Could QAM using a grid-like distribution be a quantization limitation?

Question

This is a follow up from this previous question discussing why QAM encoding is spaced in a grid like pattern like this one:

versus another with more efficient packing:

After looking at @Attie and @MarcusMüller's awesome answers and rewatching the video I have a leading idea that may explain a grid based QAM.

Question:

If the broadcasted signal is the result of 2 waves' constructive and destructive interference and only each wave's amplitude is modulated... could it be a quantization issue?

To clarify: I imagine we don't have a broadcasting device that has a perfectly continuous output and therefore can't output unlimited resolution for these signals. If that is the case, a combination of 2 discrete amplitudes would always result in a grid like pattern:

So an equilateral or irregular symbol distribution would sacrifice a discrete step in order to conform to the non-grid standard. Example created in Desmos:

Each purple line represents a discrete amplitude a broadcast device could reliably and accurately achieve. The example should show that the purple grid interests at points the equilateral distribution does not, yet the same resolution is required for both schemes.

Answer 1

You're overthinking. As explained in the answers to the other question, the reason square QAM predominates (sometimes with the corners chopped off at higher orders) is that it's less complex to process and decode, and for most applications this outweighs any slight gain in power efficiency that slightly tighter hexagonal packing would achieve.

(Note also that most QAM systems don't just send raw bits but use error protecting codes which are often tightly coupled to the constellation pattern. There is a huge body of theory on this subject.)

To clarify I imagine we don't have a broadcasting device that has a perfectly continuous output and therefore can't output unlimited resolution for these signals.

In a typical radio transmitter the raw I and Q symbols are digitally pulse-shaped and interpolated to high accuracy before being sent to the analog RF modulator, in order to meet requirements for adjacent channel power and suchlike. There's no practical limit to symbol level resolution in the way you're thinking.

Answer 2

A typical system involves a DAC (implied: can be more bits than required for the constellation), and either direct synthesis (sine tables in RAM, digital mixer, etc.), or baseband or IF synthesis with the remaining oscillators, mixers, etc. in analog, or other synthesis and modulation mechanisms. (Oscillators would most likely be PLL disciplined to an external crystal or other reference, but can otherwise be fairly ordinary LC or ring oscillators.) In any case, other than for FM, the signal at the final / power output is an analog signal, and a linear amplifier is used. (The signal chain may include predistortion to correct for amplifier nonlinearity, or certain modulations, FM for example, may not depend upon linearity, and a class C amplifier might be used, at higher efficiency.)

The DAC, or NCO accumulators, or look-up tables, etc., might all have more bit depth or buffer size than strictly required for a given modulation, to account for spur reduction, or to provide more flexibility (modulation controlled by onboard radio firmware), or to give lower noise and distortion than would otherwise be possible; DDS (at IF or RF) will be more demanding than baseband I/Q + analog mixing, but analog is more difficult to synthesize and tune on a monolithic process.

Or in more direct terms: your speculative reasoning would be trivially solved by increasing the bit depth by one or two, in a digital I-Q system. This is likely pretty cheap to do, so would not be a fundamental limitation. The limitation does not exist in the DDS case, where the carrier's dithering (except at the 0,0 coordinate) stimulates quantization noise, which, in aggregate effect over multiple cycles, manifests as a noise floor, or maybe more likely a lumpy forest (spurious tones).

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This is something of a partial answer, and would be improved by including definitions, references, diagrams illustrating example systems, etc.. But I'm hardly an expert on these systems, and merely introducing the terms/keywords, and rough examples of use, seems adequate for the similarly informal-level question. Cheers!

Answer 3

Binary signals are 0 or 1 - a state will be assigned to each of these values. They come in the form of digital logic / voltage levels (e.g: On = 3.3v = Logic 1, and Off = 0v = Logic 0), but also in modulated signals, such as On-Off Keying (where the presence or absence of a tone or carrier indicates the state, e.g: Morse Code). These are very simple to deal with, but still need to be assigned into one of the two buckets - noise and other factors can make this difficult, and assigning samples to the wrong bucket results in corruption, errors, retransmits, etc...

Extending binary signalling slightly, and you get Pulse Amplitude Modulation where the number of discrete levels is increased from 2, to n... For example, the ubiqutous Gigabit Ethernet employs PAM-5 (i.e: 5 discrete signal levels). An RF equivelant is Amplitude Modulation (and ASK), where the carrier tone is altered in amplitude only.

Another option is Frequency Modulation (including things like FSK and LoRa), where the carrier tone is altered in frequency only... but that's not what we're focusing on here.

Pairing a shift in amplitude with a shift in phase, allows more data to be conveyed for any given symbol - all of a sudden we have two useful dimensions without leaning even harder on determining slight changes in amplitude. Note that the baudrate (i.e: rate at which the signal changes from one state to the next) will generally be many times the carrier's period (many wavelengths)... this is impractical to show in examples, so the signal is often truncated to a very small ratio between the two (1:1 in Matt's video).

Now, with those two useful dimensions, aside from a portion of the frontend, we can easily have an identical pair of signal paths - one that handles the I component (In-Phase, 0°) , and one that handles the Q (Quadrature, 90°), as discussed in my answer to your original question. Both paths are able to place symbols into buckets, with the same number of buckets, the same spacing between buckets, etc...

When dealing with digital systems, it's also incredibly useful to have n equal to a power of two... so for 16-QAM, we have 2x dimensions, each with 4x buckets, giving us a 4-bit value per symbol.

If we were to use something that packed points more tightly into a given 2D space, like your suggestion of an equilateral triangle, there are a few significant changes to the encoding:

• Each axis now has different number of levels (7 on the X, and only 4 on the Y), with different gaps between each point... that increase in levels converts directly into an increase in errors. If you can tolerate that, then why not increase the points on both axes?
• Each one of the axes is able to detect and categorize more levels than before, but also, many of the resulting combinations are now invalid, which is inefficient.
• If we consider the X axis to be -1.0 to +1.0, then the Y axis is only -0.86 to +0.86, which isn't optimal.
• This constellation now gives us ~3.74 bits of information per symbol (a reduction when compared to 4), which would need to be rounded down to 3-bits (thus points discarded or trimmed), or combined with a more complex coding scheme on top.