1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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

    User Avatar
    Science Advisor
    Homework Helper

    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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Fourier transform help ?
  1. Fourier transform help (Replies: 12)

Loading...