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Cross-correlation

There are more than 1 type of cross-correlation: * Cross-correlation of deterministic signals * Cross-correlation of random vectors * Cross-correlation of stochastic processes

Cross-correlation of deterministic signals

In signal processing, given the continuous functions \(f\) and \(g\), the cross-correlation is defined as

\[ (f\star g)(\tau )\ \triangleq \int _{-\infty }^{\infty }{\overline {f(t)}}g(t+\tau )\,dt \]

where \(\overline {f(t)}\) denotes the complex conjugate of \(f(t)\).

Similarly, for discrete functions, the cross-correlation is defined as

\[ (f\star g)[n]\ \triangleq \sum _{m=-\infty }^{\infty }{\overline {f[m]}}g[m+n] \]

Cross-correlation of random vectors

In probability and statistics, the cross-correlation is used for referring to the correlations between the entries of two random vectors \(\mathbf{X} = (X_{1},\ldots ,X_{m})^{\rm{T}}\) and \(\mathbf{Y} = (Y_{1},\ldots ,Y_{n})^{\rm{T}}\), which forms the cross-correlation matrix of \(\mathbf{X}\) and \(\mathbf{Y}\) that is defined as

\[ \begin{align*} \operatorname{R}_{\mathbf{X} \mathbf{Y}} &\triangleq \ \operatorname{E} [\mathbf{X} \mathbf{Y}^{\rm {T}}] \\ &= {\begin{bmatrix}\operatorname {E} [X_{1}Y_{1}]&\operatorname {E} [X_{1}Y_{2}]&\cdots &\operatorname {E} [X_{1}Y_{n}]\\\\\operatorname {E} [X_{2}Y_{1}]&\operatorname {E} [X_{2}Y_{2}]&\cdots &\operatorname {E} [X_{2}Y_{n}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [X_{m}Y_{1}]&\operatorname {E} [X_{m}Y_{2}]&\cdots &\operatorname {E} [X_{m}Y_{n}]\\\\\end{bmatrix}} \end{align*} \]

Cross-correlation of stochastic processes

In time series analysis and statistics, the cross-correlation of a pair of random process is the correlation between values of the processes at different times.

Given a pair of random processes \((X_t, Y_t)\), if both \(X_t\) and \(Y_t\) have means and variances at time \(t\) for any \(t\), then the cross-correlation between \(t_{1}\) and \(t_{2}\) is defined as

\[ \operatorname{R}_{XY}(t_{1},t_{2}) = \operatorname{E}[X_{t_{1}}{\overline{Y_{t_{2}}}}] \]

Note that this expression may be not defined (if the means and variances are not defined)

If \((X_t, Y_t)\) are jointly wide-sense stationary, then the cross-correlation function can be re-written as

\[ \operatorname{R}_{XY}(\tau) = \operatorname{E} \left[X_{t}{\overline{Y_{t+\tau}}}\right] \]

Reference