Find the AutoCorrelation function of the following gt t fin

Find the Auto-Correlation function of the following, g(t) = t, find R (T)?

1. Calculate the following Integral: Int Pi/2 to 0 2 sin(x) cos(x) dx ?

2. Calculate the Fourier Transform of the following modulating signal,

m(t) = Cos (1000 PI*t) and Sketch the Signal Spectrum ?

3. Generalize Parseval’s Theorem to show that for real, Fourier transformable signals, g1(t) and g2(t),

Int g1(t)g2(t) dt =Int G1(-f)G2(f) df = Int G1(f) G2(-f) df

Int = Integral of function

4. Let the modulated signal be s(t) = 100sin(wc+wa)t +500 cos (wct) - 100 sin(wc-wa)t.

Find the spectrum of this signal by two methods:

Fourier Transform of s(t) 5pts

By using theorem that if s(t) = Re{g(t) exp(jwct), then

S(f) = ½{ G(f-fc) + G* (-f-fc)}, wc is the carrier angular frequency.

%Auto correlation using DFT.

% to write a program in matlab to perform auto correlation using DFT.

clear all;

close all;

clc;

x1=input(\'enter the first sequence\');

x2=fliplr(x1);

l=length(x1);

x1=fft(x1);

x2=fft(x2);

f=x1.*x2;

j=ifft(f);

subplot(2,2,1);

stem(x1);

xlabel(\'input sequence\');

ylabel(\'amplitude\');

title(\'input sequence\');

subplot(2,2,2);

stem(j);

xlabel(\'time\');

ylabel(\'amplitude\');

title(\'autocorrelated sequence\')