# QAM bit error probability for AWGN channel

**Assumption: one bit error per symbol error**

The bit error probability ${p}_{b}$ of M-QAM over an AWGN channel is given by

${p}_{B}=\frac{4}{n}\left(1-\frac{1}{\sqrt{M}}\right)Q\left(\sqrt{\frac{3n}{M-1}\cdot \frac{{E}_{B}}{{N}_{0}}}\right)$ (see QAM - Wikipedia)

or

${p}_{b}=\frac{2}{ld\left(M\right)}\left(1-\frac{1}{\sqrt{M}}\right)erfc\left(\sqrt{\frac{3ld\left(M\right)}{2\left(M-1\right)}\cdot \frac{{E}_{B}}{{N}_{0}}}\right)$ .

See Error function and Q-function.

${E}_{b}$ | Energy per bit | |

${N}_{0}$ | Noise power spectral density | |

$n$ | Number of bits per symbol, e.g. $n=2$ for 4-QAM | $$n=ld(M)={\mathrm{log}}_{2}(M)$$ |

$M$ | Number of different modulation symbols, size of modulation constellation | $M={2}^{n}$ |

$$\frac{{E}_{b}}{{N}_{0}}$$
[dB] |
4 QAM
${p}_{b}$ |
16QAM
${p}_{b}$ |
64QAM ${p}_{b}$ |
256QAM
${p}_{b}$ |
---|---|---|---|---|

-2 | 1,306E-01 | 1,790E-01 | 1,957E-01 | 1,893E-01 |

0 | 7,865E-02 | 1,392E-01 | 1,730E-01 | 1,779E-01 |

2 | 3,751E-02 | 9,756E-02 | 1,461E-01 | 1,639E-01 |

4 | 1,250E-02 | 5,862E-02 | 1,158E-01 | 1,469E-01 |

6 | 2,388E-03 | 2,787E-02 | 8,347E-02 | 1,267E-01 |

8 | 1,909E-04 | 9,247E-03 | 5,232E-02 | 1,033E-01 |

10 | 3,872E-06 | 1,754E-03 | 2,653E-02 | 7,781E-02 |

Note that the real BER is significantly greater than this analytical bit error probability in cases of small ${E}_{b}/{N}_{0}$ and big M (see marked cells in table). In contrast to the assumption of only one bit error per symbol error, a considerable number of multiple bit errors per symbol error occur in this scenario.

**Consider
${2}^{nd}$
bit error of symbol errors**

The following analytical bit error probability considers a
${2}^{nd}$
bit error occurring when non-adjacent constellation points are detected. The probability of this event is added to the probability above.

This approach is a much better approximation in cases of low signal quality. However, the assumption of two bit errors in the case of non-adjacent symbol errors is an approximation as well. In the case of three points further there might be one or three bit errors. Simulation suggests that this bit error probability is an upper limit.

$$\frac{{E}_{b}}{{N}_{0}}$$
[dB] |
4 QAM
${p}_{b}$ |
16QAM
${p}_{b}$ |
64QAM
${p}_{b}$ |
256QAM
${p}_{b}$ |
---|---|---|---|---|

-2 | 1,306E-01 | 1,873E-01 | 2,464E-01 | 2,909E-01 |

0 | 7,865E-02 | 1,410E-01 | 2,002E-01 | 2,561E-01 |

2 | 3,751E-02 | 9,774E-02 | 1,570E-01 | 2,178E-01 |

4 | 1,250E-02 | 5,862E-02 | 1,185E-01 | 1,786E-01 |

6 | 2,388E-03 | 2,787E-02 | 8,382E-02 | 1,412E-01 |

8 | 1,909E-04 | 9,247E-03 | 5,233E-02 | 1,079E-01 |

10 | 3,872E-06 | 1,754E-03 | 2,653E-02 | 7,860E-02 |

In this experiment the bit error rate (BER) vs ${E}_{b}/{N}_{0}$ of M-QAM over an AWGN channel is analyzed.

## Start

The simulation starts with 4-QAM and $\frac{{E}_{b}}{{N}_{0}}=0dB$ - see marked cell in table below.

$$\frac{{E}_{b}}{{N}_{0}}$$
[dB] |
4-QAM
${p}_{b}$ |
16-QAM
${p}_{b}$ |
64-QAM
${p}_{b}$ |
256-QAM
${p}_{b}$ |
---|---|---|---|---|

-2 | 1,306E-01 | 1,873E-01 | 2,464E-01 | 2,909E-01 |

0 | 7,865E-0 | 1,410E-01 | 2,002E-01 | 2,561E-01 |

2 | 3,751E-02 | 9,774E-02 | 1,570E-01 | 2,178E-01 |

4 | 1,250E-02 | 5,862E-02 | 1,185E-01 | 1,786E-01 |

6 | 2,388E-03 | 2,787E-02 | 8,382E-02 | 1,412E-01 |

8 | 1,909E-04 | 9,247E-03 | 5,233E-02 | 1,079E-01 |

10 | 3,872E-06 | 1,754E-03 | 2,653E-02 | 7,860E-02 |

## Experiment

Adjust ${E}_{b}/{N}_{0}$ and M. Measure the corresponding BER and compare it to the analytical bit error probability.

## Note

- This BER simulation is quite slow as it implements quadrature modulation and pulse shaping.
- Adjusting
${E}_{b}/{N}_{0}$
adapts the noise power spectral density. These settings are not modified:
Transmitting power $1{V}^{2}$ Bit duration ${T}_{bit}$ $1\mu s$

Energy per bit ${E}_{b}$ $1\mu V{s}^{2}$

## Next steps

- The analytical bit error probability ${p}_{b}$ considering ${2}^{nd}$ bit errors is a much better approximation in cases of low signal quality. For instance, ${E}_{b}/{N}_{0}=-2\text{\hspace{0.17em}}dB$ and 16-QAM: ${p}_{b}=\mathrm{0,1873}$

Select the modulation scheme, enter Eb/N0 and start the QAM baseband transmission. This simulation app implements an M-QAM baseband transmission. The bit error rate over an AWGN channel can be measured. The baseband simulation is highly performant.

Adjust ${E}_{b}/{N}_{0}$ and measure the corresponding BER.