GATE 2022 EC Q.63: solution and answer

GATE 2022 · ECQ.63 · 2 marks · NAT

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: $f_{Y ∣ x_{A}} (y) = e^{- (y + 1)} u (y + 1)$ , $f_{Y ∣ x_{B}} (y) = e^{(y - 1)} (1 - u (y - 1))$ , 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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Answer and solution

ANSWER

The probability of symbol error is $e^{- 1} (1 - e^{- 1}) \approx 0.23$ .

Official answer: 0.22 to 0.25Source: Official answer key, GATE 2022 EC (IIT Kharagpur)

Equally likely symbols are assumed (the question gives no priors, and ML detection is the same as MAP for equal priors). The supports are $f_{A} (y) = e^{- (y + 1)}$ for $y > - 1$ (zero otherwise) and $f_{B} (y) = e^{y - 1}$ for $y < 1$ (zero otherwise).
ML rule: decide $x_{A}$ if $f_{A} (y) > f_{B} (y)$ , otherwise decide $x_{B}$ .
For $y < - 1$ only $f_{B}$ is nonzero, so decide $x_{B}$ . For $y > 1$ only $f_{A}$ is nonzero, so decide $x_{A}$ .
In the overlap $- 1 < y < 1$ : $f_{A} > f_{B} ⟺ - (y + 1) > y - 1 ⟺ y < 0$ . So decide $x_{A}$ for $- 1 < y < 0$ and $x_{B}$ for $0 < y < 1$ .
Decision regions: $x_{A}$ on $(- 1, 0) \cup (1, \infty)$ and $x_{B}$ on $(- \infty, - 1) \cup (0, 1)$ .
Error given $x_{A}$ : $Y$ lands in the $x_{B}$ region. Only $(0, 1)$ has nonzero probability under $f_{A}$ , so $P (e ∣ x_{A}) = \int_{0}^{1} e^{- (y + 1)} d y = e^{- 1} (1 - e^{- 1}) \approx 0.2325.$
Error given $x_{B}$ : $Y$ lands in the $x_{A}$ region. Only $(- 1, 0)$ has nonzero probability under $f_{B}$ , so $P (e ∣ x_{B}) = \int_{- 1}^{0} e^{y - 1} d y = e^{- 1} (1 - e^{- 1}) \approx 0.2325.$
Average over the equal priors: $P_{e} = \frac{1}{2} (0.2325) + \frac{1}{2} (0.2325) = e^{- 1} (1 - e^{- 1}) \approx 0.23.$

KEY IDEA

The ML detector picks the symbol with the larger likelihood at each $y$ . Compare the two densities on their overlap, then integrate each density over the other symbol's decision region.

WATCH OUT FOR

Forgetting that each density is zero outside its support. Students then integrate over the wrong region, for example $(- \infty, - 1)$ for $x_{A}$ , or $(1, \infty)$ for $x_{B}$ , or they pick the wrong threshold. Another slip is to treat the decision boundary as $y = \pm 1$ instead of solving $f_{A} = f_{B}$ , which gives $y = 0$ .

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GATE 2022 EC Q.63: solution and answer

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