Cross-Correlation

Definition: The circular cross-correlation of two signals $x$ and $y$ in ${\bf C}^N$ may be defined by

$$\zbox {{\hat r}_{xy}(l) \isdef \frac{1}{N}(x\star y)(l) \isdef \frac{1}{N}\sum_{n=0}^{N-1}\overline{x(n)} y(n+l), \; l=0,1,2,\ldots,N-1.}$$

(Note that the ``lag'' $l$ is an integer variable, not the constant $1$.)

The term ``cross-correlation'' comes from statistics, and what we have defined here is more properly called a ``sample cross-correlation.'' That is, ${\hat r}_{xy}(l)$ is an estimator^8.4 of the true cross-correlation $r_{xy}(l)$ 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.

The DFT of the cross-correlation may be called the cross-spectral density, or ``cross-power spectrum,'' or even simply ``cross-spectrum'':

$${\hat R}_{xy}(\omega_k) \isdef \hbox{\sc DFT}_k({\hat r}_{xy}) = \frac{\overline{X(\omega_k)}Y(\omega_k)}{N}$$

The last equality above follows from the correlation theorem (§7.4.7).

Recall that the cross-correlation operator is cyclic (circular) since $n+l$ is interpreted modulo $N$. In practice, we are normally interested in estimating the acyclic cross-correlation between two signals. For this (more realistic) case, we may define instead the unbiased sample cross-correlation

$$\zbox {{\hat r}^u_{xy}(l) \isdef \frac{1}{N-l}\sum_{n=0}^{N-1-l} \overline{x(n)} y(n+l),\; l = 0,1,2,\ldots,L-1}$$

where we choose $L\ll N$ (e.g., $L\approx\sqrt{N}$) in order to have enough lagged products at the highest lag that a reasonably accurate average is obtained. The term ``unbiased'' refers to the fact that the expected value^8.5[32] of ${\hat r}^u_{xy}(l)$ is the true cross-correlation $r_{xy}(l)$ of $x$ and $y$ (assumed to be samples from stationary stochastic processes).