Symbol Error rate for QAM (16, 64, 256,.., M-QAM)

In May 2008, we derived the theoretical symbol error rate for a general M-QAM modulation (inEmbedded.com,DSPDesignLine.comand dsplog.com) under Additive White Gaussian Noise. While re-reading that post, felt that the article is nice and warrants a re-run, using OFDM as the underlying physical layer. This post discuss the derivation of symbol error rate for a general M-QAM modulation. The companion Matlab script compares the theoretical and the simulated symbol error rate for 16QAM, 64QAM and 256QAM over OFDM in AWGN channel.

Enjoy and HAPPY NEW YEAR 2012 !!!

Quadrature Amplitude Modulation (QAM) schemes like 16-QAM, 64-QAM are used in typical wireless digital communications specifications likeIEEE802.11a,IEEE802.16d. In this post let us derive the equation for probability of symbol being in error for a general M-QAM constellation, given that the signal (symbol) to noise ratio is.

The general M-QAM constellation

The number of points in the constellation is defined as,whereis the number of bits in each constellation symbol. In this analysis, it is desirable to restrictto be an even number for the following reasons (Refer Sec 5.2.2 in[DIG-COMM-BARRY-LEE-MESSERSCHMITT]):

1. Half the bits are represented on the real axis and half the bits are represented on imaginary axis. The in-phase and quadrature signals are independentlevel Pulse Amplitude Modulation (PAM) signals. This simplifies the design of mapper.

2. 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.

Average energy of an M-QAM constellation

In a general M-QAM constellation whereandthe number of bits in each constellation is even, the alphabets used are:

, where.

For example, considering a 64-QAM () constellation,

and the alphabets are

.

For computing the average energy of the M-QAM constellation, let us proceed as follows:

(a) Find the sum of energy of the individual alphabets

(b) Each alphabet is usedtimes in the M-QAM constellation.

(c) So, to find the average energy fromconstellation symbols, divide the product of (a) and (b) by.The average energy is,

.

Plugging in the number for 64-QAM,

.

Plugging in the number for 16-QAM,

.

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

For computing 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: 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:

(a) Constellation points in the corner (red-square)

The number of constellation points in the corner in any M-QAM constellation is always 4, i.e

(b) Constellation points in the inside (magneta-diamond)

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