QAM Constellation Construction - Basic 4/8 QAM

Question

I wanted to know how to construct the set of amplitudes/phase for QAM modulation. I saw that a QAM signal can be represented as: $$ s_m(t)=A_{mi}g(t)\cos(2\pi f_c t)-A_{mq}g(t)\sin(2\pi f_c t) $$ where $A_{mi}$ and $A_{mq}$are two amplitude levels and $g(t)$ is the pulse. The resultant amplitude is given by: $$ A_{ra}=\sqrt{A_{mi}^2+A_{mq}^2} $$

And the resultant phase is given by: $$ \theta_m=\arctan\left(\frac{A_{mi}}{A_{mq}}\right) $$

Given the above, what values should I take for the amplitude levels so that I will get the resultant amplitude and phase?

I tried generating the amplitude and phase values for simple 4 QAM by taking 2 different amplitude levels and 2 different phase levels like below: $$A_{mi}=\left\{-1, 1\right\}, \quad A_{mq}=\left\{-1, 1\right\}$$

Then plotted the 4 possibilities (which seems pretty overlapping): Its clear that I need to take different amplitude levels for inphase and quadrature carriers. But is there any generalized formula for $A_{mi}$ and $A_{mq}$ so that I can generate my resultant amplitude and phase for QAM?

$$ \begin{array}{|c|c|c|c|} \hline A_{mi} & A_{mq} & A_{ra} & \theta_m\\\hline -1 & -1 & \sqrt 2 & \frac \pi4\\\hline -1 & 1 & \sqrt 2 & -\frac \pi4\\\hline 1 & -1 & \sqrt 2 & -\frac \pi4\\\hline 1 & 1 & \sqrt 2 & \frac \pi4\\\hline \end{array} $$

Answer 1

One can choose any mapping desired from the unique data sequences to the symbol locations on the QAM constellation. It is common and usually best to Gray-code the mapping such that only one bit changes between adjacent constellation locations (those with minimum Euclidean distance). This way if there is a symbol error due to noise crossing an immediate decision threshold, there will only be one bit error.

See figure below showing the common mapping for QAM using Gray Coding from Wikipedia under "Gray Coding". Notice that the adjacent constellation points only differ by one bit. Under AWGN conditions at threshold error conditions with equi-probable symbols only one bit error would result with this mapping approach (the optimum error boundaries with equi-probable constellation points are mid way between the points shown).

For Gray-coded QPSK the mapping is quite trivial as illustrated by the figure from this link:

And here is a link to a simulation showing the significance in BER performance when Gray-coding QPSK.

The amplitude and phase for each possible symbol is directly taken from the constellation's I and Q values as shown in the diagrams above, and as traditionally done for any complex I, Q pair:

The amplitude is

$$A= \sqrt{I^2+Q^2}$$

And the phase is:

$$\phi = \mathrm{atan2}(Q,I)$$