In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.

In probability and statistics, the term cross-correlations refers to the correlations between the entries of two random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$, while the correlations of a random vector ${\displaystyle \mathbf {X} }$ are the correlations between the entries of ${\displaystyle \mathbf {X} }$ itself, those forming the correlation matrix of ${\displaystyle \mathbf {X} }$. If each of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of ${\displaystyle \mathbf {X} }$ are known as autocorrelations of ${\displaystyle \mathbf {X} }$, and the cross-correlations of ${\displaystyle \mathbf {X} }$ with ${\displaystyle \mathbf {Y} }$ across time are temporal cross-correlations. In probability and statistics, the definition of correlation always includes a standardising factor in such a way that correlations have values between −1 and +1.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are two independent random variables with probability density functions ${\displaystyle f}$ and ${\displaystyle g}$, respectively, then the probability density of the difference ${\displaystyle Y-X}$ is formally given by the cross-correlation (in the signal-processing sense) ${\displaystyle f\star g}$; however, this terminology is not used in probability and statistics. In contrast, the convolution ${\displaystyle f*g}$ (equivalent to the cross-correlation of ${\displaystyle {\overline {f(-t)}}}$ and ${\displaystyle g(t)}$) gives the probability density function of the sum ${\displaystyle X+Y}$.

For continuous functions ${\displaystyle f}$ and ${\displaystyle g}$, the cross-correlation is defined as:$${\displaystyle (f\star g)(\tau )\ \triangleq \int _{-\infty }^{\infty }{\overline {f(t)}}g(t+\tau )\,dt}$$which is equivalent to$${\displaystyle (f\star g)(\tau )\ \triangleq \int _{-\infty }^{\infty }{\overline {f(t-\tau )}}g(t)\,dt}$$where ${\displaystyle {\overline {f(t)}}}$ denotes the complex conjugate of ${\displaystyle f(t)}$, and ${\displaystyle \tau }$ is called displacement or lag.

For highly-correlated ${\displaystyle f}$ and ${\displaystyle g}$ which have a maximum cross-correlation at a particular ${\displaystyle \tau }$, a feature in ${\displaystyle f}$ at ${\displaystyle t}$ also occurs later in ${\displaystyle g}$ at ${\displaystyle t+\tau }$, hence ${\displaystyle g}$ could be described to lag ${\displaystyle f}$ by ${\displaystyle \tau }$.

If ${\displaystyle f}$ and ${\displaystyle g}$ are both continuous periodic functions of period ${\displaystyle T}$, the integration from ${\displaystyle -\infty }$ to ${\displaystyle \infty }$ is replaced by integration over any interval ${\displaystyle [t_{0},t_{0}+T]}$ of length ${\displaystyle T}$:$${\displaystyle (f\star g)(\tau )\ \triangleq \int _{t_{0}}^{t_{0}+T}{\overline {f(t)}}g(t+\tau )\,dt}$$which is equivalent to$${\displaystyle (f\star g)(\tau )\ \triangleq \int _{t_{0}}^{t_{0}+T}{\overline {f(t-\tau )}}g(t)\,dt}$$Similarly, for discrete functions, the cross-correlation is defined as:$${\displaystyle (f\star g)[n]\ \triangleq \sum _{m=-\infty }^{\infty }{\overline {f[m]}}g[m+n]}$$which is equivalent to:$${\displaystyle (f\star g)[n]\ \triangleq \sum _{m=-\infty }^{\infty }{\overline {f[m-n]}}g[m]}$$For finite discrete functions ${\displaystyle f,g\in \mathbb {C} ^{N}}$, the (circular) cross-correlation is defined as:$${\displaystyle (f\star g)[n]\ \triangleq \sum _{m=0}^{N-1}{\overline {f[m]}}g[(m+n)_{{\text{mod}}~N}]}$$which is equivalent to:$${\displaystyle (f\star g)[n]\ \triangleq \sum _{m=0}^{N-1}{\overline {f[(m-n)_{{\text{mod}}~N}]}}g[m]}$$For finite discrete functions ${\displaystyle f\in \mathbb {C} ^{N}}$, ${\displaystyle g\in \mathbb {C} ^{M}}$, the kernel cross-correlation is defined as:$${\displaystyle (f\star g)[n]\ \triangleq \sum _{m=0}^{N-1}{\overline {f[m]}}K_{g}[(m+n)_{{\text{mod}}~N}]}$$where ${\displaystyle K_{g}=[k(g,T_{0}(g)),k(g,T_{1}(g)),\dots ,k(g,T_{N-1}(g))]}$ is a vector of kernel functions ${\displaystyle k(\cdot ,\cdot )\colon \mathbb {C} ^{M}\times \mathbb {C} ^{M}\to \mathbb {R} }$ and ${\displaystyle T_{i}(\cdot )\colon \mathbb {C} ^{M}\to \mathbb {C} ^{M}}$ is an affine transform.

Specifically, ${\displaystyle T_{i}(\cdot )}$ can be circular translation transform, rotation transform, or scale transform, etc. The kernel cross-correlation extends cross-correlation from linear space to kernel space. Cross-correlation is equivariant to translation; kernel cross-correlation is equivariant to any affine transforms, including translation, rotation, and scale, etc.

As an example, consider two real valued functions ${\displaystyle f}$ and ${\displaystyle g}$ differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much ${\displaystyle g}$ must be shifted along the x-axis to make it identical to ${\displaystyle f}$. The formula essentially slides the ${\displaystyle g}$ function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of ${\displaystyle (f\star g)}$ is maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive.