### Cross-Correlation

**Definition:**The

*circular cross-correlation*of two signals and in may be defined by

*DFT correlation operator*`' was first defined in §7.2.5. The term ``cross-correlation'' comes from

*statistics*, and what we have defined here is more properly called a ``sample cross-correlation.'' That is, is an

*estimator*

^{8.8}of the true cross-correlation which is an assumed statistical property of the signal itself. This definition of a sample cross-correlation is only valid for

*stationary*stochastic processes,

*e.g.*, ``steady noises'' that sound unchanged over time. The statistics of a stationary stochastic process are by definition

*time invariant*, thereby allowing

*time-averages*to be used for estimating statistics such as cross-correlations. For brevity below, we will typically

*not*include ``sample'' qualifier, because all computational methods discussed will be sample-based methods intended for use on stationary data segments. The DFT of the cross-correlation may be called the

*cross-spectral density*, or ``cross-power spectrum,'' or even simply ``cross-spectrum'':

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Unbiased Cross-Correlation

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Phase Response