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How can a phone wire have multiple frequencies?
In my Networks Textbook about DSL vs Dial Up it says the following:

The residential telephone line carries both data and traditional telephone signals simultaneously, which are encoded at different frequencies:

• A high-speed downstream channel, in the 50 kHz to 1 MHz band

• A medium-speed upstream channel, in the 4 kHz to 50 kHz band

• An ordinary two-way telephone channel, in the 0 to 4 kHz band

From my basic knowledge of physics, the frequency of a wire is the rate at which it reverses the polarity. So if you have one wire, how can the electrons be simultaneously changing polarity 4,000 times/second (for talking on the phone) and also 50,000 times/second (for using DSL)?

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    \$\begingroup\$ How can air do it? \$\endgroup\$ – Octopus Jan 16 '15 at 23:36
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    \$\begingroup\$ If you don't own a text on Fourier Theory written prior to the invention of practical digital computers, go hit the oldest used-book stores you can find until you find one. It should cost practically nothing and will be worth its weight in gold. If you're in the US, and can find one, the early 1940's ones in those weird wartime sizes will be excellent, as the War Department really pushed to make sure American mathematicians understood this stuff. To see why, you might find this-- the world's first secure digital voice communications system-- interesting: en.wikipedia.org/wiki/SIGSALY \$\endgroup\$ – user64585 Jan 17 '15 at 10:43
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    \$\begingroup\$ How can an orchestra play more than one note at once? \$\endgroup\$ – Phil Frost Jan 18 '15 at 5:32
  • \$\begingroup\$ Here's a very nice demonstration of how multiple frequencies combine. (Be sure to right-click for additional settings.) \$\endgroup\$ – Jeanne Pindar Jan 18 '15 at 18:16
  • \$\begingroup\$ The polarity of an electron's charge never changes. It is always negative. Current is the flow of electrons (or, charge, carried by electrons). The flow may change direction. \$\endgroup\$ – naught101 Jan 18 '15 at 22:50
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The underlying assumption in your question - that the frequency being measured is the rate at which electrons reverse polarity - is incorrect. The frequency of a signal at the transmitter, receiver, or anywhere in between physically corresponds to the cyclic arrival of a voltage.

For example, in a digital application using amplitude modulation (let's assume on-off keying for simplicity), you could measure frequency by the number of 'on' pulses you detect per unit time. In RF communications, this might correspond to a logic-high voltage, or in optical communications it might correspond to the arrival of a large number of photons. In the ideal case, a logic-low or off state would correspond to a voltage of zero or the arrival of no photons, but dark currents and the imperfections of modulators rarely make that the case.

In terms of implementation, a straightforward and simple implementation to the transmission of two separate RF frequencies on a single medium (a copper wire) is by the use of two complete transmitter chains to encode the data at the two distinct carrier frequencies, and then the use of an RF combiner to get the two outputs from the transmitters onto a single copper wire. The receiver can be implemented a number of ways, but a simplistic method would be to use an RF power divider to create two copies of the signal, and then use a high pass filter on one and a low pass filter on the other. You can then continue with the normal receiver chain.

As others have said, multiple frequencies can be present on a wire at the same time. The instantaneous presence of multiple frequencies does not indicate multiple voltages though; there will necessarily be a single voltage at any given point on the wire (so long as the voltage is defined between that point and a common reference, typically ground). Over a span of time though, you can construct a signal by sampling at regular intervals. That signal will not look like a normal sinusoidal wave if multiple frequencies are present though, due to the principle of superposition. If you pick two carrier frequencies, let's say 5 kHz and 5 MHz, modulate data onto both and then sum the resultant modulated signals, you might be presented with a very peculiar signal in the time domain. If you apply a Fourier Transform though and look at the signal in the frequency domain, you would see a strong signal at 5 kHz, a strong signal at 5 MHz, and then a smattering of other frequencies around the carrier frequencies to account for the modulated data.

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    \$\begingroup\$ Now you have enough reputation! \$\endgroup\$ – Greg d'Eon Jan 17 '15 at 12:10
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On 'one wire' there can only be one voltage present at any instance in time at a certain point on that wire. So if you add two sine waves the sum is no longer a sine wave but something else. The electrons move in the same complex fashion as well. Observe the animation beat acoustics source.

The more frequencies you add the more complex the signal becomes. From a certain number of frequencies such as is the case for ADSL/VDSL, the combined signal appears as noise on a spectrum analyser or oscilloscope and becomes unintelligible to the human brain.

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  • \$\begingroup\$ Whoa... Trippy. \$\endgroup\$ – naught101 Jan 18 '15 at 22:37
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How can the multitude of frequencies that make up a piece of music be transmitted successfully to a loud speaker and reproduced largely without error?

Speakers are connected with wires and so are microphones - there is absolutely no difference in principle. It so happens that a telephone wire carries much higher frequencies, but the principle is the same.

Any medium that carries a single frequency is usually capable of carrying a multitude of frequencies. Air for instance - you can speak to your neighbour and the speech pattern you produce is a multitude of ever-changing frequencies.

Radio transmitters all share the same medium and there is no problem distinguishing one transmission at 98.4MHz and another at 99MHz.

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You need to look at superposition and linear systems. As one example of multiple frequencies on a wire, a square wave has lots of harmonics.

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  • \$\begingroup\$ High frequency harmonics are almost a good example of why you would think wire would not be good at handling several at once. Why don't the hi-freq harmonics from the lower frequency signals interfere with the high freq signlas? Just saying. \$\endgroup\$ – Octopus Jan 16 '15 at 23:38
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    \$\begingroup\$ @Octopus, yeah that's called inter-modulation distortion,en.wikipedia.org/wiki/Intermodulation it certainly happens when things get non-linear... mixer's and all that. \$\endgroup\$ – George Herold Jan 16 '15 at 23:59
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There's an even more fundamental problem in your question than pointed out in the other answers.

"Simultaneously" is a time-domain concept. Frequency is a frequency-domain concept.
These are Fourier transforms of one another, so they're "dual" concepts, not orthogonal concepts.

It's certainly possible to have a signal with two frequencies: just add two cosines of different frequencies together; the signal "simultaneously" has two frequencies.

But saying the signal "simultaneously" has two frequencies would be meaningless because "simultaneously" refers to a single instant in time, and if you restrict yourself to a single instant in time then you cannot possibly know anything about the various frequencies present.
(This is the time-frequency uncertainty principle, which should remind you of the Heisenberg uncertainty principle.)

Once you start looking at all possible frequencies then the notion of time becomes meaningless.

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  • \$\begingroup\$ Thanks, that was probably the fastest upvote I'd ever seen. \$\endgroup\$ – Mehrdad Jan 16 '15 at 20:01
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    \$\begingroup\$ I was in the neighbourhood :P \$\endgroup\$ – Lightness Races in Orbit Jan 16 '15 at 20:03
  • \$\begingroup\$ This answer is correct, but there is a rigorous way of telling what the frequency domain looks like at a single point in time: use the dirac distribution as a window function. That would give us a completely useless answer, but it is an answer nonetheless. \$\endgroup\$ – Tim Seguine Jan 16 '15 at 22:06
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    \$\begingroup\$ You are arguing with semantics. Maybe instead of "simultaneously" you should use "concurrently". \$\endgroup\$ – Octopus Jan 16 '15 at 23:40
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    \$\begingroup\$ @Octopus: That would be just as meaningless, either way you're including a notion of time, which does not exist in the frequency domain. \$\endgroup\$ – Mehrdad Jan 16 '15 at 23:50
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Within just a single phone conversation are many many frequencies (which changes with your voice pitch if nothing else)! Waves at different frequencies are superimposed to create the resulting waveform. If this did not work then the only sound you'd be able to hear, ever, is sine waves of various pitch.

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A wire can carry multiple electrical signals just like the air can carry multiple sounds.

Imagine you're in a quiet room and a violin starts playing a note. A single frequency that you hear through the vibrations in the air.

Then it's joined by a cello. Now you have two frequencies travelling through one medium to your eardrums. You can hear that they're different and with training could tell which note each was playing.

It works exactly the same in the wire only with electrons rather than air molecules.

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  • \$\begingroup\$ Even a single note on a single instrument isn't a single frequency. \$\endgroup\$ – Octopus Jan 16 '15 at 23:42
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    \$\begingroup\$ @Octopus: Agreed, I'm simplifying for effect. \$\endgroup\$ – Daniel Jan 17 '15 at 0:28
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After modulation and transmission from the source, the final signal on the wire IS a single signal. Just try to go back to pre-digital age of cable where you hooked up your cable tv providers wire directly to your TV and were able to watch any channel.

And if you had two TV's those times, you can watch two different channels at the same time that were present on the same wire. Note that I am talking about old times where you did NOT need a box from your cable company to see the channels.

Now back to the single signal on the wire. It is always just a single signal. The magic happens at the receiving end. You can feed the same single signal to different receivers. For successful and clear reception and processing, you will need a circuit to TUNE to the frequency of your choice. These are called band-pass filters. These circuits process the single complex signal, but they only respond to certain timing characteristics of the input signal. Anything that does not confirm to this timing is dropped (proper term is attenuated). The part of the signal that corresponds to the timing is allowed to keep its signal strength. Output of this circuit is now just the signal that the device wants to process.

The same single signal can be fed to another device tuned to another frequency. Then its output will be the second frequency that it was tuned to.

Neither the first output nor the second output, now have the other signals in them. If you try to feed these outputs to another device and tune for another frequency, you will get nothing.

For detailed explanation, you will need to google and understand how LC (RC also) circuits work. The combined charge and discharge characteristics of the LC components is what determines the tuning frequency.

There is also the other way of tuning called the band-stop filter.

Now how does the transmitter is able to get that many signals combined on to one wire, is a whole separate field.

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