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Given x(t) find X(f) | Given x(t) find X(f) | ||
| − | <math> | + | <math>x(t) \,\!= \cos(\frac{\pi t}{2})rect(\frac{t}{2}) \quad (1)</math> |
Using the convolution property | Using the convolution property | ||
| − | <math> | + | <math>X(f) = \mathcal{F} (cos(\frac{\pi t}{2}))* \mathcal{F}(rect(\frac{t}{2}))</math> |
where | where | ||
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substituting the known transforms into <math>\quad (1)</math> | substituting the known transforms into <math>\quad (1)</math> | ||
| − | <math> | + | <math>X(f) = \frac{1}{2} [\delta(f - \frac{1}{4}) + \delta(f + \frac{1}{4})] * 2 sinc( 2 f) </math> |
Evaluating the statement ( using sifting ) | Evaluating the statement ( using sifting ) | ||
| − | <math> | + | <math>X(f) = sinc(2 (f - \frac{1}{4})) + sinc( 2(f+\frac{1}{4}))</math> |
*<span style="color:red"> Nice and clean justification. Does anybody see a mistake?</span> --[[User:Mboutin|Mboutin]] 16:40, 9 February 2009 (UTC) | *<span style="color:red"> Nice and clean justification. Does anybody see a mistake?</span> --[[User:Mboutin|Mboutin]] 16:40, 9 February 2009 (UTC) | ||
Latest revision as of 18:21, 10 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)
- Was it the missing t in equation (1)?--Mlo 23:11, 9 February 2009 (UTC)
