inverse Discrete-Time-Fourier-Transform of $F(\Omega) = \frac{1}{1+a^2-2a\cos(\Omega)}$

Question

Find Inverse Discrete-Time Fourier Transform of:

$$F(\Omega) = \frac{1}{1+a^2-2a\cos(\Omega)}~~~~~~~~|\Omega| < \pi$$ What's the technique for finding the DTFT of the above function?

If I want to do this the quick way, is there a DTFT table entry for this type of Function?

Answer 1

Lets simplify the expression:

$\cos(\Omega)=\frac{e^{j\Omega}+e^{-j\Omega}}{2} \rightarrow F(\Omega)=\frac{1}{1+a^2-2a\cos(\Omega)}=\frac{1}{1+a^2-ae^{j\Omega}-ae^{-j\Omega}}=\frac{1}{(a-e^{j\Omega})(a-e^{-j\Omega})}$

Now we can use the the convolution properties of Fourier.

$x[n]*y[n]=X(\Omega)Y(\Omega)$

As a result:

$x[n]=\frac{1}{a}a^n u[-n]$ and $y[n]=\frac{1}{a}(\frac{1}{a})^nu[n]$ according to the Fourier table. (Verify my calculation)