(New page: == LECTURE on September 11, 2009 == The perfect reconstruction of '''<math>{x(t)}</math>''' from '''<math>x_s(t)</math>''' is possible if '''<math>X(f) = 0</math>''' when '''<math>|f| \ge...) |
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== LECTURE on September 11, 2009 == | == LECTURE on September 11, 2009 == | ||
| − | The perfect reconstruction of '''<math>{x(t)}</math>''' from '''<math>x_s(t)</math>''' is possible if '''<math>X(f) = 0</math>''' when '''<math>|f| \ge {1 | + | The perfect reconstruction of '''<math>{x(t)}</math>''' from '''<math>x_s(t)</math>''' is possible if '''<math>X(f) = 0</math>''' when '''<math>|f| \ge \frac{1}{|2T|}</math>''' |
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| + | '''PROOF:''' Look at the graph of '''<math>X_s(f)</math>''' | ||
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| + | *** Image goes here*** | ||
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| + | To avoid aliasing, | ||
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| + | '''<math>\frac{1}{T}\ - f_M \ge f_M</math>''' '''<math>\iff</math>''' '''<math>\frac{1}{T}\ \ge 2f_M</math>''' | ||
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| + | To recover the signal, we will require a low pass filter with gain '''<math>T</math>''' and cutoff, '''<math>\frac{1}{2T}</math>''' | ||
Revision as of 03:21, 23 September 2009
LECTURE on September 11, 2009
The perfect reconstruction of $ {x(t)} $ from $ x_s(t) $ is possible if $ X(f) = 0 $ when $ |f| \ge \frac{1}{|2T|} $
PROOF: Look at the graph of $ X_s(f) $
- Image goes here***
To avoid aliasing,
$ \frac{1}{T}\ - f_M \ge f_M $ $ \iff $ $ \frac{1}{T}\ \ge 2f_M $
To recover the signal, we will require a low pass filter with gain $ T $ and cutoff, $ \frac{1}{2T} $
