Sampling Theorem Verification, Noise removal by using cross-correlation/autocorrelation

Sampling theorem:

Sampling theorem is used to convert analog signal into discrete time signal.it has applications in digital signal processing and digital communication. Sampling theorem is determined has that if the sampling rate in any pulse modulation system exceeds twice the maximum frequency of the message signal, with this the receiver will reconstructed the signal with minimum distortion.

Whereas,

 f_s is the sampling rate

f_m is the maximum frequency of message signal

Then sampling theorem can be expressed in mathematical form as

F_s≥ 2f_m

Well 2f_m is determined as Nyquist rate.

It the condition is

F_s≤2f_m

It is defined as aliasing effect it occurred when the sampling rate is less than twice the maximum frequency of the message signal.

F_s=sampling rate

Program:

clc;

clear all;

close all;

t=-10:.01:10;

T=4;

fm=1/T;

x=cos(2*pi*fm*t);

subplot(2,2,1);

plot(t,x);

xlabel(‘time’);

ylabel(‘x(t)’);

title(‘continous time signal’);

grid;

n1=-4:1:4;

fs1=1.6*fm;

fs2=2*fm;

fs3=8*fm;

x1=cos(2*pi*fm/fs1*n1);

subplot(2,2,2);

stem(n1,x1);

xlabel(‘time’);

ylabel(‘x(n)’);

title(‘discrete time signal with fs<2fm’);

hold on;

subplot(2,2,2);

plot(n1,x1);

grid;

n2=-5:1:5;

x2=cos(2*pi*fm/fs2*n2);

subplot(2,2,3);

stem(n2,x2);

xlabel(‘time’);

ylabel(‘x(n)’);

title(‘discrete time signal with fs=2fm’);

hold on;

subplot(2,2,3);

plot(n2,x2)

grid;

n3=-20:1:20;

x3=cos(2*pi*fm/fs3*n3);

subplot(2,2,4);

stem(n3,x3);

xlabel(‘time’);

ylabel(‘x(n)’);

title(‘discrete time signal with fs>2fm’)

hold on;

subplot(2,2,4);

plot(n3,x3)

grid;

Noise removal by using cross-correlation/autocorrelation

Detection of the periodic signal conceal by random noise is of great importance .The noise signal encountered in practice may be a signal with random amplitude variations. A signal is uncorrelated with any periodic signal. If s(t) is a periodic signal and n(t) is a noise signal then.

clc;

clear all;

close all;

t=0:0.2:pi*8;

%input signal

s=sin(t);

subplot(6,1,1);

plot(s);

title(‘signal s’);

xlabel(‘t’);

ylabel(‘amplitude’);

%generating noise

n = randn([1 126]);

%noisy signal

f=s+n;

subplot(6,1,2)

plot(f);

title(‘signal f=s+n’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of input signal

as=xcorr(s,s);

subplot(6,1,3);

plot(as);

title(‘auto correlation of s’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of noise signal

an=xcorr(n,n);

subplot(6,1,4)

plot(an);

title(‘auto correlation of n’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of transmitted signal

cff=xcorr(f,f);

subplot(6,1,5)

plot(cff);

title(‘auto correlation of f’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of received signal

hh=as+an;

subplot(6,1,6)

plot(hh);

title(‘addition of as+an’);

xlabel(‘t’);

ylabel(‘amplitude’);