Given that we have discussed symbol error rate probability for a 4-PAM modulation, let us know focus on finding the symbol error probability for a QPSK (4-QAM) modulation scheme.
Consider that the alphabets used for a QPSK (4-QAM) is (Refer example 5-35 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).
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Figure: Constellation plot for QPSK (4-QAM) constellation
The scaling factor of is for normalizing the average energy of the transmitted symbols to 1, assuming that all the constellation points are equally likely.
Assuming that the additive noise follows the Gaussian probability distribution function,
with and .
Computing the probability of error
Consider the symbol
The conditional probability distribution function (PDF) of given was transmitted is:
Figure: Probability density function for QPSK (4QAM) modulation
As can be seen from the above figure, the symbol is decoded correctly only if falls in the area in the hashed region i.e.
Probability of real component of greater than 0, given was transmitted is (i.e. area outside the red region)
the complementary error function, .
Similarly, probability of imaginary component of greater than 0, given was transmitted is (i.e. area outside the blue region).
The probability of being decoded correctly is,
Total symbol error probability
The symbol will be in error, it atleast one of the symbol is decoded incorrectly. The probability of symbol error is,
For higher values of , the second term in the equation becomes negligible and the probability of error can be approximated as,
Simple Matlab/Octave script for generating QPSK transmission, adding white Gaussian noise and decoding the received symbol for various values.
Click here to download: Matlab/Octave script for computing the symbol error rate for QPSK modulation
Figure: Symbol Error Rate for QPSK (4QAM) modulation
1. Can see good agreement between the simulated and theoretical plots for 4-QAM modulation
2. When compared with 4-PAM modulation, the 4-QAM modulation requires only around 2dB lower for achieving a symbol error rate of .
Hope this helps.