I have a signal of the form $f(\omega,x) = g(\omega) e^{i \omega (x +c)},$ for $g: [\omega_1 , \omega_2] \rightarrow C, \, x\in R$ and some constant $c.$ I want to get the function $$ F (\omega,x) = \mathcal{H} ( \Re \{f(\cdot,x)\}) (\omega), $$ where $\mathcal{H}$ denotes the Hilbert transform with respect to frequency $\omega,$ and $\Re$ the real part. I need the function $F$ to be again symbolic on $x$ such that I can differentiate it later.

I am trying to implement it in Matlab using:

L = 1e-2; K = 100; c = 1;
omega = (1/L)*(-K:K-1);

g = exp(-omega.^2/4);

syms x; f(x) = real(g.*exp(1i*omega*(x+c)));

but then the hilbert build-in function is not working. Do you have any idea how to do it?

Thanks in advance!

I have a signal of the form $f(\omega,x) = g(\omega) e^{i \omega (x +c)},$ for $g: [\omega_1 , \omega_2] \rightarrow C, \, x\in R$ and some constant $c.$ I want to get the function $$ F (\omega,x) = \mathcal{H} ( \Re \{f(\cdot,x)\}) (\omega), $$ where $\mathcal{H}$ denotes the Hilbert transform with respect to frequency $\omega,$ and $\Re$ the real part. I need the function $F$ to be again symbolic on $x$ such that I can differentiate it later.

I am trying to implement it in Matlab using:

L = 1e-2; K = 100; c = 1;
omega = (1/L)*(-K:K-1);

g = exp(-omega.^2/4);

syms x; f(x) = real(g.*exp(1i*omega*(x+c)));

but then the hilbert build-in function is not working. Do you have any idea how to do it?