White gaussian noise analysis deduction

Question

I´m stuck in a deduction analysis of the variance of a gaussian white noise signal in a "integrate-and-dump detector" of a baseband data transmission receiver, where $n(t)$ is white noise with double-sided power spectral density $N_0/2$ [W/Hz]

I can understand all the steps except when they deduce

How do you get to this last expression? Thank you.

Answer 1

They say that $n(t)$ is white noise with a double-sided power spectral density (PSD) of $N_0/2$, i.e., the PSD is given by

$$S_n(f)=\frac{N_0}{2}\tag{1}$$

The auto-correlation function is the inverse Fourier transform of $(1)$, which is

$$R_n(\tau)=E\big\{n(t)n(t+\tau)\big\}=\mathcal{F}^{-1}\big\{S_n(f)\big\}=\frac{N_0}{2}\delta(\tau)\tag{2}$$