# QPSK BER for AWGN channel

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## QPSK BER for AWGN channel

The bit error probability of QPSK over an AWGN channel is given by:

${p}_{b}=\frac{1}{2}erfc\left(\sqrt{\frac{{E}_{b}}{{N}_{0}}}\right)$

${p}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right)$

$\frac{{E}_{b}}{{N}_{0}}$

[dB]

Bit error probability

${p}_{b}$

-2 1,306E-01
0 7,865E-02
2 3,751E-02
4 1,250E-02
6 2,388E-03
8 1,909E-04
10 3,872E-06
 $\frac{{E}_{b}}{{N}_{0}}$ Energy per bit to noise power spectral density ratio $\frac{{E}_{b}}{{N}_{0}}=\frac{1}{n}\frac{{E}_{s}}{{N}_{0}}$ $\frac{{E}_{s}}{{N}_{0}}$ Energy per symbol to noise power spectral density $n$ Number of bits per symbol, e.g. $n=2$ for QPSK $n={\mathrm{log}}_{2}m$ $m$ Number of different modulation symbols $m={2}^{n}$
Energy to noise power spectral density

 ${N}_{0}$ Noise power spectral density ${N}_{0}=\frac{N}{B}$ $N$ noise power $B$ bandwidth
Noise
QPSK BER for AWGN channel

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

The simulation starts with a setting of ${E}_{b}/{N}_{0}=0dB$ (see marked line in the table below). This yields to this bit error probability:

 ${p}_{b}=\frac{1}{2}erfc\left(\sqrt{\frac{{E}_{b}}{{N}_{0}}}\right)=\frac{1}{2}erfc\left(1\right)=7,865E-02$ Measured BER approximates the analytical bit error probability ${p}_{b}$
$\frac{{E}_{b}}{{N}_{0}}$

[dB]

Bit error probability

${p}_{b}$

-2 1,306E-01
0 7,865E-02
2 3,751E-02
4 1,250E-02
6 2,388E-03
8 1,909E-04
10 3,872E-06
Analytical bit error probability ${p}_{b}$ for QPSK

## Experiment

Now adjust ${E}_{b}/{N}_{0}$ . Measure the corresponding BER and compare it to the analytical bit error probability. Simulation - Settings (F11)   Measured BER approximates the analytical bit error probability: ${E}_{b}/{N}_{0}=-2,2,4dB$

## Note

• 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 ${E}_{b}$ $1\mu V{s}^{2}$

The initial noise power-spectral-density of ${N}_{0}=1\mu {V}^{2}/Hz$ results in:

$\frac{{E}_{b}}{{N}_{0}}=\frac{1\mu V{s}^{2}}{1\mu {V}^{2}/Hz}=1=0dB$
• This BER simulation is quite slow as it implements quadrature modulation and pulse shaping. To boost simulation speed you might change to QPSK BER - equivalent baseband without pulse shaping.

## Next steps

• Vary the pulse shape. Note that the pulse shape has no effect on the BER but the transmit spectrum. Simulation - Setup (F12)
• Switch to equivalent baseband. Note that the modem doesn’t affect the BER performance but the shifts the transmit spectrum to the base band. • Switch to modem none. Note that the modem doesn’t affect the BER performance. There’s only one sample per symbol and no pulse shaping, so the simulation is much faster. Adjust ${E}_{b}/{N}_{0}$ and measure the corresponding BER. Simulation - Settings (F11)   Measured BER approximates the analytical bit error probability: ${E}_{b}/{N}_{0}=-2,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}4\text{\hspace{0.17em}}dB$ Resize the BER-meter to see the number of bit errors and the total number of transferred bits