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As we know there are two kinds of signals: analog signals and digital signals.
And there are two common techniques for multiplexing the signals: Time Division Multiplexing (TDM) and Frequency Division Multiplexing (FDM).
As I understand it, FDM is commonly used to multiples analog signals, while TDM is commonly used to multiplex digital signals.
What I don't understand is: Why do we not (commonly, at least) use FDM to multiplex digital signals or TDM to multiplex analog signals?
First off, you need to understand what an "analog" signal is, and what a "digital" signal is, how they are different, and how they are similar.
The term "analog" comes from the old distinction between "analog" and "digital" computers. A "digital computer", even a very primitive one of decades ago, has always been more or less what we know now -- a box that manipulates "bits" (or, in very old systems, decimal digits), where the information is represented by numbers encoded in the bits.
An "analog computer", on the other hand, used voltage values in the computer to represent numeric quantities. Ie, where the digital computer might represent the value "6" as the binary bits 110, the analog computer might represent the value "6" as a voltage of 6 volts.
(I could go on here for several paragraphs explaining more about how an analog computer works, but it suffices to say that, except in very limited applications, analog computers are no longer used -- where needed their function can be simulated with digital computers more cheaply and accurately.)
But the point is that an "analog signal" is called that because the voltage levels in the signal are analogies ("analogs") for some "real-life" phenomenon. Eg, in an analog audio signal the voltage level is an analog for the sound pressure that would create the sound -- a higher voltage level means more sound pressure, etc. Further (and it will be important later), the "X-axis" -- the time dimension -- is also "analog". That is, in most cases (except where, eg, a recording has been sped up or slowed down) one second of time in the analog signal corresponds to one second in "real life".
In the image below, peaks in the signal represent higher air pressure (and valleys represent lower air pressure).
On the other hand, the ups and downs of a digital signal have no direct relationship to "real life". Rather, they represent binary digits (hence "digital") in a numeric code used to represent whatever "real life" phenomenon the signal corresponds to. In the diagram below, the bits (each line represents a separate bit stream) might represent characters of ASCII text, or they might represent readings from a heart rate monitor. Looking at the data you can't even tell if it might be a pleasant sound (like the guitar), or a discordant sound (like the cymbal).
Also note that the X dimension is not an analog for anything -- an entire symphony might be transmitted as a digital signal in ten seconds, or a high-resolution photograph that was snapped in an instant may be transmitted, bit-by-bit, over a period of seconds or minutes. (The old Mariner space probe would take hours to transmit a single high-resolution image, because the bit rate was so slow at that distance.)
But note that both types of signals can be represented by a fluctuating voltage on a wire, and, in fact, you can hook a digital signal (if the data rate is slow enough) directly up to an ("analog") audio amplifier and listen to it. (It will come through as a noisy squeal). So at this fundamental level -- a voltage that fluctuates over time -- analog and digital signals are basically the same.
(It also needs to be noted that, from the standpoint of "information theory", the maximum amount of information that can be encoded into a signal occupying a given bandwidth is the essentially the same, regardless of whether it's digital or analog, or how it may be multiplexed. However, while one could spend days discussing information theory and entropy I won't go into it further, because it's for the theoreticians, and because it makes my head hurt to think about it.)
Next we need to understand "multiplexing". In the US, at least, a "theatre multiplex" is a group of motion picture theatres in a single building. Similarly, when you "multiplex" signals you combine several signals into one all-encompassing signal. But just as you wouldn't want to go to a motion picture theatre and see 6 movies all projected onto a single screen simultaneously, when we multiplex signals we do so in a way that the receiving station can separate the combined signals back into the individual ones.
The earliest form of signal multiplexing(*) was common radio. Several radio stations in the same area can all broadcast their signals into the same "ether", but each uses a different assigned frequency, so that, by tuning your radio receiver to a specific frequency (actually a frequency band) you can receive the "hard rock" station you want to hear, and not have to simultaneously hear the "easy listening" station your parents listen to.
This, in essence, is Frequency Division Multiplexing (FDM) -- the available spectrum is divided into frequency bands and each signal is combined with a "carrier" frequency to occupy the appropriate frequency band. (The original signal, before multiplexing, is referred to as a "baseband" signal, because it has not been combined with a carrier. This sort of contrasts with "broadband", where multiple signals have been multiplexed, but unfortunately the term "broadband" has multiple confusing definitions.)
FDM used to be used extensively in the phone system, for long-distance lines. A single pair of wires (and associated amplifiers), which might have a frequency response from 20Hz to 100KHz, carried a dozen or so audio signals, each at an assigned frequency just like broadcast radio or TV. Later coaxial cables were used, increasing the available bandwidth to several MHz and allowing several hundred audio signals per cable.
(A significant drawback of this scheme was that many amplifiers were required -- one every ten miles at least, and one every mile for some schemes.)
Now, it should be clear that, given that there's no fundamental difference (electrically) between an analog signal and a digital signal, digital signals can indeed be broadcast using FDM. In fact, if you have a cable modem, the cable, with over a hundred analog channels, is frequency multiplexed (just as over-the-air TV is). Your cable modem signal occupies (sharing with your neighbors) one of those "analog" channels. So your cable modem signal is being transmitted via FDM (though with several added twists).
Now for Time Division Multiplexing (TDM).
Time division multiplexing involves what it name implies -- the time axis of the combined signal is divided into discrete segments, and different individual signals are transmitted in the different segments.
TDM actually goes back to before regular digital data. The Bell Telephone company in the US experimented in the 50s with chopping up analog signals into lots of thin slices and interleaving the slices from several different analog signals together, allowing the several signals to be sent over a single pair of wires without having to resort to FDM. So in this case analog signals were transmitted via TDM.
But doing this was quite complicated. The analog signal needed to be "sampled" rapidly enough that the signal didn't change much between samples. For an audio signal with a maximum frequency of about 3.5KHz (the range for common telephone lines) sampling must be done at at least twice that frequency (so about 7 thousand "slices" per second) and then those slices must be interleaved with others (maybe five or ten). (This was known as "pulse amplitude modulation".) At the other end the slices need to be separated, and the missing space between slices filled in by "smoothing" the signal. And all this needs to be done without seriously changing the amplitude (sound volume) of individual slices.
Of course, the above scheme was very difficult to get right and didn't really provide much advantage relative to FDM over the same wires, so when true digital capabilities came around they dropped it like a hot rock.
So far I've been showing how there's really no difference between analog and digital. Now I'll reverse course -- digital is really different in some very important ways.
First off, digital signals don't degrade like analog signals. Old analog telephone networks introduced a lot of distortion into the sound as it was transmitted cross-country and reamplified many times (as often as once every mile). A digital signal does develop distortion in transmission, but since it's digital it can be "regenerated" at the end of each hop. This alone made TDM much more practical for digital, since there was no longer a need to maintain accurate signal amplitude, and the cables and amplifiers become simpler and less expensive (and farther apart).
But that's not all.
Consider, for instance, if you use Skype to talk to someone on the Internet. Your voice is first converted to an analog signal by the microphone in your computer. But it's then immediately digitized, and the digitized bits are "buffered" -- stored up awaiting transmission. Algorithms in Skype will examine the digitized sound and discard periods of silence (since there's no need to transmit silence), and they'll otherwise process the sound to minimize the number of bits needed to represent it.
Eventually (after a seeming eternity of computer instructions, but in an instant from your ear's point of view) a group of sound bits will be formed into a block, have the appropriate internet headers appended to the front, and be queued for transmission over your Internet connection. But, since you happen to be simultaneously using your web browser, the sound data may have to wait a few more milliseconds for your web request to be transmitted.
On the receiving end, Skype accepts the packet of transmitted data from the receiving computer's Internet connection and sticks it back in a queue, taking care to insert it in the right order in the queue based on a "timestamp" in the data packet. (This extra care is necessary because data packets may be delivered out of order in some cases.) Then another part of the Skype program removes packets from the front of the queue and "schedules" them to play back the recorded sound at the appropriate time based on the timestamps. So the received data is not played immediately, but instead waits (perhaps a few more milliseconds) until "the time is right" to be played, so that the time dimension of the original data (and hence the fiction of continuous voiced transmission) is maintained.
So what happened here? The voice data from Skype was time-division multiplexed with your web browser requests, and, once out on the Internet trunk lines, both were time-division multiplexed with thousands of other messages. But, critically, the multiplexing was not based on a fixed schedule (eg, dividing the time dimension into 10 millisecond slices and "dedicating" the channel to transmitting each signal for 10 milliseconds), but instead an "adaptive" algorithm was used, where each signal only occupied the channel for as long as it needed. Even though this technique requires that each data packet contain an address to identify it's destination (whereas with a fixed multiplexing scheme the destination would be defined by the slice location), it's still far, far more efficient than a scheme that relies on fixed slices, either in time (primitive TDM) or frequency (FDM).
Now note the differences between TDM (as used in networking systems) and FDM. FDM has the advantage of being relatively simple (witness standard AM radio). It can carry analog or digital signals equally well, but, because it does either equally well, it offers no great advantages to digital.
TDM is only marginally practical for (undigitized) analog signals -- the complexity and potential loss of signal quality are major negatives. But digital, it seems, was "made for" TDM -- the data is already "sliced", and accurately maintaining amplitude or time reference is not required. And since "dead air" is represented by a complete absence of data packets, not by simple silence on a phone line, "adaptive" multiplexing, where only "useful" data is transmitted, and not on a rigid schedule, is much more practical. For these reasons much more digital data can be transmitted in a given bandwidth using (adaptive) TDM vs FDM.
So it's not that digital CAN'T be transmitted using FDM and analog CAN'T be transmitted using TDM, but rather there are practical reasons why doing so would be more expensive, less efficient, and less reliable.
(I need to add that even given all the above FDM is used for digital data in some circumstances. For instance, the link to/from communications satellites is very "broadband", and no single data stream could practically use the entire bandwidth. So the overall channel width is sliced into frequency bands, with each occupied by a separate data stream. In this case, since the digital data stream is already time-division multiplexed and there is very little "dead air", the advantage of further TDM would be slight.)
(*) There were some crude TDM schemes used for telegraph signals as far back as the 1850s, but that's mostly lost in the mists of history.
Answer to "Why do we not (commonly, at least) use .... TDM to multiplex analog signals?
If you have $N$ analog (meaning continuous-time) signals to transmit over a common channel, then with TDM, each signal is transmitted for $(1/N)$-th of the time, e.g. $1000$ signals are transmitted for one millisecond each in non-overlapping time slots during a $1$-second interval. So what ends up at the receiver is $1$-millisecond clips, spaced $1$ second apart, of each of the $1000$ signals. This is probably not what is desired for any of the signals, but YMMV.
Answer to "Why do we not (commonly, at least) use FDM to multiplex digital signals ....?
Digital signals are transmitted using analog (meaning continuous-time) signals using a process called digital modulation. For example, analog frequency modulation (FM) is used for commercial broadcasting of speech and music signals. A digital signal input would result in a signal more commonly referred to as frequency shift keying (FSK), and it is quite easy, if not common, to transmit these FSK signals via FDM.
The spectrum of a baseband digital stream can be extremely wideband (e.g. the spectrum can cover a huge range of "frequencies" depending on the edge rate and distribution of ones and zeros, all the way down to DC, not just at the bit rate).
FDM systems usually have various spectrum limitations (FCC regulations, properties of the transducers or transmission cables, interference, etc.), so the signal is usually modulated to convert the wideband spectrum into one better suited for the chosen transmission environment. Also, focusing the spectrum into a narrower band may improve the signal-to-noise ratio.
Inside the box, the transmission medium (traces between gates/chips, etc.) is usually capable of a wider bandwidth, sometimes down to DC, as well as shielded from noise and creating RFI.
