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.

  • $\begingroup$ Have you tried using $\frac{1}{1-x}=1+x+x^2+\cdots$ on the problem? $\endgroup$ – Dilip Sarwate May 27 '15 at 2:01

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 $\tfrac{7}{4}(-\frac{1}{2})^{n}u(n)+\tfrac{5}{4}(\frac{1}{2})^{n}u(n)$

|improve this answer|||||
  • $\begingroup$ Thx! :) didn't see that coming. :D $\endgroup$ – mrdoubtful May 28 '15 at 17:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for?Browse other questions tagged or ask your own question.