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Homework Help: Fourier transform help ?

  1. Dec 31, 2011 #1
    Fourier transform help urgent?

    dear friends,

    sorry to bug you all with things that are lengthy and rather tedious. please help me solve these questions if possible.
    i have tried them but i just cant do the integrals. please help

    FIND THE FOURIER TRANSFORMS OF THE FOLLOWING.

    f(t) = ( 1 - 1/t ) 0 <= t <= 2
    ( 1 + 1/t ) -2 <= t <= 0
    ( 0 ) otherwise

    f(t) = ( e^-t cost ) t >= 0
    ( 0 ) t < 0

    f(t) = e^at -2 <= t <= 2
    0 elsewhere
     
  2. jcsd
  3. Dec 31, 2011 #2

    vela

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    Re: Fourier transform help urgent?

    You can't do this one because 1/0 isn't defined.

    Show us the integrals and your work so far in evaluating them.
     
  4. Jan 1, 2012 #3
    Re: Fourier transform help urgent?

    dear vela,
    thanks you for your prompt response.

    the question

    f(t) = ( 1 - 1/t ) 0 <= t <= 2
    ( 1 + 1/t ) -2 <= t <= 0
    ( 0 ) otherwise

    is doable and has the following answer.

    ( 1 - cos(2ω) ) / ω^2

    next MY ATTEMPT on the first two questions.

    f(t) = ( e^-t cos t ) >= 0
    ( 0 ) t < 0

    0 - ∞∫e^-t cos t e^-jωt dt

    = ∫ e^(-1-jω)tcos t dt * my problem is purely integration. you can use integration by parts but then they just keep repeating without end. SO HOW TO DO?*

    the second question was

    f(t) = e^at -2 <= t <= 2, where a is a positive constant
    0 elsewhere

    applying F TRANSFORM.

    ∫e^(a-jω)t dt

    = [ -1/jω * e^(a-jω)t ] 2 < -> (-2)

    = [ -1/jω * e^2(a-jω)] - [ -1/jω * e^-2(a-jω)]

    = now notice the different SIGNS on the two constant a's, that is the problem

    = [ (e^-2a * e^2jω) - (e^2a * e^-2jω) ] / jω

    = I cannot factor out the constant e^2a and e^-2a to make (sin jω). so what do i do ? ? ? ? ?

    = i am stuck here so please help me ASAP. thank you so much.
    just help me get these three questions cleared.
     
    Last edited: Jan 1, 2012
  5. Jan 1, 2012 #4

    vela

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    Re: Fourier transform help urgent?

    As you've written f(t), it's not because f(0) doesn't exist. 0 can't be part of the domain. But what's more problematic is that the transform of the supposed answer doesn't look anything like your f(t).

    Let ##I=\int_0^\infty \cos t\exp[-t(1+j\omega)]\,dt##. At some point, you end up with something of the form I = (stuff)+(constant)I. Just solve for I.

    You've already made a mistake by this point. The constant out front isn't correct. You need to fix that first.

     
  6. Jan 1, 2012 #5

    Ray Vickson

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    Re: Fourier transform help urgent?

    No matter what I do, I get a divergent result for the FT of your first function (essentially due to the 1/|t| behaviour near t = 0). Direct computation of the integral from -2 to 2 yields a divergent result; indirect computation by combining the integral from -2 to -a and from a to 2, then letting a --> 0+ also yields divergence. So, show us your work on the first example.

    RGV
     
  7. Jan 1, 2012 #6
    Re: Fourier transform help urgent?

    f(t) = ( e^-t cos t ) >= 0
    ( 0 ) t < 0

    0 - ∞∫e^-t cos t e^-jωt dt

    For this question I ended up with [tex]\frac {1+ix}{(1+ix)^2+1}[/tex]

    the second question was

    f(t) = e^at -2 <= t <= 2, where a is a positive constant
    0 elsewhere

    applying F TRANSFORM.

    ∫e^(a-jω)t dt

    = [ -1/jω * e^(a-jω)t ] 2 < -> (-2)

    For this one above, the [tex] \Phi (f(t))= \frac {1}{a-ix} (sinh(2a)cos(-2x)+i*cosh(2a)sin(-2x) [/tex] from the identity, sinh(z)=sinh(x)cos(y)+i cosh(x)sin(y).
     
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