Short note: Autocorrelation function and properties

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

It tells how fast a signal changes: a slowly varying signal stays correlated for large $\tau$, a rapidly varying one loses correlation quickly. It is used to find the PSD, to detect periodic signals buried in noise, and in correlation receivers and radar.

Properties:

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 $\tau\to\infty$.

Example: white noise has $R(\tau)=\frac{N_0}{2}\delta(\tau)$; samples at different times are uncorrelated.