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

$p_{B} = \frac{4}{n} (1 - \frac{1}{\sqrt{M}}) Q (\sqrt{\frac{3 n}{M - 1} \cdot \frac{E_{B}}{N_{0}}})$ (see QAM - Wikipedia)

or

$p_{b} = \frac{2}{l d (M)} (1 - \frac{1}{\sqrt{M}}) e r f c (\sqrt{\frac{3 l d (M)}{2 (M - 1)} \cdot \frac{E_{B}}{N_{0}}})$ .

See QAM bit error probability calculator and 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	$n = l d (M) = \log_{2} (M)$
$M$	Number of different modulation symbols, size of modulation constellation	$M = 2^{n}$

Definitions

$\frac{E_{b}}{N_{0}}$ [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

Analytical bit error probability

p_{b}

for M-QAM

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 $2^{n d}$ bit error of symbol errors

The following analytical bit error probability considers a $2^{n d}$ 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

Analytical bit error probability

p_{b}

for M-QAM considering

2^{n d}

bit errors

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

Quadrature amplitude modulation (QAM) transmission

Start

The simulation starts with 4-QAM and $\frac{E_{b}}{N_{0}} = 0 d B$ - see marked cell in table below.

Measured BER for 4-QAM, E<sub>b</sub>/N<sub>0</sub> = 0dB

Measured BER approximates the analytical bit error probability

p_{b}

$\frac{E_{b}}{N_{0}}$ [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

Analytical bit error probability

p_{b}

for M-QAM considering

2^{n d}

bit errors

Experiment

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

Simulation - Settings (F11)

Simulation - Setup (F12): Set M, the size of the modulation constellation.

Note

This BER simulation is quite slow as it implements quadrature modulation and pulse shaping.

To boost simulation speed switch to modem none in Simulation - Setup (F12)

Adjusting

E_{b} / N_{0}

adapts the noise power spectral density. These settings are not modified:

Transmitting power	$1 V^{2}$
Bit duration $T_{b i t}$	$1 μ s$
Energy per bit $E_{b}$	$1 μ V s^{2}$

Next steps

The analytical bit error probability $p_{b}$ considering $2^{n d}$ bit errors is a much better approximation in cases of low signal quality. For instance, $E_{b} / N_{0} = - 2 d B$ and 16-QAM: $p_{b} = 0,1873$

Measured BER

Quadrature amplitude modulation (QAM) transmission

Select the modulation scheme, enter E_b/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.

QAM modulator demodulator baseband

Adjust $E_{b} / N_{0}$ and measure the corresponding BER.

Simulation - Settings (F11)

Simulation - Setup (F12): Set M, the size of the modulation constellation

QAM BER for AWGN channel

QAM bit error probability for AWGN channel

Assumption: one bit error per symbol error

Consider $2^{n d}$ bit error of symbol errors

Start

Experiment

Note

Next steps

Start the QAM baseband transmission