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Calculate the Fourier Transform using theorems

  1. Nov 20, 2014 #1

    grandpa2390

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    1. The problem statement, all variables and given/known data
    Using a theorem (state which theorem you are using and give the formula), Calculate the Fourier Transform of
    1. rect(x)triangle(x)
    2.cos(pi*x)sinc(x)
    3.rect(x)exp(-pi*x^2)
    4.sinc(x)sin(pi*x)
    5. exp(-pi*x^2)cos(pi*x)

    2. Relevant equations
    not sure what theorem to use for the first one.

    3. The attempt at a solution
    Well I am thinkng that since the triangle function is an even function, that I could use the power theorem which states that f(x)g(-x) = F(s)G(s)
    so since triangle(-x)=triangle(x) I can just take the transform of rect(x) and multiply by the transform of triangle(x)

    I should be able to do the same for the rest of them. take the function that is even and make it g(-x)? or does it matter if the function is even?
    maybe it is just saying that given f(x) and g(x) reverse g(x) and multiply them together to get F(s) x G(s)?

    so I got for number 1 : [sinc(pi * s) / (pi * s)]^3
    do I integrate that or is that the answer?
     
    Last edited: Nov 20, 2014
  2. jcsd
  3. Nov 20, 2014 #2

    BiGyElLoWhAt

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    I think (at least for me) some clarification would help.
    What is rect(x)? Is that a square wave function? What is triangle(x)? Is that a triangular wave function? Is rect(x)triangel(x) their products? Not really sure what to do with this at this point.

    Also, I should mention, I've never done fourier transforms with theorems. I didn't even know there were any. I've always justg done it the long way.
     
  4. Nov 20, 2014 #3

    grandpa2390

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    I figured it out. I am supposed to use the convolution theorem.
     
  5. Nov 20, 2014 #4

    grandpa2390

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    yes and yes. turned out I am supposed to do the convolution theorem which states that the fourier transform of (f(x) times g(x)) is equal to the convolution (F(s) convolved with G(s)) :)
     
  6. Nov 20, 2014 #5

    BiGyElLoWhAt

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    Oh. Cool.
     
  7. Nov 20, 2014 #6

    grandpa2390

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    any tips for doing the convolution of sinc(x) and sinc^2(x) ?

    I converted sinc and sinc^2 into sin (pi x) / pi x

    and I multiply them together but I can't do the integral... :(
     
  8. Nov 21, 2014 #7

    BiGyElLoWhAt

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    for 2? so you have cos(pi*x)sin(pi*x)/(pi*x)?
    u-sub u=sin(pi*x)/pi
     
    Last edited: Nov 21, 2014
  9. Nov 21, 2014 #8

    BiGyElLoWhAt

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    or are you using sinc for the triangle wave?
     
  10. Nov 21, 2014 #9

    BiGyElLoWhAt

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    ##\frac{sin^2(x)}{x}+\frac{cos^2(x)}{x} = \frac{1}{x} (sin^2 +cos^2) = \frac{1}{x}(1) = \frac{1}{x}##
    still a u sub, just a different u. If you have sinc^3, that's sinc^2*sinc which is (1/(argument)-cosc^2)sinc. I already feel like I'm giving out too much, so I'm gonna stop here and let you take over.
     
  11. Nov 21, 2014 #10

    BiGyElLoWhAt

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