GATE 2022 EC – Question 63
Consider a channel over which either symbol $x_A$ or symbol $x_B$ is transmitted. Let the output of the channel $Y$ be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for $Y$ given $x_A$ and $x_B$ are:
$$f_{Y|x_A}(y)=e^{-(y+1)}u(y+1),$$
$$f_{Y|x_B}(y)=e^{(y-1)}\left(1-u(y-1)\right),$$
where $u(\cdot)$ is the standard unit step function. The probability of symbol error for this system is ________ (rounded off to two decimal places).
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Show answer and explanation
Correct answer: 0.22 to 0.24
Explanation
Assuming equally likely symbols, ML decides the symbol with the larger density. For $y<-1$ only $x_B$ has non-zero density, and for $y>1$ only $x_A$. For $-1<y<1$, $f_A>f_B$ when $-(y+1)>y-1$, i.e. $y<0$. So the decision is $A$ for $-1<y<0$ or $y>1$, and $B$ otherwise. $P(\text{error}|A)=\int_0^1e^{-(y+1)}dy=e^{-1}(1-e^{-1})=0.2325$, and $P(\text{error}|B)=\int_{-1}^0e^{y-1}dy=e^{-1}(1-e^{-1})=0.2325$. So $P_e=0.23$.