The signal x(t) has Fourier Transform X(f) given for all f by X(f) = 0.5rect(f/2

Question

The signal x(t) has Fourier Transform X(f) given for all f by X(f) = 0.5rect(f/2000) + 0.5tri(f/500). What is the signal x(t)? Use MATLAB to plot it as a function of time over a sufficiently long time interval. If the signal x(t) is ideally sampled to form the signal x_s(t), sketch the Fourier Transform X_s(f) of x_s(t), for four cases: f_s = 3500 sps, f_s = 2500 sps, f_s = 1500 sps and f_s = 500 sps. You will need some math to do this but the bulk of the work involving sketching spectra should be done with MATLAB.

Explanation / Answer

FFT signal for the fs=3500 and fs=2500 and fs=1500 and fs=500 can be evalulated fom the following matlab code

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%

fs = 1e10; % 20 GHz sampling
ts = 1/fs;
t = 0:ts:10*1e-9;
Pulse = rectpuls(t); % 10ns Rect. pulse
plot(linspace(-5*1e9,5*1e9,512),fftshift(abs(fft(Pulse,512))))

%
format long;

itteration_count = 50;
sum_count = 50;
peroid = 2*pi;
origin = 0;

t = origin:peroid/itteration_count:peroid;
f = 1:itteration_count+1;

for k = 0:itteration_count
sum = 0;
for n = 1:2:sum_count
sum = sum + (((((-1)^((n-1)/2))/n^2)*sin((n*pi*t(k+1)))/peroid));
end
f(k+1) = sum*(8/(pi^2));
end

plot(t,f)

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