Wiener-Khinchin theorem verification

Wiener-Khinchin:

The Wiener-Khinchin theorem is a fundamental theorem useful for the stochastic processes. The Wiener-Khinchin theorem provides a simple relationship between time and frequency representations of a fluctuating process. Its great practical importance comes from the fact that the autocorrelation function may be directly determined from the measured power spectral density, and vice versa. The wiener-Khinchin is widely used theorem to states the power spectral density of a wide sense stationary random process of the flourier transform for the corresponding autocorrelation function.

Program:

clc;

clear all;

close all;

t=0:0.1:2*pi;

%input signal

x=sin(2*t);

subplot(3,1,1);

plot(x);

xlabel(‘time’);

ylabel(‘amplitude’);

title(‘input signal’);

%autocorrelation of input signal

xu=xcorr(x,x);

%fft of autocorrelation signal

y=fft(xu);

subplot(3,1,2);

plot(abs(y));

xlabel(‘f’);

ylabel(‘amplitude’);

title(‘fft of autocorrelation of input signal’);

%fourier transform of input signal

y1=fft(x);

%finding the power spectral density

y2=(abs(y1)).^2;

subplot(3,1,3);

plot(y2);

xlabel(‘f’);

ylabel(‘magnitude’);

title(‘PSD of input signal’);

Extraction Of Periodic Signal Using Correlation:

A signal is masked by noise are often detected either by correlation techniques or by filtering. Actually, the two techniques are equivalent. The correlation technique may be a measure of extraction of a given signal within the time domain whereas filtering achieves precisely the same leads to frequency domain.

Program:

clear all;

close all;

clc;

t=0:0.1: pi*4;

%input signal1

s=sin(t);

subplot(7,1,1)

plot(s);

title(‘signal s’);

xlabel(‘t’);

ylabel(‘amplitude’);

%input signal2

c=cos(t);

subplot(7,1,2)

plot(c);

title(‘signal c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%generating noise signal

n = randn([1 126]);

%signal+noise

f=s+n;

subplot(7,1,3);

plot(f);

title(‘signal f=s+n’);

xlabel(‘t’);

ylabel(‘amplitude’);

%cross correlation of signal1&signal2

asc=xcorr(s,c);

subplot(7,1,4)

plot(asc);

title(‘ correlation of s and c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%cross correlation of noise&signal2

anc=xcorr(n,c);

subplot(7,1,5)

plot(anc);

title(‘ correlation of n and c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%cross correlation of f&signal2

cfc=xcorr(f,c);

subplot(7,1,6)

plot(cfc);

title(‘ correlation of f and c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%extracting periodic signal

hh=asc+anc;

subplot(7,1,7)

plot(hh);

title(‘addition of sc+nc’);

xlabel(‘t’);

ylabel(‘amplitude’);