GATE EC 2024 Set 1 Q24 — Information Theory

Let the relevant bandwidth $(B)$ of a digital communication system be $1 ~\text{MHz}$ and $k T=-174 \mathrm{~dBm} / \mathrm{Hz}$, where $k$ is Boltzmann's constant and '$T$' is equivalent noise temperature of the receiver. The power $(S)$ of signal received through an additive Gaussian channel is $-80 ~\text{dBm}.$ Which of the following …

Consider an additive white Gaussian noise (AWGN) channel with bandwidth $W$ and noise power spectral density $\frac{N_{0}}{2}$. Let $P_{a v}$ denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity $C$ on the bandwidth $W$ (keeping $P_{a v}$ and $N_{0}$ fixed)?

$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by: $$\begin{array}{l} P(X=0, Y=0)=0.06 \\ P(X=0, Y=1)=0.14 \\ P(X=1, Y=0)=0.24 \\ P(X=1, Y=1)=0.56 \end{array}$$ The mutual information $I(X ; Y)$ is $\_\_\_\_\_\_\_\_$ (rounded off to two decimal places).

The frequency of occurrence of $8$ symbols $(a-h)$ is shown in the table below. A symbol is chosen and it is determined by asking a series of $\text{"yes/no"}$ questions which are assumed to be truthfully answered. The average number of questions when asked in the most efficient sequence, to determine the chosen symbol, is _______________ (rounded off …

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 …

A speech signal, band limited to $4\:\text{kHz}$, is sampled at $1.25$ times the Nyquist rate. The speech samples, assumed to be statistically independent and uniformly distributed in the range $-5\:V$ to $+5\:V$, are subsequently quantized in an $8$-bit uniform quantizer and then transmitted over a voice-grade $\text{AWGN}$ telephone channel. If …