For a real function $f(x)=A(x)e^{ix}+\overline{A}(x)e^{-ix}$, where $\overline{A}$ represents complex conjugate of $A$. Note that $A(x)$ itself is a complex function $A(x)=A_r(x)+i A_i(x)$. It seems that the envelope of $f(x)$ should be $\pm 2\vert A \vert$, where $\vert \cdot \vert$ repersents the modulus of a complex number. That is, the envelope of $f(x)$ is double amplitude of $A$ if $f$ can be written as $A(x)e^{ix}+c.c.$
Now, I am confusing with the envelope of the real function $g=ff_x$ since it cannot be written in a similar form. Substituting $f(x)$ into $g=ff_x$, one obtains $g=A(A_x+iA)e^{2ix}+A\overline{A}_x+c.c.$, where c.c. repersents the complex conjugate of the preceding terms. How can I express the amplitude of the envelope of $g$ with $A$, $\overline{A}$ and their derivatives?
For the 1st term and its conjugate, the envelope should be $\pm 2 \vert A(A_x+iA)\vert$. But I do not know how to include the 2nd term and its c.c. Can anyone please give me some suggestions? Thank you in advance!