GATE 2022 EC – Question 64
Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, $f(x)$, as shown in the figure.
Consider a 1 bit quantizer that maps positive samples to value $\alpha$ and others to value $\beta$. If $\alpha^*$ and $\beta^*$ are the respective choices for $\alpha$ and $\beta$ that minimize the mean square quantization error, then $(\alpha^*-\beta^*)=$ ________ (rounded off to two decimal places).
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Show answer and explanation
Correct answer: 1.16 to 1.18
Explanation
The total area is 1: the triangle on $[-2,0]$ plus the rectangle on $[0,1]$ of height $h$ gives $\frac12(2h)+h=2h=1$, so $h=0.5$. The MSE-optimal reconstruction levels are the centroids of each cell. For $x>0$ the density is uniform on $[0,1]$, so $\alpha^*=0.5$. For $x<0$ the density is $\frac{x+2}{4}$ with mass $0.5$ and $\int_{-2}^0x\frac{x+2}{4}dx=-\frac13$, so $\beta^*=\frac{-1/3}{0.5}=-\frac23$. Hence $\alpha^*-\beta^*=0.5+0.667=1.17$.