Obtain the complex envelope of a two-tone signal

Question

I know it is possible to represent a single-tone signal by its complex envelope: \begin{align} x(t) &= A(t)\cos(2\pi f_c t + \Phi(t))\\ x(t) &= \operatorname{Re}\left\{A(t)\mathrm{e}^{j \Phi(t)}\,\mathrm{e}^{j 2\pi f_c t}\right\}\\ x(t) &= \operatorname{Re}\left\{\tilde{x}(t)\,\mathrm{e}^{2\pi f_c t}\right\} \end{align}

I think it should also be possible to represent a two-tone signal by its complex envelope:

$$x_2(t) = A_1(t)\cos\left(2\pi f_1 t + \Phi_1(t)\right) + A_2(t)\cos\left(2\pi f_2 t + \Phi_2(t)\right)$$

Can somebody show me the necessary steps to obtain the complex envelope of $x_2$ ?

Answer 1

You can represent any bandpass signal by its complex envelope.

You must, however, pick a frequency for which you do that. In your first example, that frequency is $f_c$; in your second example, you need to pick $f_1$, $f_2$ or any other frequency as $f_c$ – and then represent your cosines into products of $e^{j2\pi (f_1-f_c) t}$ and $e^{j2\pi f_c t}$.

Any textbook that introduces the concept of complex envelope (and, usually, complex baseband) should introduce that concept; if yours doesn't, it might be worth having a second textbook for cross-literature.