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 with two functions

  1. Apr 6, 2012 #1
    [itex]\geq[/itex]1. The problem statement, all variables and given/known data
    Find the Fourier Transform of

    y = exp([itex]^{}-at[/itex])sin([itex]\omega[/itex][itex]_{}0[/itex]t) for t ≥ 0
    and = 0 for t < 0

    Find the amplitudes C([itex]\omega[/itex], S([itex]\omega[/itex]), and energy spectrum [itex]\Phi[/itex]' for [itex]\omega[/itex] > 0 if the term that peaks at negative frequency can be disregarded for pos frequency.

    2. Relevant equations

    Y([itex]\omega[/itex]) = C([itex]\omega[/itex]) - iS([itex]\omega[/itex])
    [itex]\Phi[/itex]' = C^2([itex]\omega[/itex]) + iS^2([itex]\omega[/itex])
    C([itex]\omega[/itex])= [itex]\int[/itex]( y(t)cosw([itex]\omega[/itex]t)dt from -∞ -> ∞
    S([itex]\omega[/itex])= [itex]\int[/itex] ( y(t)sin([itex]\omega[/itex]t)dt from -∞ -> ∞

    3. The attempt at a solution

    I have page after page of trying to simplify the algebra down with no luck. In my text it writes "y" in the equation with no function of ( t ) or ([itex]\omega[/itex]) for most every other equation I see contains either of those. Is there something different about y?

    It looks like I have to take the fourier of two functions exp([itex]^{}-at[/itex]) and sin([itex]\omega[/itex][itex]_{}0[/itex]t) over t = 0 -> [itex]\infty[/itex]

    I try exp(-at)[/itex]sin[itex]\omega[/itex][itex]_{}0[/itex]tcos([itex]\omega[/itex]t) using sin(bx)=(exp(ibx)-exp(-ibx)/2i

    Am I missing something? Are there any algebraic tricks I may be missing? Thanks !




    C([itex]\omega[/itex])= [itex]\int[/itex]( y(t)cosw([itex]\omega[/itex]t)dt from 0 -> ∞ Since y(t) = 0 for negative t

    = [itex]\int[/itex] exp(-at)sin(ω0)t)cos(ω0t)dt
    = [itex]\int[/itex] exp(-at)[ (1/2i) ( exp(iω0t) - exp(-iω0t) ) (1/2) ( exp(iωt) + exp(-iωt))] dt
     
    Last edited: Apr 6, 2012
  2. jcsd
  3. Apr 6, 2012 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In this case, it's much easier to compute [itex]Y(\omega)[/itex] directly and determine [itex]C(\omega), S(\omega)[/itex] as the real and imaginary parts. You will want to use the identity

    [tex] \sin z = \frac{e^{iz} - e^{-iz}}{2i}.[/tex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Fourier Transform with two functions
Loading...