Determining the spectral density?

Suppose you have a process $X_{t} = 0.5X_{t-1} + w_{t}$ where $w_{t}$ is $WN(0,\sigma^{2})$. How does one determine the spectral density of the process? Do you first find the ACF of the process and then take the fourier transform of it? So, for this process, would I just calculate $\int_{-0.5}^{0.5}\frac{4}{3}\phi^{|h|}\sigma^{2}e^{-2\pi hi\epsilon}dh$