Autocorrelation Function

The Autocorrelation $R_x(T)$$R_x(T)$ of a signal $x(t)$$x(t)$ is defined as,

Periodic Signals

In the special case where $x(t)$$x(t)$ is a periodic signal, the Autocorrelation is defined as,

Wiener-Khinchin Theorem

The inverse Fourier transform of the Power Spectral Density of signal $x(t)$$x(t)$ is equal to the Autocorrelation of signal $x(t)$$x(t)$:

Linear Time-Invariant System

For a linear time-invariant system $y(t)$$y(t)$ with input $x(t)$$x(t)$ and impulse response h(t),

Properties

Symmetry

• $R_x^{\ast }(\tau )=R_x(-\tau )$$R_x^*(\tau)=R_x(-\tau)$

Mean-Squared Value

• $R_x(\tau ){|}_{\tau =0}=P_x$$R_x(\tau)|_{\tau=0} = P_x$

Periodicity

• If $x(t)$$x(t)$ is periodic with time period $T$$T$, then $R_x(\tau +T)=R_x(\tau )$$R_x(\tau+T) = R_x(\tau)$

Maximum Value

• $|R_x(\tau )|\le R_x(0)$$|R_x(\tau)| \le R_x(0)$

Crosscorrelation Function

The Crosscorrelation Function $R_{xy}(\tau )$$R_{xy}(\tau)$ of two signals $x(t)$$x(t)$ and $y(t)$$y(t)$ is defined as,

The Crosscorrelation Function measures the similarity between two signals. $R_{xy}(\tau )=0$$R_{xy}(\tau) = 0$ implies $x(t)$$x(t)$ and $y(t)$$y(t)$ are uncorrelated.