How to calculate correlation between two signals?
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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:[1][2][3]
{\displaystyle (f\star g)(\tau )\ {\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }f^{*}(t)\ g(t+\tau )\,dt,}
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 f(t)}.
Similarly, for discrete functions, the cross-correlation is defined as:[4][5]
{\displaystyle (f\star g)[n]\ {\stackrel {\mathrm {def} }{=}}\sum _{m=-\infty }^{\infty }f^{*}[m]\ g[m+n].}

Visual comparison of convolution, cross-correlation and autocorrelation.
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 is used for referring to the correlations between the entries of two random vectors X and Y, while the correlationsof a random vector X are considered to be the correlations between the entries of X itself, those forming the correlation matrix (matrix of correlations) of X. If each of X and Y is a scalar random variable which is realized repeatedly in temporal sequence (a time series), then the correlations of the various temporal instances of X are known as autocorrelations of X, and the cross-correlations of X with Y across time are temporal cross-correlations.
Furthermore, 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 fand 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 f(t) and g(−t) ) gives the probability density function of the sum {\displaystyle X+Y}.
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:[1][2][3]
{\displaystyle (f\star g)(\tau )\ {\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }f^{*}(t)\ g(t+\tau )\,dt,}
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 f(t)}.
Similarly, for discrete functions, the cross-correlation is defined as:[4][5]
{\displaystyle (f\star g)[n]\ {\stackrel {\mathrm {def} }{=}}\sum _{m=-\infty }^{\infty }f^{*}[m]\ g[m+n].}

Visual comparison of convolution, cross-correlation and autocorrelation.
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 is used for referring to the correlations between the entries of two random vectors X and Y, while the correlationsof a random vector X are considered to be the correlations between the entries of X itself, those forming the correlation matrix (matrix of correlations) of X. If each of X and Y is a scalar random variable which is realized repeatedly in temporal sequence (a time series), then the correlations of the various temporal instances of X are known as autocorrelations of X, and the cross-correlations of X with Y across time are temporal cross-correlations.
Furthermore, 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 fand 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 f(t) and g(−t) ) gives the probability density function of the sum {\displaystyle X+Y}.
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