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Homework Help: 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


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    Science Advisor
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    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]
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