QPSK BER - equivalent baseband

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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.

QPSK / 4-QAM transmission - Monte Carlo simulation of bit error rate for AWGN channel. For best simulation performance this equivalent baseband simulation applies no pulse shaping and uses one sample per symbol.

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.