TITLE: How Does My Radio Work?
QUESTION [65 upvotes]: 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.

REPLY [11 votes]: I've been trained as an engineer, and I can tell you that engineers have a somewhat simplified view of the matter. (But, not only a simplified view, of course.) The other answers fill in some detail, but I think a higher-level view is useful.
There is no such thing as perfect recovery of the transmitted signal. The best you can hope is to bound the error.
For most modulation techniques the basic idea is that the spectrum $X$ of a signal $x$ is nearly 0 outside a narrow band: if $|f-f_0|\gt B$ then $X(f)\approx0$. Both AM and FM are essentially means of transforming a spectrum centered around $0$ into one centered around $f_0$. So, in order to recover a signal, the main concern is to make sure that the spectrums $X_1$, $X_2$, …, $X_n$ do not overlap. This is achieved in a rather uninteresting way: regulation. Then you can extract one signal by shifting $f_0$ to $0$ (convolution with a Dirac impulse in frequency domain, meaning multiplication with a harmonic signal in the time domain), and then applying a low pass filter (multiplication with a rectangular function in the frequency domain, meaning convolution with a sinc in the time domain). See also this related question.
There are broad-spectrum modulation techniques, which are used for example in fourth generation mobile-phone networks, that do not rely on the assumption that the signal covers a narrow band. The two main ones are frequency hoping (use some narrow-band modulation technique but change $f_0$ often in some pseudorandom sequence) and spread spectrum (multiply the signal with a pseudorandom sequence before using a narrow-band modulation technique). The signals obtained thru such methods have a wide band, but are bounded $|X(f)| < c$ for some $c$ for all $f$. This way they behave as background noise as far as demodulating any narrow-band signal is concerned.