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You would use the complex result which shows the magnitude and phase of the cross correlation (both are important depending on the use of the result). For example, if this was used to determine carrier phase in a receiver where there is an offset in frequency between the transmitted signal and the first estimate of its carrier in the receiver, then by ...


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Perceval's theorem is satisfied with you apply scale factor of $1/\sqrt(N)$ for both the forward and inverse transform. I.e. If $$X(k) = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} x(n) e^{-j2\pi\frac{kn}{N}}$$ then $$x(n) = \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1} x(n) e^{j2\pi\frac{kn}{N}}$$ and $$\sum_{n=0}^{N-1}|x(n)|^2 = \sum_{k=0}^{N-1}|X(k)|^2 $$


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