White noise: relation between PSDF and AC function

White noise

White noise is an idealised random noise whose power spectral density is flat (constant) over all frequencies, like white light containing all colours. Thermal and shot noise are nearly white up to very high frequencies ($\sim 10^{12}$ Hz), so the model is accurate for any practical receiver bandwidth. It is usually also assumed Gaussian, zero-mean (AWGN).

Relation between PSDF and AC function

By the Wiener–Khinchin theorem, PSD and autocorrelation are a Fourier transform pair:

$$S(f)=\int_{-\infty}^{\infty}R(\tau)e^{-j2\pi f\tau}d\tau,\qquad R(\tau)=\int_{-\infty}^{\infty}S(f)e^{j2\pi f\tau}df$$

PSD (two-sided):

$$S_W(f)=\frac{N_0}{2}\quad\text{for all } f$$

where $N_0=kT_e$ (W/Hz), $k$ = Boltzmann constant and $T_e$ = equivalent noise temperature.

Autocorrelation (inverse Fourier transform of PSD, using $\mathcal{F}^{-1}\{1\}=\delta(\tau)$):

$$R_W(\tau)=\int_{-\infty}^{\infty}\frac{N_0}{2}e^{j2\pi f\tau}df=\frac{N_0}{2}\,\delta(\tau)$$

   S_W(f)                      R_W(tau)
     |                             ^  (N0/2) delta
 N0/2+-----------------            |
     |                             |
 ----+-----------------> f   ------+------> tau
     0                             0

Meaning: $R_W(\tau)=0$ for every $\tau\ne0$, so any two samples of white noise, however close, are uncorrelated (and independent if Gaussian). The total power $R_W(0)=\int S_W(f)df$ is infinite, which is why white noise is only a model; after any filter with finite bandwidth the power becomes finite.