Quadrature Amplitude Modulation (QAM) schemes like 4-QAM (QPSK), 16-QAM and 64-QAM are used in typical wireless digital communications specifications like IEEE802.11a, IEEE802.16d. In this post we'll derive the probability of a symbol being in error for a general M-QAM constellation, given that the signal (symbol) to noise ratio is  .
Defining the general M-QAM constellation
The number of points in the constellation is defined as,  where  is the number of bits in each constellation symbol. In this analysis, it is desirable to restrict to an even number for the following reasons (Refer Sec 5.2.2 in [1]):
- Half the bits are represented on the real axis and half the bits are represented on the imaginary axis. The in-phase and quadrature signals are independent
 level Pulse Amplitude Modulation (PAM) signals. This simplifies the design of the mapper.
- For decoding, symbol decisions may be applied independently on the real and imaginary axis, simplifying the receiver implementation.
Note that the above square constellation is not the most optimal scheme for a given signal to noise ratio. However, considering that typical implementations prefer the reduced complexity, let us keep this assumption.
Average energy of an M-QAM constellation
In a general M-QAM constellation where and are even, the alphabets used are:
\pm(2m-1)j\right}) , where  .
For example, considering a 64-QAM ( ) constellation,  and the alphabets are
 .
To compute the average energy of the M-QAM constellation:
- Find the sum of energy of the individual alphabets
+j(2m-1)\right|^2\\&=&\frac{\sqrt{M}}{3}M-1\end{eqnarray})
- Each alphabet is used
 times in the M-QAM constellation.
Thus, to find the average energy from  constellation symbols, divide the product of (1) and (2) above by. The average energy is,
 .
Plugging in the number for 64-QAM,
=42) .
Plugging in the number for 16-QAM,
=10) .
From the above explanations, it is reasonably intuitive to guess that the scaling factor of  ,  which is seen along with 16-QAM, 64-QAM constellations, respectively, is for normalizing the average transmit power to unity.
Finding the symbol error rate
To compute the symbol error rate for an M-QAM modulation, let us consider the 64-QAM constellation as shown in the figure below and extend it to the M-QAM case.
Figure 1. Constellation plot for 64-QAM modulation (without the scaling factor of )
As can be seen from the above figure, there are three types of constellation points in a general M-QAM constellation:
- Constellation points in the corner (red squares). The number of constellation points in the corner in any M-QAM constellation is always 4, i.e.,
- Constellation points in the inside (magneta diamonds). The number of constellation points in the inside is,
(\sqrt{M}-2)) .
For example with M=64, there are 36 constellation points in the inside.
- Constellation points neither at the corner, nor at the center (blue stasr). The number of constellation points of this category is,
) .
For example with M=64, there are 24 constellation points in the inside.
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