(New page: = Practice Question on sampling and reconstruction (related to Nyquist rate) = The signal <math> x(t)= \cos ( \pi t ) \frac{\sin (3 \pi t)}{\pi t} </math> is sampled with a sampling p...) |
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= Practice Question on sampling and reconstruction (related to Nyquist rate) = | = Practice Question on sampling and reconstruction (related to Nyquist rate) = | ||
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The signal | The signal | ||
<math> x(t)= \cos ( \pi t ) \frac{\sin (3 \pi t)}{\pi t} </math> | <math> x(t)= \cos ( \pi t ) \frac{\sin (3 \pi t)}{\pi t} </math> | ||
| − | is sampled with a sampling period < | + | is sampled with a sampling period <span class="texhtml">''T''</span>. For what values of T is it possible to reconstruct the signal from its sampling? |
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== Share your answers below == | == Share your answers below == | ||
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You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too! | You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too! | ||
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=== Answer 1 === | === Answer 1 === | ||
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| + | X(w) = F(cos(pi t)) * F(sin(3pi t) / (pi t)) | ||
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| + | = pi[delta(w-pi) + delta(w+pi)] * [u(w+3pi) - u(w-3pi)] | ||
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| + | <br> | ||
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| + | Let Z(w) = [u(w+3pi) - u(w-3pi)] | ||
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| + | X(w) = pi[delta(w-pi) + delta(w+pi)] * Z(w) | ||
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| + | = pi[Z(w-pi) + Z(w+pi)] | ||
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| + | = pi[u(w+2pi) - u(w-4pi) + u(w+4pi) - u(w-2pi)] | ||
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| + | <br> | ||
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| + | Thus the signal is band limited abs(w<sub>m</sub>) < 4pi | ||
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| + | Nyquist rate = 2w<sub>m</sub> = 8pi | ||
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| + | w<sub>s</sub> must be greater than 8pi | ||
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| + | T = 2pi/w<sub>s</sub> < 2pi/8pi = 1/4 | ||
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| + | T < 1/4 | ||
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=== Answer 2 === | === Answer 2 === | ||
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Write it here | Write it here | ||
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=== Answer 3 === | === Answer 3 === | ||
| − | Write it here. | + | |
| + | Write it here. | ||
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Revision as of 08:34, 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?
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
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
Answer 2
Write it here
Answer 3
Write it here.
