QPSK BER for AWGN channel

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Quadrature phase-shift keying (QPSK) transmission - bit error rate for AWGN channel

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

$p b = 1 2 erfc( E b N 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaa caaIYaaaaiaadwgacaWGYbGaamOzaiaadogadaqadaqaamaakaaaba WaaSaaaeaacaWGfbWaaSbaaSqaaiaadkgaaeqaaaGcbaGaamOtamaa BaaaleaacaaIWaaabeaaaaaabeaaaOGaayjkaiaawMcaaaaa@44C0@$

$p b =Q( 2 E b N 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaGccqGH9aqpcaWGrbWaaeWaaeaadaGc aaqaamaalaaabaGaaGOmaiaadweadaWgaaWcbaGaamOyaaqabaaake aacaWGobWaaSbaaSqaaiaaicdaaeqaaaaaaeqaaaGccaGLOaGaayzk aaaaaa@4117@$

$E b N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaala aabaGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWc baGaaGimaaqabaaaaaaa@3ACA@$

[dB]

Bit error probability

$p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaaaaa@3922@$

6 2,388E-03
 $E b N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaala aabaGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWc baGaaGimaaqabaaaaaaa@3ACA@$ Energy per bit to noise power spectral density ratio $N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eadaWgaaWcbaGaaGimaaqabaaaaa@38D3@$ Noise power spectral density $N 0 = N B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eadaWgaaWcbaGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaad6eaaeaa caWGcbaaaaaa@3B8D@$ $N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eaaaa@37ED@$ Noise power $B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadk eaaaa@37E1@$ Bandwidth

In this experiment the bit error rate (BER) vs $E b / N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaGccaGGVaGaamOtamaaBaaaleaacaaI Waaabeaaaaa@3B6D@$ of QPSK over an AWGN channel is analyzed.

Quadrature phase-shift keying (QPSK) transmission - Monte Carlo simulation of bit error rate for AWGN channel

Start

The simulation starts with a setting of $E b / N 0 =0dB MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaGccaGGVaGaamOtamaaBaaaleaacaaI Waaabeaakiabg2da9iaaicdacaWGKbGaamOqaaaa@3EE7@$. This yields to this bit error probability:

 $p b = 1 2 erfc( E b N 0 )= 1 2 erfc( 1 )=7,865E−02 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaa caaIYaaaaiaadwgacaWGYbGaamOzaiaadogadaqadaqaamaakaaaba WaaSaaaeaacaWGfbWaaSbaaSqaaiaadkgaaeqaaaGcbaGaamOtamaa BaaaleaacaaIWaaabeaaaaaabeaaaOGaayjkaiaawMcaaiabg2da9m aalaaabaGaaGymaaqaaiaaikdaaaGaamyzaiaadkhacaWGMbGaam4y amaabmaabaGaaGymaaGaayjkaiaawMcaaiabg2da9iaaiEdacaGGSa GaaGioaiaaiAdacaaI1aGaamyraiabgkHiTiaaicdacaaIYaaaaa@552A@$
Measured BER approximates the analytical bit error probability $p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaaaaa@3922@$

$E b N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaala aabaGaamyramaaBaaaleaacaWGIbaabeaaaOqaaiaad6eadaWgaaWc baGaaGimaaqabaaaaaaa@3ACA@$

[dB]

Bit error probability

$p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaaaaa@3922@$

-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 $p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadc hadaWgaaWcbaGaamOyaaqabaaaaa@3922@$ for QPSK

Experiment

Now adjust $E b / N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaGccaGGVaGaamOtamaaBaaaleaacaaI Waaabeaaaaa@3B6D@$ . Measure the corresponding BER and compare it to the analytical bit error probability.

Simulation - Settings (F11)

Measured BER approximates the analytical bit error probability: ${E}_{b}/{N}_{0}=-2,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}4dB$

Note

• Adjusting $E b / N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaGccaGGVaGaamOtamaaBaaaleaacaaI Waaabeaaaaa@3B6D@$ adapts the noise power spectral density. These settings are not modified:
 Transmitting power $1 V 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaik dacaWGwbWaaWbaaSqabeaacaaIYaaaaaaa@399A@$ Bit duration $T b i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaads fadaWgaaWcbaGaamOyaiaadMgacaWG0baabeaaaaa@3AED@$ $1 μ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaik dacqaH8oqBcaWGZbaaaa@3A84@$ Energy per bit $E b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaaaaa@38F7@$ $1 μ V s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaik dacqaH8oqBcaWGwbGaam4CamaaCaaaleqabaGaaGOmaaaaaaa@3C48@$

The initial noise power-spectral-density of $N 0 = 1 μ V 2 / H z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6 eadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIXaGaeqiVd0MaamOv amaaCaaaleqabaGaaGOmaaaakiaac+cacaWGibGaamOEaaaa@40A1@$ results in:

$\frac{{E}_{b}}{{N}_{0}}=\frac{1\mu V{s}^{2}}{1\mu {V}^{2}/Hz}=1=0\text{\hspace{0.17em}}dB$
• 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 $E b / N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaGccaGGVaGaamOtamaaBaaaleaacaaI Waaabeaaaaa@3B6D@$ and measure the corresponding BER.

Simulation - Settings (F11)

Measured BER approximates the analytical bit error probability: $E b / N 0 =−2, 2, 4 dB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBamXvP5wqonvsaeHbd9wDYLwzYbqe e0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9fr Fj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadw eadaWgaaWcbaGaamOyaaqabaGccaGGVaGaamOtamaaBaaaleaacaaI Waaabeaakiabg2da9iabgkHiTiaaikdacaGGSaGaaGjbVlaaikdaca GGSaGaaGjbVlaaisdacaaMc8Uaamizaiaadkeaaaa@4754@$
Resize the BER-meter to see the number of bit errors and the total number of transferred bits