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# Symbol Error Rate (SER) for QPSK (4-QAM) modulation

by on November 6, 2007

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.

Background

Consider that the alphabets used for a QPSK (4-QAM) is $\alpha_{QPSK}=\left{\pm 1\ \pm 1j\right}$ (Refer example 5-35 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]).

Figure: Constellation plot for QPSK (4-QAM) constellation

The scaling factor of $\sqrt{\frac{E_s}{2}}$ is for normalizing the average energy of the transmitted symbols to 1, assuming that all the constellation points are equally likely.

## Noise model

Assuming that the additive noise $n$ follows the Gaussian probability distribution function,

$p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}$ with $\mu=0$ and $\sigma^2 = \frac{N_0}{2}$.

## Computing the probability of error

Consider the symbol $s_2$

The conditional probability distribution function (PDF) of $y$given $s_2$ was transmitted is:

$p(y|s_2) = \frac{1}{\sqrt{\pi N_0}}e^{\frac{-\left(y-\sqrt{\frac{E_s}{2}}\right)^2}{N_0}$.

Figure: Probability density function for QPSK (4QAM) modulation

As can be seen from the above figure, the symbol $s_2$ is decoded correctly only if $y$falls in the area in the hashed region i.e.

$p(c|s_2) = p(\Re{y>0}|s_2)p(\Im{y>0}|s_2)$.

Probability of real component of $y$ greater than 0, given $s_2$ was transmitted is (i.e. area outside the red region)

$p(\Re y>0|s_2) = 1-\frac{1}{\sqrt{\pi N_0}}\int_{-\infty}^0e^{\frac{-(\Re y-\sqrt{\frac{E_s}{2}})^2}{N_0}}dy=1-\frac{1}{2}erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right)$, where

the complementary error function, $erfc(x) = \frac{2}{\sqrt{\pi}}\int_x^\infty e^{-x^2}dx$.

Similarly, probability of imaginary component of $y$ greater than 0, given $s_2$ was transmitted is (i.e. area outside the blue region).

$p(\Im y>0|s_2) = 1-\frac{1}{\sqrt{\pi N_0}}\int_{-\infty}^0e^{\frac{-(\Im y-\sqrt{\frac{E_s}{2}})^2}{N_0}}dy=1-\frac{1}{2}erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right)$.

The probability of $s_2$ being decoded correctly is,

$\begin{eqnarray}p(c|s_2) & = & \left[1-\frac{1}{2}erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right)\right]^2\\ & = & \left[1-\frac{2}{2}erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right) + \frac{1}{4}erfc^2\left({\sqrt{\frac{E_s}{2N_0}}}\right)\right]\\ & = & 1-erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right) + \frac{1}{4}erfc^2\left({\sqrt{\frac{E_s}{2N_0}}}\right)\end{eqnarray}$.

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,

$\begin{eqnarray}\mathbf{P}_{QPSK} & = & 1 - p(c|s_2)\\ & = & 1 - \left[1-erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right) + \frac{1}{4}erfc^2\left({\sqrt{\frac{E_s}{2N_0}}}\right)\right]\\ & = & erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right) - \frac{1}{4}erfc^2\left({\sqrt{\frac{E_s}{2N_0}}}\right)\end{eqnarray}$.

For higher values of $\frac{E_s}{N_0}$, the second term in the equation becomes negligible and the probability of error can be approximated as,

$\begin{eqnarray}\mathbf{P}_{QPSK} \approx erfc\left({\sqrt{\frac{E_s}{2N_0}}}\right)\end{eqnarray}$.

## Simulation Model

Simple Matlab/Octave script for generating QPSK transmission, adding white Gaussian noise and decoding the received symbol for various $\frac{E_s}{N_0}$ values.

Figure: Symbol Error Rate for QPSK (4QAM) modulation

Observations

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 $\frac{E_s}{N_0}$ for achieving a symbol error rate of $10^{-3}$.

## Reference

Hope this helps.

Krishna

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