How Do Radio Receivers Isolate Specific Frequencies from a Combined Signal?

[Spoonboy]

I've been puzzling how it is that you can tune into a signal on a particular frequency. How can it be possible to get rid of all the other signals that are added together? I don't know how much of this makes any sense, but I have tried to specify the problem mathematically.

Suppose $$R_1(t), R_2(t), R_3(t)$$ are all functions of time that represent the audio going out on 3 radio stations. Suppose the carrier frequencies are $$f_1, f_2, f_3$$ respectively. Then the total signal $$R(t)$$ will be
$$R(t) = R_1(t) \sin(2 \pi f_1 t) + R_2(t) \sin(2 \pi f_2 t) + R_3(t) \sin(2 \pi f_3 t)$$.

How is it that we can recover $$R_1(t), R_2(t), R_3(t)$$, only by sampling $$R(t)$$ and given that we know $$f_1, f_2, f_3$$? What mathematical operation can we do to recover the signals?

I believe for this to work $$R_1(t), R_2(t), R_3(t)$$ cannot carry frequencies higher than half their respective carrier frequencies. Is that right?

 

[ZapperZ]

You take the time domain signal, and do a Fourier transform, so that you get the frequency domain of the signal. The "peaks" in the signal in the frequency domain will correspond to all the frequency component of the signal.

Zz.

 

[Count Iblis]

I believe for this to work $$R_1(t), R_2(t), R_3(t)$$ cannot carry frequencies higher than half their respective carrier frequencies. Is that right?

That's indeed the important part. In fact the functions R(t) cannot have frequencies larger than the frequency separation of the carriers, otherwise you'll get interference from one station if you listen to the other station.

 

[Spoonboy]

You take the time domain signal, and do a Fourier transform, so that you get the frequency domain of the signal.

How much of the time domain signal? You can't look into the future, and you don't want to be storing everything you've ever received so far.

The "peaks" in the signal in the frequency domain will correspond to all the frequency component of the signal.

So $$\mathcal{F}\{R\}(f_1) =$$ ?

 

[Mephisto]

I'm not sure how this works in FM, but for AM i believe you use resonance to amplify a particular frequency and then read of the amplitudes as the actual data. If you have an oscillator with a very high Q, and you tune it to resonate at a particular frequency, you can then pick out just one component (amplified like couple thousand times even) , and other will be negligible.

 

[Dale]

Its not that you amplify a particular frequency, you modulate it.

Say you have an audio signal that goes from 0-20 kHz. You want to broadcast this at RF so you multiply it with a 150 MHz sine wave carrier frequency. This is called modulation. You then have a RF signal that goes 149.980 MHz to 150.020 MHz (150 MHz +/- 20 kHz).

The radio station down the road wants to broadcast their own signal so they use 151 MHz carrier wave for their modulation and the same +/- 20 kHz bandwidth, so there is no overlap of the signals.

Then to get your signal back you simply do a demodulation. You again multiply your signal by the 150 MHz carrier wave. This results in the original 0-20 kHz signal being demodulated down into the audio band.

The signal from the station down the road is also demodulated down to 1 MHz +/- 20 kHz. So to get rid of it you simply apply a low-pass filter that gets rid of all signals over 100 kHz or so.

 

[rbj]

this should go into the Electrical Engineering forum. or maybe post this (without the LaTeX) to comp.dsp .

anyway, the idea is that when the carrier is "modulated" by the audio, that the resultant signal practically has finite bandwidth (and the FCC will kick your butt if you spill too much energy outside of your finite BW). for regular AM (not single-sideband, or SSB), this bandwidth extends equally on both sides of the discrete carrier frequency. if the spacing of two different transmitted signals is greater than the bandwidth, resonant filter circuits (or signal processing algs) can be designed that will pass one signal and reject (to a finite extent, theoretically something always leaks through) the other.

now how all this is done requires at least a couple of decent electrical engineering courses. to really understand this, you would likely need to take courses in Signals and Systems (we called it Linear System Theory when i was in school), basic Circuits, Electronics, and Communications Systems Engineering. somewhere in there, you would learn how to design a "filter", a device that descriminates between frequencies, passing some while rejecting (more precisely, attenuating) other frequencies. if you like math, Filter design is a nice elegant theory (both analog or digital) that has some sophistication (but, from my POV, is a helluva lot simpler than Tensors or whatever physicists deal with nowadays). this kinda theory and technology is directly down my line. this is the kinda math i deal with in my work.

 

[Count Iblis]

Can a 100% digital radio that records signals from an antenna be made for an affordable price? What I'm thinking about is the following. In an ordinary shortwave radio there are a lot of mixers, amplifiers etc. that not only introduce a lot of noise, but because of their non-linear behavior produce intermodulation effects.

So, one can think of the following solution. Take an ordinary very low noise RF amplifier. This amplifier does not have to be extremely linear. We then simply amplify the signal from an antenna, convert it to a digital signal using some AD converter and simply record the signal in the memory of a computer. If the sampling rate is sufficiently fast and we have enough memory available, one should be able to record radio signals from a large frequency range for a long time.

Now, if we simply measure the behavior of the amplifier, we should be able to calculate the real signal strength from the recorded signal, thereby undoing the nasty intermodulation products.

Using Fourier transformation we can then tune into some station and later we can listen to another station that was broadcasting on the same time on another frequency, so we don't miss anything. Also, if we record signals from different antennas, we can do interferometry, null one station to listen to a weaker station that was broadcasting on the same frequency.

 

[rbj]

Can a 100% digital radio that records signals from an antenna be made for an affordable price?

this occasionally gets talked about on comp.dsp . for the most part, the consensus is that it's still not cost effective (a better radio for less money can be made with analog parts to at least the IF stage).

What I'm thinking about is the following. In an ordinary shortwave radio there are a lot of mixers, amplifiers etc. that not only introduce a lot of noise, but because of their non-linear behavior produce intermodulation effects.

So, one can think of the following solution. Take an ordinary very low noise RF amplifier. This amplifier does not have to be extremely linear.

well, if there is some resonant circuit before the no-so-linear RF amp, then it need not be so linear. we need not worry about the images of the narrow-band RF signal of interest created by the nonlinearity, they will go to frequencies we can filter out later. but what about the images of what had been clean signals that are not the signal we want, but the nonlinearities of the RF amp creates frequency components of these deselected signals that spill into the channel of interest? once they get added to the frequencies we are interested in, it's pretty hard to un-add them.

We then simply amplify the signal from an antenna, convert it to a digital signal using some A/D converter and simply record the signal in the memory of a computer.

now the expensive part is an A/D converter that is fast enough to sample the RF signal directly. it's not so bad for commercial or shortwave AM signals (in what we call the MF band, up to 3 MHz) but can start to get pricey to directly sample HF signals (3-30 MHz) and very pricey for VHF signals (30-300 MHz). also A/D converters at those high frequencies have fewer bits of resolution (more quantization error or noise) than slower A/D converters.

also, what you can do with a DSP at higher sampling rates is less. with a given instruction rate (MIPS), the number of instructions per sample is inversely proportional to the sampling rate.

If the sampling rate is sufficiently fast and we have enough memory available, one should be able to record radio signals from a large frequency range for a long time.

Now, if we simply measure the behavior of the amplifier, we should be able to calculate the real signal strength from the recorded signal, thereby undoing the nasty intermodulation products.

if IM products fall into the passband of interest, you are already screwed (just like the problem of aliasing when some signal is undersampled. once you add two numbers together and you know only the sum, it's kinda hard to separate them again.

Using Fourier transformation we can then tune into some station and later we can listen to another station that was broadcasting on the same time on another frequency, so we don't miss anything.

if it was sampled cleanly and adequately (sufficient sampling rate), sure, you're correct. if you have a million samples per second (of a real signal), you have the total information of 1/2 MHz of spectrum. if each station was 50 kHz wide (stereo, single-sided spectrum) your million samples per second has the information of 10 separate independant RF signals. picking one signal out from the others can be done in DSP software.

Also, if we record signals from different antennas, we can do interferometry, null one station to listen to a weaker station that was broadcasting on the same frequency.

that can be done also.

Count, i think the present consensus is that it is still most cost effective to do almost everything you're talking about (broadband sampling and processing) on the IF (Intermediate Frequency) signal. that is, RF amplifier (with tank circuit, so that only the broad band of interest is amplified), a simple and clean modulator (called a "mixer" by the radio guys, not to be confused with the "mixer" that audio guys think about) bumps down the entire broad band of interest to IF, frequencies ranging from near DC to a frequency equal to the broad bandwidth. then it's sampled with a slower, less expensive A/D (with a decent number of bits in the word, at least 16) at twice that bandwidth. in DSP software, you could easily filter out all but the station of interest and (by use of Hilbert Transform) create what we call a "quadrature" signal: a complex-valued signal that could be expressed as

$$r(t) = A(t) e^{i \phi(t)} = x(t) + i \mathcal{H}\left\{x(t)\right\}$$ .

the amplitude envelope information is $A(t) = |r(t)|$ and the instantaneous frequency information is the derivative (w.r.t. time) of $\phi(t) = \arg\left\{r(t)\right\}$ (both functions are real). all this is presently very doable. one sort of holy grail for DSP guys doing this or similar, is a fast (few instructions) and decent (in terms of accuracy) implementation of the $\arctan()$ or $\arg \{ z \}$ functions. DSP guys (at least some of them) think a lot about the arctan and doing it efficiently and well.

well, lessee if my LaTeX turned out...

 

[Dale]

I agree with rbj, in my work we take a very low voltage 64 MHz or 128 MHz RF signal, amplify it, demodulate it down to about 1 MHz intermediate frequency, digitally sample it at about 10 MHz, and then digitally process it to baseband. But this is all as a small part of a multi-million dollar piece of equipment (MRI), so "affordable price" is less of a concern than getting the information out nice and clean but we still don't do anything more expensively than necessary for the quality required.