4.1 Correlation and covariance

Correlation and covariance are techniques for measuring the similarity of one signal to another. For a random process $X(t, \alpha )$ they are defined as follows.

• Auto-correlation function:
  $r_{XX}(t_1, t_2)=(X(t_1, \alpha )X(t_2, \alpha ))=\int \int x_1{x}_{2}f(x_1, x_2)\,d x_1\,d x_2$
  where the expectation is performed over all $\alpha \in $ (i.e. the whole ensemble), and $f(x_1, x_2)$ is the joint pdf when $x_1$ and $x_2$ are samples taken at times $t_1$ and $t_2$ from the same random event $\alpha$ of the random process $X$ .
• Auto-covariance function:
  $c_{XX}(t_1, t_2)=((X(t_1, \alpha )-\langle X(t_1)\rangle )(X(t_2, \alpha )-\langle X(t_2)\rangle ))=\int \int (x_1-\langle X(t_1)\rangle )(x_2-\langle X(t_2)\rangle )f(x_1, x_2)\,d x_1\,d x_2=r_{XX}(t_1, t_2)-2\langle X(t_1)\rangle \langle X(t_2)\rangle +\langle X(t_1)\rangle \langle X(t_2)\rangle =r_{XX}(t_1, t_2)-\langle X(t_1)\rangle \langle X(t_2)\rangle$
  where the same conditions apply as for auto-correlation and the means $\langle X(t_1)\rangle$ and $\langle X(t_2)\rangle$ are taken over all $\alpha \in $ . Covariances are similar to correlations except that the effects of the means are removed.
• Cross-correlation function: If we have two different processes, $X(t, \alpha )$ and $Y(t, \alpha )$ , both arising as a result of the same random event $\alpha$ , then cross-correlation is defined as
  $r_{XY}(t_1, t_2)=(X(t_1, \alpha )Y(t_2, \alpha ))=\int \int x_1{y}_{2}f(x_1, y_2)\,d x_1\,d y_2$
  where $f(x_1, y_2)$ is the joint pdf when $x_1$ and $y_2$ are samples of $X$ and $Y$ taken at times $t_1$ and $t_2$ as a result of the same random event $\alpha$ . Again the expectation is performed over all $\alpha \in $ .
• Cross-covariance function:
  $c_{XY}(t_1, t_2)=((X(t_1, \alpha )-\langle X(t_1)\rangle )(Y(t_2, \alpha )-\langle Y(t_2)\rangle ))=\int \int (x_1-\langle X(t_1)\rangle )(y_2-\langle Y(t_2)\rangle )f(x_1, y_2)\,d x_1\,d y_2=r_{XY}(t_1, t_2)-\langle X(t_1)\rangle \langle Y(t_2)\rangle$