(a) As xp(t) = x(t) p(t) By dual of convolution theorem we have .
So we first find the Fourier Transform of p(t) as follows :-
The Fourier Transform of a periodic function is an impulse train at intervals of .
Strength of impulse at being:
Thus, we have can sketch :
Thus we can also sketch and hence :
(b)To recover x(t) from xp(t):
Modulate xp(t) with .
has a spectrum with impluses of equal strength at . Thus the new signal will have copies of the original spectrum (modulated by a constant of-course) at all even multiples of . Now an appropriate Low-pass filter can extract the original spectrum!
To recover x(t) from y(t):
Here too, notice from the figures that modulation with will do the job. Here too, the modulated signal will have copies of the original spectrum at all even multiples of .
(c)So long as adjacent copies of the original spectrum do not overlap in , theoretically one can reconstruct the original signal. Therefore the condition is:
.
.
being: 
:
and
:
. 
. Thus the new signal will have copies of the original spectrum (modulated by a constant of-course) at all even multiples of 
. Now an appropriate Low-pass filter can extract the original spectrum!
