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In my digital communication, class we have covered the various modulation schemes, namely QAM and its derivatives (B(A/P)SK, PAM and (Q/M)PSK) as well as BFSK. These two schemes have one or at most two basis signals. There are many orthonormal sets whose size is larger than two. Why not modulate amplitude and phase with multiple basis functions, for eaxmple with many varying frequency sinusoids?

Although not covered in my class, I am aware that MFSK exists, and from Fourier we know that with enough frequencies the phase and amplitude will become redundant (since they can be reconstructed with more frequencies.) Surely there is some utility in achieving the same result while using less of the frequency spectrum.

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Ah! They don't generally involve only two degrees of freedom. The ones you've been looking at mostly have, but that's not because they're the only ones used or taught.

Chances are you'll meet one or multiple higher-dimensional modulation formats pretty soon!

But let's take a step back. You say "involve only two carrier signals", and every passband-oriented radio engineer will tell you that e.g. a QPSK transmission has one cosine carrier at the carrier frequency, and every baseband-oriented communications engineer will tell you that there's only one carrier, it's at frequency 0, and it contains a complex value for every time instant.

The fact that you have "two degrees of freedom" here just come from the fact that if you take the Fourier transform of any signal (for which it is defined), you get a complex number at every frequency. A complex number has a real and imaginary part, and they correspond to the even (cosine) and odd (sine) components in the time domain signal.

That's why all the QAM, PSKs you learn are "two-dimensional": Modulating something with a real harmonic oscillator (which is what we know well how to build, e.g. from capacitors, inductors, crystals and transistors) gives you a signal that when described in baseband is complex, and complex numbers have 1. a real and 2. an imaginary part.

So, your signal being physically on any single carrier, you get exactly these two degrees of freedom – which you can use. (You don't have to. BPSK at the same symbol rate takes exactly the same spectrum as QPSK, but transports half as much data – exactly because it ignores one of these dimensions.)

Math tells us that if we have a 2D base, we can't create more than two independent dimensions through any base change, so that's that.

However, if you use more than one physical carrier, sure: OFDM (Orthogonal Frequency Division Multiplexing) takes $N$ frequencies, and gives you the freedom of $N$ complex values to express your signal. Does that yield higher spectral efficiency? No. It's still just a base transform; but it makes dealing with specific types of channels easier.

Same with any other mechanism that combines multiple frequencies or time values into one "meta-symbol" that thus has a multiple of the original degrees of freedom: You only get as many degrees of freedom as you use by defining the complex values your meta-symbol has, so, no free lunch here, spectrally. You'll find such scheme as CDMA, SC-FDMA and a lot of other methods that all increase the bandwidth or duration of the channel usage by the same factor as they increase the degrees of freedom. Well, an invertible base transform is an invertible base transform.

Where things do get interesting is what happens when you start saying "wait, it's not only about getting as many bits onto the channel per channel access as possible, it's about correctly receiving them!". You would then start to look for higher-dimensional "meta-symbol" spaces, in which you can optimize the distance between symbol points, such that noise "falsifying" a symbol into another valid symbol becomes very unlikely. That just happens to be exactly what channel coding is. Take $k$ elements from some source symbol alphabet, transform them into $n$ symbols (typically, $n>k$), and do that revertibly and in a way that minimizes the chance that one of your $n$-dimensional symbols gets mistaken for a different one. While this looks strange (you make your data longer, i.e. you split the energy you have to transmit across more symbols, but that leads to fewer errors?!??!), it enables you to use less power to achieve the same data rate at a given error probability. Neat stuff, and a pretty large field of research with applications in every digital device that is more complex than a microwave timer (and probably even in that).

But here's the "signal" side of things "killer application": MIMO. Say, you have more than one antenna on the transmitter, and more than one on the receiver. "So what," you'll say, "if they use the same frequency, in the same space, you could have just added up what you transmit on both antennas and sent it from a single antenna, and would have gotten the same on the air!"
But you'd be mistaken. If the channel between the first transmit and the first receive, the first transmit and the second receive, the second transmit and the first receive and the second transmit and the second receive antenna are sufficiently independent, you do actually get two independent channels over which you can then send complex symbols. On the same frequency. At the same time. In the same place. That's free spectrum! All the magic here is basically that a full-rank square matrix can be diaogonalized, and that gives you ways to find base transforms.

So, you ask a good question! But lucky you, depending on how deep your university program goes on that, you might learn more on these things soon enough. And if it doesn't go that deep, you can still learn on these things, pretty easily. Generally, when I was still responsible for the exercises in Digital Communications I at my uni, the student that writes a post in the lecture forum (or the email, or however you communicate) asking about the things you ask here would be very much encouraged, get a nice reply, and an invitation to discuss which advanced courses they could do. Keep up the good work!

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  • $\begingroup$ On a related note, even if one is using a single transmitter, I would expect that using multiple receivers that are separated by a distance that is significant relative to the wavelength and correlating the same signals, one could compute the expected received-phase difference that should be expected for the transmitter, and correlate the received signals with the phase shift applied. Having receivers be aware of multiple transmitters could improve rejection further, but even a receiver that was simply agnostic to any transmitter other than the one of interest could still... $\endgroup$
    – supercat
    Commented Apr 5, 2023 at 15:18
  • $\begingroup$ ...have some ability to filter out other signals using the same channels; adjusting the phase shift a receiver is looking for would allow it to select which transmitter it would detect. $\endgroup$
    – supercat
    Commented Apr 5, 2023 at 15:19
  • $\begingroup$ nope; you can increase gain / SNR with that, but if the channels seen are linearly dependent (and a phase shift is just a complex factor), then they're not independently usable. That's what the rank of the channel matrix is defined to be: the number of linearly independent components. $\endgroup$ Commented Apr 5, 2023 at 15:30
  • $\begingroup$ Hmm... yeah, I guess if one had two receiving antennas and one competing transmitter (as well as the one of interest), the best reception would be probably be achieved by identifying the phase shift and amplitude ratio associated with the competing antenna and nulling that out. If one had three receiving antennas, one could null out two competing sources. For any given transmitter-of-interest placement, there would be some combinations of placements of competing transmitters which could not be nulled out without also nulling out the transmitter of interest, but with enough... $\endgroup$
    – supercat
    Commented Apr 5, 2023 at 15:35
  • $\begingroup$ ...receiving antennas, one could reduce the likelihood of competing transmitters falling into such a pattern. Alternatively, if transmitters coordinate frequency hopping, they could ensure that no "bad" combinations of transmitters would use the same frequency at the same time. $\endgroup$
    – supercat
    Commented Apr 5, 2023 at 15:37

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