Correlation

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

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

We may interpret the correlation operator as

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

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.