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Auto and Cross Correlation

June 2, 2021

Back to: Fundamentals of Signal Processing

Auto-Correlation

The auto-correlation function Rxx(m) for a real-valued sequence x(n) is defined as:

(1)   \begin{equation*} R_{xx}(n,m)=E[x(n+m)x(n)] \end{equation*}

If the data sequence x(n) is wide sense stationary, then Rxx(n,n+m) simplifies to:

(2)   \begin{equation*} R_{xx}(m)=E[x(m)x(0)] \end{equation*}

The statistical properties do not depend on absolute time n but on time difference m.

Rxx(m) displays how statistically related a data sequence is to an m-sample delayed version of itself. For example, if you look at sample x(3), how strong is its statistical relationship with x(4)? How about x(5) or x(6)?

For many real-world systems, Rxx(m) generally decreases as m increases, meaning the data sequence x becomes less statistically related to itself as the delay increases.

Two properties of Rxx(m) are its center spike and symmetry.

Center Spike

Rxx(m) has a spike at the center m=0 location because:

(3)   \begin{equation*} R_{xx}(0)=E[x(0)x^{\star}(0)] \end{equation*}

(4)   \begin{equation*} =E|x(0)|^{2} \end{equation*}

(5)   \begin{equation*} =\text{variance of }x(0) \end{equation*}

Symmetry

Rxx(m) is symmetric about the center m=0 location because:

(6)   \begin{equation*} R_{xx}(-m)=E[x(-m)x(0)] \end{equation*}

 by definition of Rxx(m)

(7)   \begin{equation*} =E[x(-m+m)x(0+m)] \end{equation*}

 by wide sense stationary property

(8)   \begin{equation*} =E[x(m)x(0)] \end{equation*}

 by commutative property

(9)   \begin{equation*} =R_{xx}(m) \end{equation*}

 by definition of Rxx(m)

Cross-Correlation

The cross-correlation function Ryx(m) is similar to the autocorrelation Rxx(m), but two sequences x and y are compared rather than solely x.

The cross-correlation function Ryx(m) for a real-valued sequence x(n) is defined as:

(10)   \begin{equation*} R_{yx}(n,m)=E[x(n+m)y(n)] \end{equation*}

If the data sequence x(n) is wide sense stationary, then Ryx(n,n+m) simplifies to:

(11)   \begin{equation*} R_{yx}(m)=E[x(m)y(0)] \end{equation*}

The statistical properties do not depend on absolute time n, only on the time difference m.

Ryx(m) displays how statistically related a data sequence x is to an m-sample delayed version of a data sequence y. For example, if you look at sample x(3), how strong is its statistical relationship with y(4)? How about y(6)?