In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. 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 also has applications in pattern recognition, single particle analysis, electron tomographic averaging, cryptanalysis, and neurophysiology.
For continuous functions, f and g, the cross-correlation is defined as:
where f * denotes the complex conjugate of f.
Similarly, for discrete functions, the cross-correlation is defined as:
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 unless the signal is a trivial zero signal.
In probability theory and statistics, correlation is always used to include a standardising factor in such a way that correlations have values between −1 and +1, and the term cross-correlation is used for referring to the correlation corr(X, Y) between two random variables X and Y, while the "correlation" of a random vector X is considered to be the correlation matrix (matrix of correlations) between the scalar elements of X.
If and are two independent random variables with probability density functions f and g, respectively, then the probability density of the difference is formally given by the cross-correlation (in the signal-processing sense) ; however this terminology is not used in probability and statistics. In contrast, the convolution (equivalent to the cross-correlation of f(t) and g(−t) ) gives the probability density function of the sum .
Read more about Cross-correlation: Explanation, Properties, Normalized Cross-correlation, Time Series Analysis, Scaled Correlation, Time Delay Analysis