I am trying to obtain spectra of derivative of a two-point correlation. What I know is that spectra of two-point correlations are computed as follows
$\phi_{a{b}} = FFT(a) \times conj(FFT(b))$
where $\phi_{a{b}}$ is spectra of correlation of $a$ and $b$, and $a$ and $b$ are two signals collected at two different locations.
I also know that FFT of derivative of a function is
$FFT(\partial f/\partial x) = i \times \kappa \times FFT( f)$
where $\kappa$ is wavenumber and $f$ is a function.
My question is that is the following statement correct or not.
$\phi_{\partial a{b}/\partial x} = i \times \kappa \times FFT(a) \times conj(FFT(b))$