Consider two communication systems $C_{1}$ and $C_{2}$ that use pulse amplitude modulation $\text{(PAM)}$, $P A M_{1}$ and $P A M_{2}$. Let the distance between any two points of $P A M_{1}$ be $d$, and $P A M_{2}$ be $2 d$, respectively. Assume that $C_{1}$ and $C_{2}$ are corrupted by additive white Gaussian noise of variance $\sigma^{2}$ and $2 \sigma^{2}$, respectively. Let $P_{1}$ and $P_{2}$ be the probability of error for $C_{1}$ and $C_{2}$. Then
- $P_{1}=P_{2}$.
- $P_{1} < P_{2}$
- $P_{1}>P_{2}$.
- $P_{1}=P_{2}+\frac{1}{2}$
- None of the above.