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For each simulation run, N data and interfering symbols are randomly generated assuming shifted 16-QAM modulation constellation is used (M=16).
We can see that the resulting modulation constellation is shifted along the horizontal axis, which increases the average energy per symbol of the modulation constellation.
This easily follows from the proof of previous step, where we showed that if the modulation constellation is symmetric then (mathcal {A}) can be represented by (32).
We assume (mathrm {g}:mathcal {K}rightarrow mathcal {K}) does not exist such that (frac {x_{k}}{x_{mathrm {g} k)}}=c), but the modulation constellation is symmetric with respect to origin.
Firstly, we need to show that if a bijective function (mathrm {g}:mathcal {K}rightarrow mathcal {K}) exists such that (frac {x_{k}}{x_{mathrm {g} k)}}=c), then the modulation constellation is symmetric with respect to origin.
Then a modulation constellation is symmetric with respect to the origin if and only if (mathrm {f} -x_{k})=-mathrm {f} -x_{k} forall x_{k} in mathcal {A})[31].
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Also, the channel response and the modulation constellation are analyzed to give a clear picture on how the proposed scheme will perform in an indoor VLC environment.
Step 3: Thirdly, we show that the condition (frac {x_{k}}{x_{mathrm {g} k)}}=c forall k in mathcal {K}) is equivalent to the modulation constellation being symmetric around the origin.
Finally, using Definition 2, we show that the condition (frac {x_{k}}{x_{mathrm {g} k)}}=c forall k in mathcal {K}) is equivalent to the modulation constellation being symmetric with respect to the origin.
The optimum design of such modulation constellations is outside the scope of this work.
Modulation constellations are adopted from IEEE 802.11a and the related parameters (symbol period, number of subcarrier) are used in calculating throughput [19].
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