[Math] Autocorrelation function of Gaussian Random Process

Question

Let $(X_t : t \in \mathbb{R})$ be a continuous time, zero mean Gaussian random process with power spectral density $$S_X(\omega) = \frac{N_0}{2}.$$ I know that the autocorrelation function of $X_t$ should be $$R_X(\tau) = \frac{N_0}{2}\delta(\tau)$$ but I am not sure how to compute this for myself. The power spectral density is defined to be $$S_X(\omega) = \mathcal{F}\{R_X(\tau)\}$$ where $\mathcal{F}$ is the fourier transform but I am not sure how to take the inverse fourier transform to get $R_X(\tau)$. How does one go about such a calculation?

Answer 1

Note that the inverse Fourier transform over $\mathbb{R}$ can be computed as an integral: $$ \mathcal{F}^{-1}[g](x)=\int_{-\infty}^\infty e^{2\pi{i}x\xi}g(\xi)d\xi$$ (Per Wikipedia). In your case, $g(\xi)=N_0/2=\text{const.}$, so you can pull it outside the integral, and you are left with: $$ \mathcal{F}^{-1}[g](x)=(N_0/2)\int_{-\infty}^\infty e^{2\pi{i}x\xi}d\xi.$$ The value of this integral is exactly what you want. See, for example, this question.