how to set the sampling frequency and the time vector for an aperiodic signal while FFT and IFFT

I need to plot x=e^(-t/2*tau) * sin(2*pi*f0*t) and then do the FFT, and IFFT to come back.
f0=1 GHz while tau=2 micro secs
While evaluating the function x, it gives me all values zero. I dont know why...
I am not sure how to set the Fs (sampling frequency) and N (the length).
Please help me as soon as you can.
Here below some code.

T = 1/150;
fs = 1 / T; %fs=150
N = 512; 
t = (0:N-1) * T;

tau = 2*10^-6; %sec
fo=1*10^9; % 1 GHz 

x = (exp(1).^( -t./(2*tau) ) ) .* sin(2*pi*fo*t); %exp(1)=e

%The DFT approximation to the CTFT is given by

X_DFT = T * fft( x );
df = fs / N;
f = df .* (0:(N-1));
w = 2 * pi * f;

%or for a symmetric spectrum

X_DFT2 = fftshift( X_DFT );
f2 = df .* ((-N/2):(N/2-1));
w2 = 2 * pi * f2;
%The exact CTFT and DTFT are
fa = linspace( 0 , 2 * fs , 1000 );
wa = 2 * pi * fa;
fa2 = linspace( -1 * fs , 1 * fs , 1000 );
wa2 = 2 * pi * fa2;
X_CTFT = 1.0 ./ ( a + 1i .* wa );
X_CTFT2 = 1.0 ./ ( a + 1i .* wa2 );
X_DTFT = T * 1.0 ./ ( 1 - exp( -1i .* wa .* T) .* exp( -a .* T ) );
X_DTFT2 = fftshift( X_DTFT ); 

%Plotting the asymmetric transforms

figure( 1 );
plot ( f , abs( X_DFT ) , 'r-o' );
hold on;
plot( fa , abs( X_CTFT ) , 'b-' );
plot( fa , abs( X_DTFT ) , 'g-' );
xlabel( 'f (Hz)' );
ylabel( '|X(f)|' );
title( '|X(f)|' );
axis( [ 0 400 0 0.12] );
legend( 'CTFT estimated from DFT' , 'Analytic CTFT' , 'Analytic DTFT' );
hold off;
print( 'ft_mag.eps' , '-depsc' ); 

figure(2);
plot(t,x);

Thanks to all for any help you can provide to me.