Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical.

I live in an area with $n$ AM radio stations and $m$ FM radio stations.

AM station number $j$ wants to send me the signal $\phi_j(t)$. FM station number $k$ wants to send me the signal $\psi_k(t)$.

Of course if they just sent those signals, my radio would recieve their sum and have no idea how to disentangle them. Therefore, the signals are first encoded. My (possibly ill-informed) understanding is that (modulo a gazillion bells and whistles), AM station $j$ sends the signal $\phi_j(t)\sin(\omega_j t)$. where $\omega_j$ is some constant, and FM station $k$ sends the signal $A_k \sin(\psi_k(t))$ where $A_k$ is some constant.

My radio then receives the signal $$\sum_{j=1}^n\phi_j(t)\sin(\omega_j t)+\sum_{k=1}^mA_k\sin(\psi_k(t))\quad\quad(1)$$

Having received this signal, and knowing the values of the $\omega_j$ and the $A_k$, my radio is then somehow able to compute any one of the signals $\phi_j(t)$ or $\psi_k(t)$ and play it for me on request. (In fact, I'm pretty sure it can recover $\phi_j$ on the basis of $\omega_j$ alone, without knowing the values of the other $\omega$'s.)

It's not obvious to me that this is mathematically possible, though my radio seems to have no problem doing it.

**Question 1 (Pure Mathematics).** For what values of $\omega_1,\ldots,\omega_n,A_1,\ldots,A_m$ is it possible to recover the functions $\phi_1,\ldots,\phi_n,\psi_1,\ldots\psi_m$ from expression (1) alone? And what assumptions are being made on the class of allowable functions from which the $\phi_j$ and $\psi_k$ are drawn?

**Question 2 (Part Engineering, part Pure Mathematics).** If (as is not impossible), AM and/or FM works entirely differently than I think it does, thus rendering Question 1 entirely unmotivated, then how **do** AM and FM work, what is the correct analogue of expression (1), and what is the right answer to the corresponding new version of Question 1?

**Edited to add:** I'm aware that there are all sorts of issues with distorted transmissions, error-correcting, etc. I want to abstract away from all of these and understand the basics.

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