List of top Signals and Systems Questions on Fourier Transform

The Fourier transform \(X(\omega)\) of the signal \(x(t)\) is given by \[ X(\omega) = \begin{cases} 1, & |\omega| < W_0
0, & |\omega| > W_0 \end{cases} \] Which one of the following statements is true?

Match List I with List II

List I (Signal)		List II (Fourier Transform)
A	$ x(t-t_0)$	I	$ \frac{dx(\omega)}{{d\omega}}$
B	$ e^{j \omega_0 t} x(t) $	II	$ e^{j \omega t_0} X(\omega) $
C	$ \frac{dx(t)}{{dt}}$	III	$ x(\omega-\omega_0)$
D	$(-jt)x(t) $	IV	$j\omega X(\omega) $

Choose the correct answer from the options given below:

The Fourier transform $X(j\omega)$ of the signal \[ x(t)=\frac{t}{(1+t^{2})^{2}} \] is ________________.