(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...) |
|||
| (4 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| − | = Practice Question on sampling and reconstruction (related to Nyquist rate) = | + | = [[:Category:Problem_solving|Practice Question]] on sampling and reconstruction (related to Nyquist rate) = |
| + | |||
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? |
| + | |||
---- | ---- | ||
| + | |||
== Share your answers below == | == Share your answers below == | ||
| + | |||
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! | ||
---- | ---- | ||
| + | |||
=== Answer 1 === | === 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)] | ||
| + | |||
| + | <br> | ||
| + | |||
| + | 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[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 | ||
| + | |||
| + | T = 2pi/w<sub>s</sub> < 2pi/8pi = 1/4 | ||
| + | |||
| + | T < 1/4 | ||
| + | |||
| + | --[[User:Ssanthak|Ssanthak]] 12:35, 20 April 2011 (UTC) | ||
| + | |||
| + | :TA's comment: The condition on the sampling frequency is correct. However you have a mistake when using the multiplication property of the FT, it is missing the <span class="texhtml">1 / 2π</span> factor. | ||
| + | |||
=== Answer 2 === | === Answer 2 === | ||
| + | |||
| + | I got the same answer, but with my FT multiplied by 1/4π. (I'm pretty sure you just flipped the 1/2π on accident.) | ||
| + | |||
| + | --[[User:Kellsper|Kellsper]] 18:34, 21 April 2011 (UTC) | ||
| + | |||
Write it here | Write it here | ||
| + | |||
=== Answer 3 === | === Answer 3 === | ||
| − | Write it here. | + | |
| + | Write it here. | ||
| + | |||
---- | ---- | ||
Latest revision as of 10:30, 11 November 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
--Ssanthak 12:35, 20 April 2011 (UTC)
- TA's comment: The condition on the sampling frequency is correct. However you have a mistake when using the multiplication property of the FT, it is missing the 1 / 2π factor.
Answer 2
I got the same answer, but with my FT multiplied by 1/4π. (I'm pretty sure you just flipped the 1/2π on accident.)
--Kellsper 18:34, 21 April 2011 (UTC)
Write it here
Answer 3
Write it here.
