Revision as of 18:40, 20 April 2011

Practice Question on Nyquist rate

What is the Nyquist rate of the signal

Failed to parse (lexing error): x(t) = \frac{ \sin ( \pi t )}{\pi t} \frac{ \sin ( \pi t )}{\pi t} ?

^ Sorry, I think I broke the equation in the problem statement but I don't know how to fix it. -ke

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Answer 1

Use CTFT to find the frequency response

Using table, we know FT(sin(pi t)/(pi t)) --> u(w+W) - u(w-W)

X(w) = [u(w+pi) - u(w-pi)] * [u(w+pi) - u(w-pi)]

= int_-infinity^infinity ( [u(W+pi) - u(W-pi)] [u(w-W+pi) - u(w-W-pi)] )dW

= int-_pi^pi (u(w-W+pi) - u(w-W-pi)) dW

since W-pi <= w < pi-W, and -pi <= W < pi

-2pi <= w < 2pi

I'm not sure if I did the convolution right... help please (if you can read it)

int_pi^-w-pidW if -2pi <= w < 0

X(w) = { int_w-pi^pidW if 0 <= w < 2pi

0 else

-w-2pi if -2pi <= w < 0

= { 2pi-w if 0 <= w < 2pi

0 else

Regardless, wm = 2pi so NR = 4pi

--Kellsper 22:36, 20 April 2011 (UTC)

Answer 2

Write it here

Answer 3

Write it here.

Back to ECE301 Spring 2011 Prof. Boutin

@@ Line 3: / Line 3: @@
 What is the Nyquist rate of the signal
-<math>x(t) = \frac{ \sin ( \pi t )}{\pi t} \frac{ \sin ( \pi t )}{\pi t} ?</math>
+'''Failed to parse (lexing error): x(t) = \frac{ \sin ( \pi t )}{\pi t} \frac{ \sin ( \pi t )}{\pi t}&nbsp;?'''
+^ Sorry, I think I broke the equation in the problem statement but I don't know how to fix it. -ke<br>
 ----
@@ Line 15: / Line 17: @@
 === Answer 1  ===
 Use CTFT to find the frequency response
-Using table, we know FT(sin(pi t)/(pi t)) --&gt; u(w+W) - u(w-W)
-X(w) =&nbsp;[u(w+pi) - u(w-pi)] * [u(w+pi) - u(w-pi)]
-&nbsp;&nbsp; &nbsp; &nbsp; = int<sub>-infinity</sub><sup>infinity</sup>&nbsp;( [u(W+pi) - u(W-pi)] [u(w-W+pi) - u(w-W-pi)] )dW<sup></sup><sub></sub><sub></sub>
+Using table, we know FT(sin(pi t)/(pi t)) --&gt; u(w+W) - u(w-W)
-&nbsp;&nbsp; &nbsp; &nbsp; =&nbsp; int-<sub>pi</sub><sup>pi</sup>&nbsp;(u(w-W+pi) - u(w-W-pi)) dW
+X(w) =&nbsp;[u(w+pi) - u(w-pi)] * [u(w+pi) - u(w-pi)]
-&nbsp;&nbsp; &nbsp; &nbsp; since &nbsp; &nbsp; &nbsp;W-pi &lt;= w &lt; pi-W, &nbsp; &nbsp; and &nbsp; &nbsp; -pi &lt;= W &lt; pi
+&nbsp;&nbsp; &nbsp; &nbsp; = int<sub>-infinity</sub><sup>infinity</sup>&nbsp;( [u(W+pi) - u(W-pi)] [u(w-W+pi) - u(w-W-pi)] )dW<sup></sup><sub></sub><sub></sub>
-&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;-2pi &lt;= w &lt; 2pi
+&nbsp;&nbsp; &nbsp; &nbsp; =&nbsp; int-<sub>pi</sub><sup>pi</sup>&nbsp;(u(w-W+pi) - u(w-W-pi)) dW
-I'm not sure if I did the convolution right... help please (if you can read it)
+&nbsp;&nbsp; &nbsp; &nbsp; since &nbsp; &nbsp; &nbsp;W-pi &lt;= w &lt; pi-W, &nbsp; &nbsp; and &nbsp; &nbsp; -pi &lt;= W &lt; pi
-&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;int<sub>pi</sub><sup>-w-pi</sup>dW &nbsp; &nbsp; if -2pi &lt;= w &lt; 0
+&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;-2pi &lt;= w &lt; 2pi
-X(w) = { &nbsp; &nbsp; int<sub>w-pi</sub><sup>pi</sup>dW &nbsp; &nbsp; &nbsp;if &nbsp;0 &lt;= w &lt; 2pi
+I'm not sure if I did the convolution right... help please (if you can read it)
-&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; else
+&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;int<sub>pi</sub><sup>-w-pi</sup>dW &nbsp; &nbsp; if -2pi &lt;= w &lt; 0
+X(w) = { &nbsp; &nbsp; int<sub>w-pi</sub><sup>pi</sup>dW &nbsp; &nbsp; &nbsp;if &nbsp;0 &lt;= w &lt; 2pi
+&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; else
-&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; -w-2pi &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;if -2pi &lt;= w &lt; 0
+<br>
-&nbsp;&nbsp; &nbsp; &nbsp; &nbsp;= { &nbsp; &nbsp; 2pi-w&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if 0 &lt;= w &lt; 2pi
+&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; -w-2pi &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;if -2pi &lt;= w &lt; 0
-&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; else
+&nbsp;&nbsp; &nbsp; &nbsp; &nbsp;= { &nbsp; &nbsp; 2pi-w&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if 0 &lt;= w &lt; 2pi
+&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; else
+<br>
 Regardless, wm = 2pi so NR = 4pi
 --[[User:Kellsper|Kellsper]] 22:36, 20 April 2011 (UTC)<br>
 === Answer 2  ===

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