Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbol interference. The constellation points are picked equally likely. Let $P\text{(QAM)}$ denote the average transmit power and $P_{e}\text{(QAM)}$ denote the average probability of error under optimal decoding.
Which of the following is TRUE?
- $P\left(\mathrm{QAM}_1\right)>P\left(\mathrm{QAM}_2\right), P_e\left(\mathrm{QAM}_1\right)<P_e\left(\mathrm{QAM}_2\right)$.
- $P\left(\mathrm{QAM}_1\right)<P\left(\mathrm{QAM}_2\right), P_e\left(\mathrm{QAM}_1\right)>P_e\left(\mathrm{QAM}_2\right)$.
- $P\left(\mathrm{QAM}_1\right)>P\left(\mathrm{QAM}_2\right), P_e\left(\mathrm{QAM}_1\right)>P_e\left(\mathrm{QAM}_2\right)$.
- $P\left(\mathrm{QAM}_1\right)<P\left(\mathrm{QAM}_2\right), P_e\left(\mathrm{QAM}_1\right)<P_e\left(\mathrm{QAM}_2\right)$.
- $P\left(\mathrm{QAM}_1\right)=P\left(\mathrm{QAM}_2\right), P_e\left(\mathrm{QAM}_1\right)=P_e\left(\mathrm{QAM}_2\right)$.