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An answer to this 1a) question is stated in the discussion [https://kiwi.ecn.purdue.edu/rhea/index.php/Talk:HW_3_Question_1] | An answer to this 1a) question is stated in the discussion [https://kiwi.ecn.purdue.edu/rhea/index.php/Talk:HW_3_Question_1] | ||
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b) | b) | ||
| − | <math> | + | This is how I came to my conclusion, I think it makes morse sense then the previous mentioned answer. |
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| + | First take the x(t) from part a and call it <math>x_1(t)</math> | ||
| + | <math>x_1(t) \,\!= \cos(\frac{\pi t}{2})rect(\frac{t}{2})</math> | ||
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| + | Now since this is a repeating function use the rep function to get | ||
| + | <math>x(t) = rep_4(x_1(t))</math> | ||
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| + | We know that the CTFT of x1(t) is: | ||
| + | <math>x_1(f) = sinc(2(f-\frac{1}{4} ) + sinc(2(f + \frac{1}{4})</math> | ||
| + | We also know from alabechs notes section 1.4.1 that | ||
| + | <math> rep_T(x_1(t)) \Rightarrow \frac{1}{T} comb_{1/T}(X_1(f))</math> | ||
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| + | Put all the pieces together and you get something that looks like<br/> | ||
| + | <math> X(f) = \frac{1}{4} comb_{1/4}(sinc(2(f-\frac{1}{4} ) + sinc(2(f + \frac{1}{4}))</math> | ||
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<math>x_(f) \,\!= \frac{1}{T}\sum_{k} ( \delta (f - \frac{1}{4}) + \delta (f + \frac{1}{4}))( \delta (f - \frac{k}{4}))</math> | <math>x_(f) \,\!= \frac{1}{T}\sum_{k} ( \delta (f - \frac{1}{4}) + \delta (f + \frac{1}{4}))( \delta (f - \frac{k}{4}))</math> | ||
*<span style="color:red"> Can you write your answer using a comb operator? </span> --[[User:Mboutin|Mboutin]] 10:45, 9 February 2009 (UTC) | *<span style="color:red"> Can you write your answer using a comb operator? </span> --[[User:Mboutin|Mboutin]] 10:45, 9 February 2009 (UTC) | ||
* <span style="color:red"> How did you get to that answer? Please add some intermediate steps. </span> --[[User:Mboutin|Mboutin]] 10:50, 9 February 2009 (UTC) | * <span style="color:red"> How did you get to that answer? Please add some intermediate steps. </span> --[[User:Mboutin|Mboutin]] 10:50, 9 February 2009 (UTC) | ||
Revision as of 17:52, 10 February 2009
1 a)
$ x_(t) \,\!= \cos(\frac{\pi}{2})rect(\frac{t}{2}) $
Based on the Prof Alen's note page 179
$ x_(f) \,\!= \frac{1}{2}( \delta (f - \frac{1}{4}) + \delta (f + \frac{1}{4}))sinc(t/2) $
- Would you know how to compute this FT without a table if asked? --Mboutin 10:45, 9 February 2009 (UTC)
An answer to this 1a) question is stated in the discussion [1]
b)
This is how I came to my conclusion, I think it makes morse sense then the previous mentioned answer.
First take the x(t) from part a and call it $ x_1(t) $ $ x_1(t) \,\!= \cos(\frac{\pi t}{2})rect(\frac{t}{2}) $
Now since this is a repeating function use the rep function to get $ x(t) = rep_4(x_1(t)) $
We know that the CTFT of x1(t) is: $ x_1(f) = sinc(2(f-\frac{1}{4} ) + sinc(2(f + \frac{1}{4}) $ We also know from alabechs notes section 1.4.1 that $ rep_T(x_1(t)) \Rightarrow \frac{1}{T} comb_{1/T}(X_1(f)) $
Put all the pieces together and you get something that looks like
$ X(f) = \frac{1}{4} comb_{1/4}(sinc(2(f-\frac{1}{4} ) + sinc(2(f + \frac{1}{4})) $
$ x_(f) \,\!= \frac{1}{T}\sum_{k} ( \delta (f - \frac{1}{4}) + \delta (f + \frac{1}{4}))( \delta (f - \frac{k}{4})) $
