How Does My Radio Work? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:51:43Z http://mathoverflow.net/feeds/question/99549 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99549/how-does-my-radio-work How Does My Radio Work? Steven Landsburg 2012-06-14T04:32:29Z 2012-06-26T10:05:01Z <p>Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical.</p> <p>I live in an area with $n$ AM radio stations and $m$ FM radio stations. </p> <p>AM station number $j$ wants to send me the signal $\phi_j(t)$. FM station number $k$ wants to send me the signal $\psi_k(t)$.</p> <p>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.</p> <p>My radio then receives the signal $$\sum_{j=1}^n\phi_j(t)\sin(\omega_j t)+\sum_{k=1}^mA_k\sin(\psi_k(t))\quad\quad(1)$$</p> <p>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.)</p> <p>It's not obvious to me that this is mathematically possible, though my radio seems to have no problem doing it.</p> <p><b>Question 1 (Pure Mathematics).</b> 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?</p> <p><b>Question 2 (Part Engineering, part Pure Mathematics).</b> 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 <b>do</b> 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?</p> <p><b>Edited to add:</b> 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.</p> http://mathoverflow.net/questions/99549/how-does-my-radio-work/99553#99553 Answer by Will Sawin for How Does My Radio Work? Will Sawin 2012-06-14T05:40:46Z 2012-06-14T05:40:46Z <p>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.</p> <p>Both engineering and mathematics: I believe that the radio recovers, approximately, the functions $\phi_j$ by convolving the signal with $e^{(-\beta+\omega_j i)t}$.</p> <p>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.</p> <p>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 $0$.</p> <p>I don't know the right solution for FM.</p> http://mathoverflow.net/questions/99549/how-does-my-radio-work/99564#99564 Answer by PaPiro for How Does My Radio Work? PaPiro 2012-06-14T07:37:06Z 2012-06-19T12:49:54Z <p>Modulation is used to reduce antenna height, noise distortion, to avoid interference... </p> <p>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.</p> <p>When you tune your radio (choosing the carrier frequency of the radio station), you're also indicating the <strong>electronic circuit</strong> to be used to demodulate (extract information from the carrier) AM or FM signals to recover the (low frequency) human audible signal. </p> <p>See any book from B.P. Lathi (e.g. <a href="http://www.amazon.com/Digital-Communication-Electrical-Computer-Engineering/dp/0195110099" rel="nofollow">Modern Digital and Analog Communication System</a>) or from Oppenheim (e.g <a href="http://www.amazon.com/Signals-Systems-Edition-Alan-Oppenheim/dp/0138147574" rel="nofollow">Signals and Systems</a>). These are classical books used in Electric Enginnering courses.</p> <p>According to Lathi, the <strong>AM</strong> 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. </p> <p>(1) Rectifier Detector: If an <strong>AM</strong> signal is applied to a diode and resistor circuit, the negative part of the <strong>AM</strong> wave will be supressed. The rectified output, $v_r(t)$, is:</p> <p>$$v_r(t) = [A+m(t)]\cos \omega_ct [1/2 + 2/\pi(\cos\omega_ct- 1/3 \cos3\omega_ct + \ldots)] = 1/\pi[A + m(t)] + hft$$ </p> <p>where $hft$ are <em>high frequency terms</em>, $m(t) = B \cos\omega_mt$ 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.</p> <p>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 $m(t)/\pi$.</p> <p>(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 <em>Lathi</em>.</p> <p>The information (low frequency signal) in <strong>FM</strong> 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 <strong>FM</strong> 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.</p> <p>There is a tutorial <a href="http://eeweb.poly.edu/~yao/EE3414/comm_analog.pdf" rel="nofollow">here</a> or <a href="http://drlohar.com/PPTs/slide3.ppt" rel="nofollow">here</a>.</p> <p><strong>BTW, this is a very beautiful (real) application of Fourier Theory</strong>.</p> <p><strong>ADDED</strong>:</p> <p>In relation to your question, if we have</p> <p>$$\phi_{AM}(t) = A \cos\omega_ct + f(t)\cos\omega_ct$$ 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 $$\phi_{FM}(t) \leftrightarrow \pi A [\delta(\omega + \omega_c) + \delta(\omega - \omega_c)] + j\frac{Ak_f}{2}[G(\omega - \omega_c) - G(\omega + \omega_c)]$$</p> <p>Considering $$v(t) = \phi_{FM}(t) + \phi_{AM}(t)$$ we have the Fourier transform:</p> <p>$$\frac{1}{2}[F(\omega + \omega_c) + F(\omega - \omega_c)] + 2 \pi A [\delta(\omega + \omega_c) + \delta(\omega - \omega_c)] + j\frac{Ak_f}{2}[G(\omega - \omega_c) - G(\omega + \omega_c)]$$</p> <p>The next step is to adapt your equation to these equations. Then, procedures (1) <strong>or</strong> (2) should be used to recover the low frequency signal in the receiver side.</p> http://mathoverflow.net/questions/99549/how-does-my-radio-work/100249#100249 Answer by Emilio Pisanty for How Does My Radio Work? Emilio Pisanty 2012-06-21T14:28:29Z 2012-06-21T14:54:24Z <p>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.)</p> <p>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!)</p> <p>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!</p> <p>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.</p> http://mathoverflow.net/questions/99549/how-does-my-radio-work/100669#100669 Answer by rgrig for How Does My Radio Work? rgrig 2012-06-26T10:05:01Z 2012-06-26T10:05:01Z <p>I've been trained as an engineer, and I can tell you that engineers have a somewhat simplified view of the matter. (But, not <em>only</em> a simplified view, of course.) The other answers fill in some detail, but I think a higher-level view is useful.</p> <p>There is no such thing as perfect recovery of the transmitted signal. The best you can hope is to bound the error.</p> <p>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 $|f \pm f_0| &lt; B$. 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 $X_1$, $X_2$, &hellip;, $X_n$ 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 <a href="http://en.wikipedia.org/wiki/Rectangular_function" rel="nofollow">rectangular function</a> in the frequency domain, meaning convolution with a sinc in the time domain). See also <a href="http://mathoverflow.net/questions/5892/what-is-convolution-intuitively" rel="nofollow">this related question</a>.</p> <p>There are broad-spectrum modulation techniques, which are used for example in fourth generation mobile-phone networks, that do <em>not</em> rely on the assumption that the signal covers a narrow band. The two main ones are <em>frequency hoping</em> (use some narrow-band modulation technique but change $f_0$ often in some pseudorandom sequence) and <em>spread spectrum</em> (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 $|X(f)| &lt; c$ for some $c$ for all $f$. This way they behave as background noise as far as demodulating any narrow-band signal is concerned.</p>