GATE EC question library

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable independent of $S$. The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$. Suppose the probability of symbol estimation error $P_{e}$ is …

Consider a real-valued random process $$f(t)=\sum_{n=1}^{N} a_{n} \: p(t-n T),$$ where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and $0$ otherwise. The coefficients $a_{n}$ are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the …

The identical MOSFETs $M_{1}$ and $M_{2}$ in the circuit given below are ideal and biased in the saturation region. $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ have a transconductance $g_{m}$ of $5$ mS. The input signals (in Volts) are: $$\begin{array}{l} V_{1}=2.5+0.01 \sin \omega t \\ V_{2}=2.5-0.01 \sin \omega t \end{array}$$ The output signal …

A $10$-bit analog-to-digital converter (ADC) has a sampling frequency of $1\: \mathrm{MHz}$ and a full scale voltage of $3.3 \: \mathrm{V}$. For an input sinusoidal signal with frequency $500 \: \mathrm{kHz}$ , the maximum SNR (in $\mathrm{dB}$ , rounded off to two decimal places) and the data rate (in $\mathrm{Mbps}$) at the output of the ADC …

A positive-edge-triggered sequential circuit is shown below. There are no timing violations in the circuit. Input $P 0$ is set to logic '$0$ ' and $P 1$ is set to logic '$1$' at all times. The timing diagram of the inputs $S E L$ and $S$ are also shown below. The sequence of output $Y$ from time $T_{0}$ to $T_{3}$ is _______.

The intrinsic carrier concentration of a semiconductor is $2.5 \times 10^{16} / \mathrm{m}^{3}$ at $300 \: \mathrm{K}$. If the electron and hole mobilities are $0.15 \mathrm{~m}^{2} / \mathrm{Vs}$ and $0.05 \mathrm{~m}^{2} / \mathrm{Vs}$, respectively, then the intrinsic resistivity of the semiconductor (in $\mathrm{k} \Omega . \mathrm{m}$ ) at …

In the circuit shown, the identical transistors $\text{Q1}$ and $\text{Q2}$ are biased in the active region with $\beta=120$. The Zener diode is in the breakdown region with $\mathrm{V}_{\mathrm{Z}}=5 \mathrm{~V}$ and $\mathrm{I}_{\mathrm{Z}}=25 \mathrm{~mA}$. If $\mathrm{I}_{\mathrm{L}}=12 \mathrm{~mA}$ and …

The electron mobility $\mu_{\mathrm{n}}$ in a non-degenerate germanium semiconductor at $300 \: \mathrm{K}$ is $0.38 \mathrm{~m}^{2} / \mathrm{Vs}$. The electron diffusivity $D_{n}$ at $300 \: \mathrm{K}$ (in $\mathrm{cm}^{2} / \mathrm{s}$, rounded off to the nearest integer) is _________. (Consider the Boltzmann constant …

A square metal sheet of $4 \mathrm{~m} \times 4 \mathrm{~m}$ is placed on the $x-y$ plane as shown in the figure below. If the surface charge density (in $\mu \mathrm{C} / \mathrm{m}^{2}$ ) on the sheet is $\rho_{s}(x, y)=4\mid y \mid$, then the total charge (in $\mu \mathrm{C}$, rounded off to the nearest integer) on the sheet is $\_\_\_\_\_\_$

An electric field of $0.01 \mathrm{~V} / \mathrm{m}$ is applied along the length of a copper wire of circular cross-section with diameter $1 \: \mathrm{mm}$. Copper has a conductivity of $5.8 \times 10^{7} \mathrm{~S} / \mathrm{m}$. The current (in Amperes, rounded off to two decimal places) flowing through the wire is $\_\_\_\_\_\_$.

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval. Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality …

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

Consider a system where $x_{1}(t), x_{2}(t)$, and $x_{3}(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by …

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)=\alpha$ and $P(X=0)=1-2 \alpha$, where $0<\alpha<1 / 2$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

Let $f(t)$ be a periodic signal with fundamental period $T_{0}>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$. The Fourier series expansions of $f(t)$ and $y(t)$ are given by …

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

All the diodes in the circuit given below are ideal. Which of the following plots is/are correct when $V_{I}$ (in Volts) is swept from $-M$ to $M$?

Two fair dice (with faces labeled $1,2,3,4,5$, and $6$) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained. The expectation of $X$ is __________ (rounded off to two decimal places).

Consider the vectors $$\boldsymbol{a}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right]$$ For real-valued scalar variable $x$, the value of $$\min _{x}\|\boldsymbol{a} x-\boldsymbol{b}\|_{2}$$ is $\_\_\_\_\_$ (rounded off to two decimal places). $\|\cdot\|_{2}$ denotes the …

$X$ 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).