Electrical – Differentiating between 32-QAM and 32-PSK

This suggests it's impossible to perfectly determine the instantaneous signal even in a theoretical channel with zero noise

I'd turn this around and say it's impossible to instantaneously determine the signal.

one has to assume that the signal is changing slowly (band-limited). How is this overcome in practice?

In practice, our message signals are bandlimited, so this is not a difficulty. In fact, our message signals generally have much less bandwidth than the carrier.

To approach your theoretical question, is a band limit strictly required, Imagine trying to modulate a 1 Hz carrier with a 1 MHz signal --- the result would be unusable. So in fact there must be some kind of limit.

A good idea is to go step by step. 8 QAM (Quadrature Amplitude Modulation) is not the easiest modulation. It combines amplitude shift keying (ASK) and phase shift keying (PSK) modulation. It is a digital modulation technique, so it is used to transmit a stream of bits in contrast to analog modulation, where the source signals are continuous.

At first we should take a look at the simplest form of digital modulation: 2-ASK, that is an ASK with two possible values per chip. In the baseband (the NOT yet high frequency modulated signal) there are two possible logic values: 0 and 1. They correspond to two input voltages (e.g. -1V and 1V). Note that we can draw these values as locations in the IQ plane. We will need this later to understand QAM.

To modulate the ASK signal (as shown in the block diagram) the I value is multiplied with the carrier cosine signal. Multiplying with 1V means a cosine impulse is transmitted, multiplying with -1V means an inverted cosine impulse is transmitted (see the table).

An interesting fact is that inverting the cosine results in the same output signal as delaying it by a half period (180 degree phase shift). This makes 2-ASK equivalent to a 2-PSK modulation. This can be seen in the IQ plane. The ASK idea is to mirror the first location (I=1V) on the Q axis to generate the second location (I=-1V). The PSK idea is to move the first location 180 degree on a circle around the IQ plane origin. Both ideas result in exactly the same locations and therefor in the same signals.

When we want to generate higher order PSK signals (more than two locations in the IQ plane - e.g. 4), we need 90, 180, 270 degree delayed versions of the carrier cosine. They are represented by 4 locations on a circle on the IQ plane. 4 locations mean we can transmit 2 bits in one chip-time, so this effectively increases the channels bitrate:

To build a delay circuit is costly, so the relation between ASK and PSK is used as a trick to build this device only once. In addition to the original carrier (I path) only one 90 degree shifted version (the Q path) is generated. Using these two signals, all other delayed versions can be created. Take a look at the IQ plane. 180 degree is at (I=-1V, Q=0V), 270 degree is at (I=0V, Q=-1V) and so on. In addition to the ASK modulator a Q path is added to generate the PSK. I and Q path are then just added to form the modulated output signal.

Each of the I and Q signals can have 3 different values: 1V, 0V, -1V But there are only 4 allowed combinations, which select one of the 4 locations. To do that we need some kind of "2 bit to 3 voltage level" converter, which converts two digital input values to the appropriate I and Q voltages as shown in the table:

bit0bit1:I  Q  
--------------------------
  0   0 : 1V0V
  0   1 : 0V1V
  1   1 : -1V0V
  1   0 : 0V-1V

In general the matching of input bit patterns to locations is arbitrary, but in theory it can be shown that the Bit Error Rate is lower, when the locations are mapped using the gray code (as in the table).

These 4 locations have the same distance to the origin of the IQ plane, so the respective output signals have the same amplitude (here: 1V). The only information of the signal is stored in its phase, ergo this is a PSK.

The only difference to PSK is that QAM also changes the amplitude of the signal:

The image shows (one of the many possible) 8 location QAM. 8 locations mean 3 bits per chip. The locations obviously have different distances to the origin. In this case we need 4 different voltages each for the I and Q signals to reach the locations. This means we need some circuit which converts 3 digital input values (2 voltage levels) to the appropriate I and Q combinations (4 voltage levels each). -> your 2 to 4 voltage converter

This is the same thing as for PSK, again. Just one more bit and one more voltage, resulting in a somewhat larger table. I and Q are then feed to the well known modulator that was already used for PSK.