Inverse DTFT Problem

Question

Having trouble finding the inverse DTFT of $\ X(\ e^{j \omega}) = \frac{3 - \frac{1}{4} e^{-j\omega}}{1 - \frac{1}{4} e^{-2j\omega}} $

Given the IDFT of $Xe^{j \omega}$ as :

$x(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\ e^{j \omega}) \cdot e^{j \omega n} d\omega$

Tried long division as an option and reduction to partial fractions but not to any good so far.

Not sure on how to proceed.

Answer 1

You can use partial fractions first to expand the $X(e^{j\omega})$ as

$$X(e^{j\omega})=\frac{\tfrac{7}{4}}{1+\frac{1}{2}e^{-j\omega}}+\frac{\tfrac{5}{4}}{1-\frac{1}{2}e^{-j\omega}} $$.

Then taking the inverse DTFT you get $\tfrac{7}{4}(-\frac{1}{2})^{n}u(n)+\tfrac{5}{4}(\frac{1}{2})^{n}u(n)$