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Definition of global correlation

Global correlation coefficient between $a(t)$ and $b(t)$ can be defined as the functional

\gamma = \frac{<a(t),b(t)>}{\sqrt{<a(t),a(t)>\,<b(t),b(t)>}}\;,
\end{displaymath} (8)

where $<x(t),y(t)>$ denotes the dot product between two signals:

<x(t),y(t)> = \int x(t)\,y(t)\,d t\;.

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 $\gamma(t)$ that identifies local changes in the signal similarity.