# Auto and Cross Correlation

June 2, 2021

Back to: Fundamentals of Signal Processing

### Auto-Correlation

The auto-correlation function R_{xx}(m) for a real-valued sequence x(n) is defined as:

(1)

If the data sequence x(n) is wide sense stationary, then R_{xx}(n,n+m) simplifies to:

(2)

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

R_{xx}(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, R_{xx}(m) generally decreases as m increases, meaning the data sequence x becomes less statistically related to itself as the delay increases.

Two properties of R_{xx}(m) are its **center spike** and **symmetry**.

#### Center Spike

R_{xx}(m) has a spike at the center m=0 location because:

(3)

(4)

(5)

#### Symmetry

R_{xx}(m) is symmetric about the center m=0 location because:

(6) |
by definition of R_{xx}(m) |

(7) |
by wide sense stationary property |

(8) |
by commutative property |

(9) |
by definition of R_{xx}(m) |

### Cross-Correlation

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

The cross-correlation function R_{yx}(m) for a real-valued sequence x(n) is defined as:

(10)

If the data sequence x(n) is wide sense stationary, then R_{yx}(n,n+m) simplifies to:

(11)

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

R_{yx}(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)?