Finding x[n] from Continuous Fourier Transform with Given Properties

[emanuel_hr]

Hello everyone(my first post here), I hope I have posted in the right section...

Homework Statement

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

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

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

find $$\inline x$$ 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)

Homework Equations

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

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

The Attempt at a Solution

It is straightforward to show that if

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

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

By applying the inverse Fourier transform of $$X_{e}(e^{j\omega})$$ we obtain:

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

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.

 

[emanuel_hr]

Any ideas?