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Fourier Transform of a Modified Impulse Train

  1. Jul 20, 2007 #1
    I hope this is OK to post here. I thought it would be better here than in the math questions forum, since you are EEs, and probably have more experience dealing with things related to the delta function.



    [tex]\hat{x}(t) = \sum_{k=-\infty}^{\infty}\delta(t-2k).[/tex]

    Now let

    [tex]x(t) = 2\hat{x}(t) + \hat{x}(t-1).[/tex]

    Find the Fourier Transform of [itex]x(t)[/itex].

    Given Solution

    Here is the official solution given by Oppenheim:

    From Table 4.2,

    [tex]\hat{X}(j\omega) = \pi \sum_{k=-\infty}^{\infty}\delta(\omega - \pi k)[/tex]


    [tex]X(j\omega)=\hat{X}(j\omega)[2+e^{-\omega}] = \pi \sum_{k=-\infty}^{\infty}\delta(\omega-\pi k)[2 + (-1)^k].[/tex]

    My Two Questions

    Question 1. When applying the time-shifting property of Fourier series, why did they multiply by [itex]e^{-\omega}[/itex] and not [itex]e^{-j\omega}[/itex]??

    Question 2. How did the [itex]e^{-\omega}[/itex] become [itex](-1)^k[/itex] in the final answer?

    Thank you!
    Last edited: Jul 20, 2007
  2. jcsd
  3. Jul 21, 2007 #2


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    it's a mistake. i don't have the book, but if you represented this correctly, they are wrong and you are right. delaying by one sample multiplies the continuous spectrum by [itex]e^{-j\omega}[/itex] . this will fix your Q2 also.
  4. Jul 21, 2007 #3
    Maybe not! How does [itex]e^{-j\omega}[/itex] become [itex](-1)^k[/itex] in the final answer? :biggrin: I know [itex]e^{-j k \pi} = (-1)^k[/itex] but how could that come from [itex]e^{-j\omega}[/itex] when omega is variable and there is no k? Is the final answer also a typo?

  5. Jul 24, 2007 #4
    Remember, you are multiplying the [itex]e^{-j \omega}[/itex] by a chain of delta functions (in [itex]\omega[/itex]-space). The only thing that matters is where the delta function is nonzero.

    Remember [itex] f(x)\delta(x-a) = f(a)\delta(x-a) [/itex]
  6. Jul 24, 2007 #5
    Of course! Which happens exactly when [itex]\omega = \pi k[/itex]. I should have realized that! :rolleyes:

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