Cross-correlation

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.

For continuous functions f and g, the cross-correlation is defined as:

where ${\displaystyle f^{*}}$ denotes the complex conjugate of ${\displaystyle f}$, and ${\displaystyle \tau }$ is the displacement, also known as lag, although a positive value of ${\displaystyle \tau }$ actually means that ${\displaystyle g(t+\tau )}$ leads ${\displaystyle g(t)}$.

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