Autocorrelation properties; AC of white noise

Properties of the autocorrelation function

The autocorrelation (AC) function measures how similar a signal is to a time-shifted copy of itself. For a power signal or a wide-sense stationary (WSS) random process $X(t)$:

$$R_X(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}x(t)\,x(t+\tau)\,dt \quad\text{or}\quad R_X(\tau)=E[X(t)X(t+\tau)]$$

For an energy signal, $R_g(\tau)=\int_{-\infty}^{\infty}g(t)g(t+\tau)\,dt$.

Important properties (any three answer the question):

1. Even symmetry: $R_X(-\tau)=R_X(\tau)$.
2. Value at origin = mean-square value (power): $R_X(0)=E[X^2(t)]=P$ (for an energy signal, $R_g(0)=E$).
3. Maximum at origin: $\lvert R_X(\tau)\rvert\le R_X(0)$ for all $\tau$.
4. Fourier pair with PSD (Wiener–Khinchin): $S_X(f)=\int_{-\infty}^{\infty}R_X(\tau)e^{-j2\pi f\tau}d\tau$ and $R_X(\tau)=\int_{-\infty}^{\infty}S_X(f)e^{j2\pi f\tau}df$.
5. Periodicity: if $x(t)$ is periodic with period $T_0$, $R_X(\tau)$ is also periodic with $T_0$.
6. DC component: if $X(t)$ has mean $m$ and no periodic part, $R_X(\tau)\to m^2$ as $\tau\to\infty$.

Autocorrelation function of 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).

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.