The probability of the symbol decoded being in error is,Īs can be seen from the above figure, the symbol in the inside is decoded correctly only if real part of lies from to and the imaginary part of lies from to. (c) Given that the total probability is always 1, for finding the probability of the real component lies within to, subtract the sum of (a) and (b) from 1. (b) Find the probability that the real component lies from to. (a) Find the probability that the real component lies from to. The probability of the real component falling with in 0 to 2 can be found by integrating the probability distribution function of two parts: The probability of correct demodulation is, The conditional probability distribution function (PDF) of given that the transmitted symbol is :Īs can be seen from the above figure, the symbol in the inside is decoded correctly only if real part of lies inbetween 0 to 2 and the imaginary part of lies inbetween 0 to 2. Additive White Gaussian Noise (AWGN) channelĪssume that the additive noise follows the Gaussian probability distribution function, (c) Constellation points neither at the corner, nor at the center (blue-star) The number of constellation points of this category is,įor example with M=64, there are 24 constellation points in the inside. The number of constellation points in the inside is,įor example with M=64, there are 36 constellation points in the inside. (b) Constellation points in the inside (magneta-diamond) The number of constellation points in the corner in any M-QAM constellation is always 4, i.e (a) Constellation points in the corner (red-square)
Finding the symbol error rateįor 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.įigure: Constellation plot for 64-QAM modulation (without the scaling factor of )Īs can be seen from the above figure, there are three types of constellation points in a general M-QAM constellation: The average energy is,įrom 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. (c) So, to find the average energy from constellation symbols, divide the product of (a) and (b) by. (b) Each alphabet is used times in the M-QAM constellation. (a) Find the sum of energy of the individual alphabets In a general M-QAM constellation where and the number of bits in each constellation is even, the alphabets used are:įor example, considering a 64-QAM ( ) constellation,įor computing the average energy of the M-QAM constellation, let us proceed as follows:
Note that the above square constellation is not the most optimal scheme for a given signal to noise ratio. For decoding, symbol decisions may be applied independently on the real and imaginary axis, simplifying the receiver implementation. The in-phase and quadrature signals are independent level Pulse Amplitude Modulation (PAM) signals. Half the bits are represented on the real axis and half the bits are represented on imaginary axis.
In this analysis, it is desirable to restrict to be an even number for the following reasons (Refer Sec 5.2.2 in ):ġ. The number of points in the constellation is defined as, where is the number of bits in each constellation symbol. 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.
Quadrature Amplitude Modulation (QAM) schemes like 16-QAM, 64-QAM are used in typical wireless digital communications specifications like IEEE802.11a, IEEE802.16d. The companion Matlab script compares the theoretical and the simulated symbol error rate for 16QAM, 64QAM and 256QAM over OFDM in AWGN channel. This post discuss the derivation of symbol error rate for a general M-QAM modulation. While re-reading that post, felt that the article is nice and warrants a re-run, using OFDM as the underlying physical layer. In May 2008, we derived the theoretical symbol error rate for a general M-QAM modulation (in, and ) under Additive White Gaussian Noise.
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