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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 $$\sum_{j=1}^n\phi_j(t)\sin(\omega_j(t))+\sum_{k=1}^mA_k\sin(\psi_k(t))\quad\quad(1)$$$\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. 1 # How Does My Radio Work? 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.