GATE 2018 EC – Question 27
A binary source generates symbols $X\in\{-1,1\}$ which are transmitted over a noisy channel. The probability of transmitting $X=1$ is 0.5. Input to the threshold detector is $R=X+N$. The probability density function $f_N(n)$ of the noise $N$ is shown below.
If the detection threshold is zero, then the probability of error (correct to two decimal places) is ________.
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Correct answer: 0.12 to 0.14
Explanation
The noise pdf is a triangle on $[-2,2]$ with peak 0.5, so $f_N(n)=0.5\left(1-\frac{|n|}{2}\right)$. An error for $X=1$ needs $N<-1$, and for $X=-1$ needs $N>1$, each with probability $\int_1^2 0.5\left(1-\frac n2\right)dn=0.125$. So $P_e=0.125\approx0.13$.