GATE EC question library

The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure. The transition probabilities are as marked. The parameter $\alpha$ lies in the interval $[0.25, 1].$ The value of $\alpha$ for which the capacity of this channel is maximized, is ___________ (rounded off to two decimal …

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 – \in),$ and flipped with probability $\in.$ For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the …

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 $\text{(ML)}$ detector at the receiver. The conditional probability density functions for $Y$ given $x_{A}$ and $x_{B}$ are: $$f_{Y|x_{A}}(y) = e^{-(y+1)}u(y+1),$$ …

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 $\alpha^{\ast}$ and $\beta^{\ast}$ are the respective choices for …

In an electrostatic field, the electric displacement density vector, $\overrightarrow{D},$ is given by $$\overrightarrow{D} (x, y, z) = (x^{3} \overrightarrow{i} + y^{3} \overrightarrow{j} + xy^{2} \overrightarrow{k}) \; \text{C/m}^{2},$$ where $\overrightarrow{i}, \overrightarrow{j}, \overrightarrow{k}$ are the unit vectors along …