I am looking to take the derivative of the Discrete-Time Fourier Transform with respect to time t. The following is what I have:
\begin{align*} \frac{d}{dt}F(\omega) &=\sum_{t=-\infty}^{\infty} {\frac{d}{dt}f(t)e^{-2\pi j\omega t}}\\ &= \sum_{t=-\infty}^{\infty} {f'(t)e^{-2\pi j\omega t}} - 2 \pi j \omega \sum_{t=-\infty}^{\infty}{f(t)e^{-2\pi j\omega t}}\ \text{(by product rule)}\\ &= F[f'(t)] - 2 \pi j \omega F(\omega)\ \text{(using Fourier identity)}\\ &= j\omega F(\omega) - 2 \pi j \omega F(\omega)\ \text{(using time derivative property)}\\ &= F(\omega) [j\omega - 2 \pi j \omega]\ \text{(factor)} \end{align*}
I don't believe this is proper. From here https://github.com/locuslab/pytorch_fft/issues/4 it's claimed the proper derivative is the Fourier transform of the input gradient, which would be only the first half of my equation under "Using Time Derivative Property". Am I incorrect in my derivation? Is there a reason the 2nd term was dropped?