Two signals have arrived with different phases. The sum signal is given by
$$x(t) = A\cos(2\pi f_ct) + B\cos(2\pi f_ct + \theta)$$
What is the complex envelope of $x(t)$? Need advice on how to get started.
What is the analytic signal $x_+(t)$? For this, can I not apply the $x(t) + \mathscr{H}\big\{x(t)\big\}$ to both $A\cos(2\pi f_ct)$ and $B\cos(2\pi f_ct + \theta)$, and just add the result?
Using the fact that
$$\cos(\omega_0t)=\textrm{Re}\left\{e^{j\omega_0t}\right\}$$
it is straightforward to write down the analytic signal of $x(t)$.
The analytic signal has the form
$$x_c(t)e^{j\omega_0t}$$
and you have
$$x(t)=\textrm{Re}\left\{x_c(t)e^{j\omega_0t}\right\}$$
where $x_c(t)$ is the complex envelope. Judging from the comments under the question, you know everything you need in order to obtain the analytic signal as well as the complex envelope.
Perhaps the third row of Table 1 at:
https://www.dsprelated.com/showarticle/635.php
would be helpful to you.
