Correlation

The correlation operator for two signals $ x$ and $ y$ in $ {\bf C}^N$ is defined as

$\displaystyle \zbox {(x\star y)_n \isdef \sum_{m=0}^{N-1}\overline{x(m)} y(m+n)}
$

We may interpret the correlation operator as

$\displaystyle (x\star y)_n = \left<\hbox{\sc Shift}_{-n}(y), x\right>
$

which is $ \vert\vert\,x\,\vert\vert ^2=N$ times the coefficient of projection onto $ x$ of $ y$ advanced by $ n$ samples (shifted circularly to the left by $ n$ samples). The time shift $ n$ is called the correlation lag, and $ \overline{x(m)}
y(m+n)$ is called a lagged product. Applications of correlation are discussed in §8.4.


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