I need to compute the Fourier transform o the time derivative of the autocorrelation function (ACF) of a discrete signal with sign changed. Lets call it $Y(\omega)$.
I had some computations problems due to noise and seasonality. In order to solve those issues using existing libraries, I employed methods that gave me the Power Spectral Density (PSD). However, it is not exactly what I need.
I tried to compute $Y(\omega)$ from the PSD. For this I applied the iFFT trying to obtain the ACF, and then: just 1) changed its sign, 2) applied the time derivative 3) applied the FFT.
My questions are the following:
1 - Should de above procedure leads me from the PSD to $Y(\omega)$?
2 - I did not get completely satisfactory result because the imaginary part of $Y(\omega)$ is not close to zero for small $\omega$s. I get something like the second image of this question. What could be the mistake?
Thanks in advance