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If I understand correctly, wikipedia defines the cross-correlation of two signals (see here) as:

$$ x \star y = \sum_{i=-\infty}^{\infty} x^\ast(i)y(n+i) $$

However, in some other sources, as matlab manual (see "More about" here), it is defined as:

$$ R_{xy}=\sum_{i=-\infty}^{\infty} x(n+i)y^\ast(i) $$

are these ones two different definitions?

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Note that $x\star y(n)=R_{yx}(n)=R_{xy}^\ast(-n)$. Convention variations such as these are common in studies of sesquilinear functions, partly because authors can't necessarily agree on whether they want the leftmost or rightmost argument to be the one in which the quantity is antilinear, partly because complex conjugation preserves sesquilinearity while changing which argument exhibits antilinearity.

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