Question

Given an energy signal x(t) = (t-5T)e-t-T) , where T-0.1 compute its energy spectrum density in Matlab 1. use two different methods to a. Get its spectrum using Fourier transformation, follovfed by the squaring its amplitude. Plot its Fourier transformation and its energy spectrum density. Get its autocorrelation function, followed by its Fourier transformation. Plot its autocorrelation function and its energy spectrum density. b. 2. In a multipath channel, the received signal y (t) x(t) 0.15x(t -6T) +0.09x(t 10.5T), plot y(t) and its energy spectrum density using either of the two methods in Problem 1. Compare the energy spectrum density of x(t) and y(t). Comment on their similarity and differences. 3. Given an energy message x(t)where T-10 ms Plot the DSB-CS modulated signal with carrier frequency of 500 Hz. Plot its spectrum. Plot the AM modulated signal with carrier frequency of 500 Hz and modulation index of 1. Plot its spectrum. Repeat problem 3.b for modulation index of 0.5 Plot the SSB-LSB modulated signal with carrier frequency of 500 Hz. Plot its spectrum. a. b. c. d.

Answer 1

solution :- 1

clear all;
close all;
clc;

%Define the energy signal x(t)
T=0.1*(10^-3)
t=0:T/100:10*T
x=(t-(5*T)).*(exp(-(((t-(5*T))/T).^2)))
subplot(321)
plot(t,x)
xlabel('t')
ylabel('x(t)')

%Fourier transform of x(t)
k=0;
for f=-15000:2:15000
k=k+1;
X(k)=trapz(t,x.*exp(-j*2*pi*f*t));
end
f=-15000:2:15000
subplot(322)
plot(f,X,'g')
xlabel('f')
ylabel('X(f)')
title('Fourier transform of x(t)')

%Energy spectral density=|X(f)|.^2
ESD=(abs(X)).^2
subplot(323)
plot(f,ESD,'m')
xlabel('f')
ylabel('|X(f)|.^2')
title('ESD of x(t)=|X(f)|.^2')

%Auto correlation of x(t)
x1=fliplr(x)
t1=-fliplr(t)
tt=min(t)+min(t1):T/100:max(t)+max(t1)
y=conv(x,x1)
subplot(324)
plot(tt,y,'r')
xlabel('t')
ylabel('Rxx(t)')
title('Autocorr of x(t)')

%ESD=FT of autocorr(x(t))
k=0;
for f=-15000:2:15000
k=k+1;
Y(k)=trapz(tt,y.*exp(-j*2*pi*f*tt));
end
f=-15000:2:15000
subplot(325)
plot(f,Y,'m')
title('ESD=FT of autocorr(x(t))')
xlabel('f')
ylabel('ESD')

Fourier transform of x(t) 15000-1000D-3000 sc00 15000 1500010000-5000DDO 10000 15000

Fourier transform of x(t) 1 1500010000-5000DDO 10000 15000