## Correlation Analysis

Correlation analysis applies only to *stationary* stochastic
processes (§C.1.5).

### Cross-Correlation

**Definition: **The *cross-correlation* of two signals
and
may be defined by

(C.24) |

*I.e.*, it is the

*expected value*(§C.1.6) of the lagged products in random signals and .

### Cross-Power Spectral Density

The DTFT of the cross-correlation is called the *cross-power
spectral density*, or ``cross-spectral density,'' ``cross-power
spectrum,'' or even simply ``cross-spectrum.''

### Autocorrelation

The cross-correlation of a signal with itself gives the
*autocorrelation function* of that signal:

(C.25) |

Note that the autocorrelation function is Hermitian:

When is real, its autocorrelation is

*symmetric*. More specifically, it is

*real and even*.

### Sample Autocorrelation

See §6.4.

### Power Spectral Density

The Fourier transform of the autocorrelation function
is
called the *power spectral density* (PSD), or *power
spectrum*, and may be denoted

When the signal is real, its PSD is real and even, like its autocorrelation function.

### Sample Power Spectral Density

See §6.5.

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White Noise

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Random Variables & Stochastic Processes