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Given x(t) find X(f)
 
Given x(t) find X(f)
  
<math>x_(t) \,\!= \cos(\frac{\pi}{2})rect(\frac{t}{2}) \quad (1)</math>  
+
<math>x_(t) \,\!= \cos(\frac{\pi t}{2})rect(\frac{t}{2}) \quad (1)</math>  
  
 
Using the convolution property
 
Using the convolution property

Revision as of 19:03, 9 February 2009

1a/

Given x(t) find X(f)

$ x_(t) \,\!= \cos(\frac{\pi t}{2})rect(\frac{t}{2}) \quad (1) $

Using the convolution property

$ X_(f) = \mathcal{F} (cos(\frac{\pi t}{2}))* \mathcal{F}(rect(\frac{t}{2})) $

where

$ \mathcal{F} (cos(\frac{\pi t}{2})) = \frac{1}{2} [\delta(f - \frac{1}{4}) + \delta(f + \frac{1}{4})] $

and

$ \mathcal{F}(rect(\frac{t}{2})) = 2 sinc( 2 f) $

substituting the known transforms into $ \quad (1) $

$ X_(f) = \frac{1}{2} [\delta(f - \frac{1}{4}) + \delta(f + \frac{1}{4})] * 2 sinc( 2 f) $

Evaluating the statement ( using sifting )

$ X_(f) = sinc(2 (f - \frac{1}{4})) + sinc( 2(f+\frac{1}{4})) $

  • Nice and clean justification. Does anybody see a mistake? --Mboutin 16:40, 9 February 2009 (UTC)

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