# Relationship between FRF, FFT, AutoPower, & CrossPower

Given A is an input signal & B is an output signal: I know that the frequency response function (FRF) is defined as the CrossPower of AxB divided by the AutoPower AxA.
Logic tells me I should be able to get the same results using FFT of both signals by taking FFT(B)/FFT(A).
When I do this and compare results between using cross(A,B)/auto(A) and FFT(B)/FFT(A), I get NEARLY the same FRF result... magnitude is the same but the phase is inverted. (i.e. The real parts of both calculations are identical, but the imaginary parts are opposite.)
Any thoughts on why?

• I am not 100% sure what you mean by cross, since a cross-product can only be applied to vectors in 3D. As far as I know the crosspower would be defined as $B\bar{A}$, where $\bar{A}$ is the complex conjugate of $A$ (in frequency domain). Maybe you are looking at the wrong half of the FFT, since it also gives values for negative frequencies, which is equal to the complex conjugate of the positive frequency when the time domain signal is real. – fibonatic Feb 19 '15 at 4:36