
x=A*cos(2*pi*1.2*n+phi);
subplot(2,2,4);
stem(n,x);

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