Autocorrelation of a 3D-signal with FFT

Hey,

This is probably more of a mathematical question than Matlab exclusive, but which isn't?

I want to compute the Autocorrelation of a signal, which is a vector varying over time.

The Autocorrelation is described as:

This Autocorrelation can be computed by the dotproduct(A,A) or A * 'A and doing a moving window. But I was wondering if there was a faster method and found the following:

For the 1D case one can calculate the Autocorrelation of A(t) via the FFT as noted on Wikipedia and multiple other sources (" * " is complex conjugation):

Now here is the question: Is it possible to expand this method for a "vector"-signal and how would you implement that?

Thanks for your help!