GATE 2020 EC – Question 58
In a digital communication system, a symbol $S$ randomly chosen from the set $\{s_1,s_2,s_3,s_4\}$ is transmitted. It is given that $s_1=-3$, $s_2=-1$, $s_3=+1$ and $s_4=+2$. The received symbol is $Y=S+W$. $W$ is a zero-mean unit-variance Gaussian random variable and is independent of $S$. $P_i$ is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol $S=s_i$. The index $i$ for which the conditional symbol error probability $P_i$ is the highest is ________.
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Correct answer: 3
Explanation
The ML decision thresholds are the midpoints $-2$, $0$ and $1.5$. Then $P_1=Q(1)$, $P_2=2Q(1)$, $P_3=Q(1)+Q(0.5)$ (the nearer threshold at distance 0.5 on the right and 1 on the left), and $P_4=Q(0.5)$. With $Q(1)=0.159$ and $Q(0.5)=0.309$: $P_1=0.159$, $P_2=0.317$, $P_3=0.467$, $P_4=0.309$. The highest is $i=3$.