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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 $$\sum_{j=1}^n\phi_j(t)\sin(\omega_j t)+\sum_{k=1}^mA_k\sin(\psi_k(t))\quad\quad(1)$$

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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Follow up question: "... and can you guys fix it?" –  Vidit Nanda Jun 14 '12 at 4:42
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Have you tried turning it off and on again? –  François G. Dorais Jun 14 '12 at 4:58
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...or just take a very simple circuit diagram for a crystal diode radio and write down the ODE for the response in the earphones from the earphones with the given signal. This should extract out frequencies close to the one to which you are tuned and perform the (amplitude) demodulation. –  George Lowther Jun 14 '12 at 8:57
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@David Roberts' points out that you did not quite express FM correctly – you did not incorporate the base frequency, whereas I care whether I'm listening to 88.5 or 89.1. According to Wikipedia, FM radio transmits the data $\psi_k(t)$ via an encoding of the form $A_k\sin(\lambda_k t + \Psi_k(t))$, where $\lambda_k$ is a frequency much higher (three orders of magnitude, or so) than any of the Fourier modes in $\psi_k(t)$, and $\Psi_k(t)$ is (a multiple of – the units would be off otherwise) an antiderivative of $\psi_k(t)$. –  Theo Johnson-Freyd Jun 15 '12 at 2:35
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Also, again according to Wikipedia, AM is more accurately described via a signal of the form $\Phi_j(t) \sin(\omega_j t)$, where $\Phi_j(t) = A_j(1 + M_j\phi_j(t))$. Here $A$ and $M$ are unit-full quantities, usually set so that $1 + M\phi_j(t)$ is always positive (otherwise the signal is distorted), but has minimum value $1/c$ for some single-digit number $c$. –  Theo Johnson-Freyd Jun 15 '12 at 2:49
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4 Answers 4

Modulation is used to reduce antenna height, noise distortion, to avoid interference...

The low frequency signal (e.g., human voice) is superimposed to a high frequency signal (the carrier) and transmited. Every radio station have its own high frequency (carrier frequency) of transmission.

When you tune your radio (choosing the carrier frequency of the radio station), you're also indicating the electronic circuit to be used to demodulate (extract information from the carrier) AM or FM signals to recover the (low frequency) human audible signal.

See any book from B.P. Lathi (e.g. Modern Digital and Analog Communication System) or from Oppenheim (e.g Signals and Systems). These are classical books used in Electric Enginnering courses.

According to Lathi, the AM signal can be demodulated coherently (demodulation synchronous) or noncoherently (demodulation assynchronous). In practice, two demodulation noncoherent methods are used: (1) rectifier detection and (2) envelope detection.

(1) Rectifier Detector: If an AM signal is applied to a diode and resistor circuit, the negative part of the AM wave will be supressed. The rectified output, $v_r(t)$, is:

$$v_r(t) = [A+m(t)]\cos \omega_ct [1/2 + 2/\pi(\cos\omega_ct- 1/3 \cos3\omega_ct + \ldots)] = 1/\pi[A + m(t)] + hft$$

where $hft$ are high frequency terms, $m(t) = B \cos\omega_mt$ is the information (low frequency signal), $\omega_m$, information frequency, $\omega_c$, carrier frequency, $A$ and $B$, amplitude of carrier and low frequency signals, respectively.

When $v_r(t)$ is applied to a low-pass filter of cutoff $B$ Hz, the output is $[A +m(t)]/\pi$, and all the other terms in $v_r$ of frequencies higher than $B$ Hz are supressed. The dc term $a/\pi$ may be blocked by a capacitor to give the desired output $m(t)/\pi$.

(2) Envelope Detector: the output of the detector follows the envelop of the modulated signal. The circuit is a diode followed by a RC-filter. Mathematical details in Lathi.

The information (low frequency signal) in FM resides in the instantaneous frequency $\omega_i = \omega_c + k_f m(t)$, $k_f$ is a modulation index. A frequency-selective network with a transfer function $$|H(\omega)| = a\omega + b$$ over the FM band would yield an output proportional to the instantaneous frequency, $\omega_i$. There are several possible networks with such characteristics, the simplest is an ideal differentiator with transfer function $j\omega$. The mathematical details can be seen in Lathi.

There is a tutorial here or here.

BTW, this is a very beautiful (real) application of Fourier Theory.

ADDED:

In relation to your question, if we have

$$\phi_{AM}(t) = A \cos\omega_ct + f(t)\cos\omega_ct$$ representing the AM signal and $$\phi_{FM}(t) = A \cos\omega_ct - A k_f g(t)\sin\omega_ct$$ representing the FM signal, then $$\phi_{AM}(t) \leftrightarrow \frac{1}{2}[F(\omega + \omega_c) + F(\omega - \omega_c)] + \pi A [\delta(\omega + \omega_c) + \delta(\omega - \omega_c)]$$ and $$\phi_{FM}(t) \leftrightarrow \pi A [\delta(\omega + \omega_c) + \delta(\omega - \omega_c)] + j\frac{Ak_f}{2}[G(\omega - \omega_c) - G(\omega + \omega_c)] $$

Considering $$v(t) = \phi_{FM}(t) + \phi_{AM}(t)$$ we have the Fourier transform:

$$\frac{1}{2}[F(\omega + \omega_c) + F(\omega - \omega_c)] + 2 \pi A [\delta(\omega + \omega_c) + \delta(\omega - \omega_c)] + j\frac{Ak_f}{2}[G(\omega - \omega_c) - G(\omega + \omega_c)]$$

The next step is to adapt your equation to these equations. Then, procedures (1) or (2) should be used to recover the low frequency signal in the receiver side.

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Contrary to your second sentence, the low-frequency signal is not superimposed on the high-frequency signal. Superimposing means addition. Modulation maintains the signal within a small neighborhood of the carrier in frequency space. –  S. Carnahan Jun 20 '12 at 8:47
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Entirely engineering: I believe "different frequency bands" means that the Fourier transform of $\sin (\phi_k(t))$ is small in some interval containing the $\omega_j$. Note that I am completely unqualified to speak about the engineering here.

Both engineering and mathematics: I believe that the radio recovers, approximately, the functions $\phi_j$ by convolving the signal with $e^{(-\beta+\omega_j i)t}$.

Mathematics: One cannot recover the signal exactly without strong assumptions on the $\phi_j$. Take the signal $\sin(\omega_1 t )\sin(\omega_2 t)$. This could represent $\phi_1=\sin(\omega_2 t)$, $\phi_2=0$, or $\phi_1=0$, $\phi_2=\sin(\omega_1 t)$ or any linear combination.

Assume that the $\phi_i$ have Fourier transforms supported on the interval $[-\delta,\delta]$ and the $\omega_i$ are separated by gaps of size at least $2\delta$. Assume no FM stations. Then we can recover each AM station by taking the Fourier transform and setting everything outside $[\omega_i -\delta,\omega_i+\delta]$ to $0$.

I don't know the right solution for FM.

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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: $X(f)\approx0$ when $|f \pm f_0| < B$. 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.

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Your expression for FM transmissions is not quite right - it's missing the radio frequency! The simple model that captures the essentials of what FM station $k$ is sending you is the function $$B_k\sin\left((\omega_k+\gamma_k\psi_k(t))t\right),$$ where $\gamma_k\psi_k$ never gets close to $\omega_k$ (so you're only modulating the frequency and not completely disrupting it). If the interesting signal $\psi_k$ is a pure note at frequency $\omega$, then the spectrum of the actual radio signal can be found in terms of Bessel functions and consists of sidebands separated from the carrier by spacing $\omega$. (The number of sidebands is controlled by how large $\gamma_k$ is.)

The real radio signal your device is getting, then, is $$F(t)=\sum_{j=1}^n\phi_j(t)\sin(\omega_j t)+\sum_{k=1}^m B_k\sin\left((\omega_k+\gamma_k\psi_k(t))t\right).$$ Because of the modulation, none of the stations' radio signals are single peaks; instead they are spread over a bandwidth roughly given by the frequency content of the audio signals they encode. (For comparison, human hearing can detect 16 Hz to roughly 20,000 Hz, AM frequencies are medium-wave radio at 520 kHz to 1,610 kHz, and FM stations run at 87.5 to 108 MHz. Thus in reality the peaks are quite narrow!)

To detect a signal, your device uses a combination of antennas, loops of wire, parallel plates, and the like, which contrive to give to the decoding device (the one that takes a radio signal and gives you an audio one) a voltage $f$ that's controlled by a damped harmonic oscillator equation of the form $$\frac{d^2}{dt^2}f-2\gamma\frac{d}{dt}f+\omega_0^2f=F,$$ where the resonance frequency $\omega_0$ is controlled by a knob on the device. The spectral response of this dynamical system is routinely evaluated in college ODE courses, and comes out as a Lorentzian bell-shaped curve centred at $\omega_0$ and of width $\gamma$. Choose $\gamma$ to match the spectral width of the typical radio station, and you've got a fantastic filter!

EDIT: After doing some looking up, I find that the $\psi_k$ here is not exactly the audio signal the station is trying to encode, but rather something like its average over the interval $[0,t]$, so it is equivalent to it up to simple mathematical operations performed at the decoder.

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