Fourier transform of Autocorrelation function, what am I missing here? (Power spectrum)
That question was posted quite some time ago! Have you tried the discrete time Fourier transform of $R_{XX}[m]$, instead of the Fourier transform? I believe this is what was meant, there (the text talks about a "sequence"). 

An "easy" way to solve this, according to me, is to compute the Z-transform, then identify $S_{XX}(\omega) = Z[R_{XX}](z=e^{j\omega})$. The Z-transform is given as:
\begin{align}
Z[R_{XX}](z) &amp;= \sum_{m=-\infty}^{\infty} R_{XX}[m] z^{-m} \\
   &amp;= \sum_m e^{-\alpha |m|} z^{-m} \\
   &amp;= \frac{1 - e^{-2\alpha}}{(1-e^{-\alpha}z^{-1})(1 - e^{-\alpha}z)} 
\end{align}

Hope that still helps!