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=== Answer 1 === | === Answer 1 === | ||
| − | X(w) = F(cos(pi t)) * F(sin(3pi t) / (pi t)) | + | X(w) = F(cos(pi t)) * F(sin(3pi t) / (pi t)) |
| − | = pi[delta(w-pi) + delta(w+pi)] * [u(w+3pi) - u(w-3pi)] | + | = pi[delta(w-pi) + delta(w+pi)] * [u(w+3pi) - u(w-3pi)] |
| − | <br> | + | <br> |
| − | Let Z(w) = [u(w+3pi) - u(w-3pi)] | + | Let Z(w) = [u(w+3pi) - u(w-3pi)] |
| + | <br> | ||
| + | X(w) = pi[delta(w-pi) + delta(w+pi)] * Z(w) | ||
| − | + | = pi[Z(w-pi) + Z(w+pi)] | |
| − | = pi[ | + | = pi[u(w+2pi) - u(w-4pi) + u(w+4pi) - u(w-2pi)] |
| − | + | <br> | |
| − | < | + | Thus the signal is band limited abs(w<sub>m</sub>) < 4pi |
| − | + | Nyquist rate = 2w<sub>m</sub> = 8pi | |
| − | + | w<sub>s</sub> must be greater than 8pi | |
| − | w<sub>s</sub> | + | T = 2pi/w<sub>s</sub> < 2pi/8pi = 1/4 |
| − | T | + | T < 1/4 |
| + | |||
| + | --[[User:Ssanthak|Ssanthak]] 12:35, 20 April 2011 (UTC) | ||
| − | |||
=== Answer 2 === | === Answer 2 === | ||
Revision as of 08:35, 20 April 2011
Contents
The signal
$ x(t)= \cos ( \pi t ) \frac{\sin (3 \pi t)}{\pi t} $
is sampled with a sampling period T. For what values of T is it possible to reconstruct the signal from its sampling?
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Answer 1
X(w) = F(cos(pi t)) * F(sin(3pi t) / (pi t))
= pi[delta(w-pi) + delta(w+pi)] * [u(w+3pi) - u(w-3pi)]
Let Z(w) = [u(w+3pi) - u(w-3pi)]
X(w) = pi[delta(w-pi) + delta(w+pi)] * Z(w)
= pi[Z(w-pi) + Z(w+pi)]
= pi[u(w+2pi) - u(w-4pi) + u(w+4pi) - u(w-2pi)]
Thus the signal is band limited abs(wm) < 4pi
Nyquist rate = 2wm = 8pi
ws must be greater than 8pi
T = 2pi/ws < 2pi/8pi = 1/4
T < 1/4
--Ssanthak 12:35, 20 April 2011 (UTC)
Answer 2
Write it here
Answer 3
Write it here.
