Hello to my Math Fellows,
Problem: I am looking for a way to calculate w-derivative of Fourier transform, i.e. $\frac{\mathrm{d}}{\mathrm{d}\omega} F\{x(t)\}(\omega)$, in terms of regular Fourier transform, $X(\omega)=F\{x(t)\}(\omega)$.
Definition Based Solution (not good enough): from the Fourier Transform Definition, I can find that the $\omega$-derivative of Fourier transform $x(t)$ is the Fourier transform of $t\cdot x(t)$ multiplied by $-j$: $$ \frac{\mathrm{d}}{\mathrm{d}\omega} F\{x(t)\}(\omega) = \frac{\mathrm{d}}{\mathrm{d}\omega}X(\omega )=-j F\{t x(t)\}(\omega) $$
Question: Taking into account the differentiation and duality properties of Fourier transform: Fourier Transform Properties
is it possible to express the derivative $\frac{\mathrm{d}}{\mathrm{d}\omega} F\{x(t)\}(\omega)$ in frequency domain using terms of $X(\omega)$ ???
Many Thanks, Desperate Engineer.