Explanation
As an example, consider two real valued functions and differing only by an unknown shift along the x-axis. One can use the cross-correlation to find how much must be shifted along the x-axis to make it identical to . The formula essentially slides the function along the x-axis, calculating the integral of their product at each position. When the functions match, the value of is maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive.
With complex-valued functions and, taking the conjugate of ensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral.
In econometrics, lagged cross-correlation is sometimes referred to as cross-autocorrelation
Read more about this topic: Cross-correlation
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