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Homework Help: Fourier analysis problem

  1. Jan 31, 2010 #1
    Hello everyone(my first post here), I hope I have posted in the right section...

    1. The problem statement, all variables and given/known data

    Given [tex]x[n][/tex] is a discrete stable(absolutely summable) sequence and its continuous Fourier transform [tex]X(e^{j\omega})[/tex] having the following properties:

    [tex]x[n]=0, \ \ \ \forall n<1[/tex] and

    [tex]Re\{X(e^{j\omega })\}=\frac{3}{2\cos \omega -\frac{5}{2}}, \ \ \ \forall \omega \in \mathbb{R}[/tex]

    find [tex]\inline x[/tex] as good as possible(I don't know how to state this any better, basically one should find x if possible, if not a sequence that resembles x as good as possible)

    2. Relevant equations

    [tex]X(e^{j\omega})=\sum_{n\in \mathbb{Z}}^{ } x[n]e^{-j\omega n}, \ \ \ \forall \omega \in \mathbb{R };[/tex]

    [tex]x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi} X(e^{j\omega})e^{j\omega n}d\omega,\ \ \ \forall n \in \mathbb{Z }.[/tex]

    3. The attempt at a solution
    It is straightforward to show that if

    [tex]x_{e}[n] = \frac{x[n]+\overline{x[-n]}}{2}, \ \ \forall n \in \mathbb{Z}, \ \textup{then} \ X_{e}(e^{j\omega})= Re\{X(e^{j\omega})\}, \ \forall \omega \in \mathbb{R}[/tex]

    Given the fact that [tex]x[n]=0, \forall n<1[/tex] by finding [tex]x_{e}[n] [/tex] we can also find [tex]x[n][/tex]

    By applying the inverse fourier transform of [tex]X_{e}(e^{j\omega})[/tex] we obtain:

    [tex]x_{e}[n] = \frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{3}{2\cos \omega -\frac{5}{2}}\cdot e^{j\omega n}d\omega[/tex]

    And at this point I'm stuck, i have no idea how to evaluate that integral(I've tried the usual tricks but none seem to work). Maybe the approach is not the best one, I don't know.
    Thanks in advance for any advice.
    Last edited: Jan 31, 2010
  2. jcsd
  3. Feb 4, 2010 #2
    Any ideas?
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