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Global correlation coefficient between and can be
defined as the functional
where denotes the dot product between two signals:
According to definition 8, the correlation
coefficient of two identical signals is equal to one, and the
correlation of two signals with opposite polarity is minus one. In all
the other cases, the correlation will be less then one in magnitude
thanks to the Cauchy-Schwartz inequality.
The global measure 8 is inconvenient because it
supplies only one number for the whole signal. The goal of local
analysis is to turn the functional into an operator and to produce
local correlation as a variable function that identifies
local changes in the signal similarity.