GATE EC question library

The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion $$x(t)=5\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{3}+7\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{2}$$ would be

A uniformly distributed random variable $X$ with probability density function $$f_{\gamma}(x)=\frac{1}{10}(t r(x+5)-u(x-5))$$ where $u(.)$ is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random variable $Y$ would be

A system with input $x[n]$ and output $y[n]$ is given as $y[n]=\left(\sin \frac{5}{6} \pi n\right) x(n)$. The system is

The unit-step response of a system starting from rest is given by $\qquad C(t)=1-e^{-2 t} \text{ for } t \geq 0$ The transfer function of the system is

The Nyquist plot of $\mathrm{G}(j \omega) \mathrm{H}(j \omega)$ for a closed loop control system, passes through $(-1, j 0)$ point in the $\mathrm{GH}$ plane. The gain margin of the system in $\mathrm{dB}$ is equal to

The positive values of $\text{“K”}$ and “$a$” so that the system shown in the figure below oscillates at a frequency of $2 \mathrm{rad} / \mathrm{sec}$ respectively are

The unit impulse response of a system is $$h(t)=e^{-t}, t \geq 0$$ For this system, the steady-state value of the output for unit step input is equal to

The transfer function of a phase-lead compensator is given by $$G_{c}(s)=\frac{1+3 T s}{1+T s} \text{ where } T>0$$ The maximum phase-shift provided by such a compensator is

A linear system is described by the following state equation $$X(t)=\mathrm{AX}(t)+\mathrm{BU}(t), \mathrm{A}=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0\end{array}\right]$$ The state-transition matrix of the system is

The minimum step-size required for a Delta-Modulator operating at $32 \mathrm{~K}$ samples/ $\mathrm{sec}$ to track the signal (here $u(t)$ is the unit-step function) $\begin{array}{l}x(t)=125 t(u(t)-u(t-1))+(250-125 t)(u(t-1) -u(t-2)) \end{array}$ so that slope-overload is avoided, would be

A zero- mean white Gaussian noise is passed through an ideal lowpass filter of ban width $10 \; \mathrm{kHz}$. The output is the uniformly sampled with sampling period $t_{s}=0.03 \; \mathrm{msec}$. The samples so obtained would be

A source generates three symbols with probabilities $0.25,0.25,0.50$ at a rate of $3000$ symbols per second. Assuming independent generation of symbols, the most efficient source encoder would have average bit rate as

The diagonal clipping in Amplitude Demodulation (using envelope detector) can be avoided if RC time-constant of the envelope detector satisfies the following condition, (here $W$ is message bandwidth and $\omega_{c}$ is carrier frequency both in rad/sec)

In the following figure the minimum value of the constant ' $C$ ', which is to be added to $y_{1}(t)$ such that $y_{1}(t)$ and $y_{2}(t)$ are different, is

A message signal with bandwidth $10 \; \mathrm{kHz}$ is Lower-Side Band SSB modulated with carrier frequency $f_{c 1}=10^{6} \mathrm{~Hz}$. The resulting signal is then passed through a Narrow-Band Frequency Modulator with carrier frequency $f_{c_{2}}=10^{9} \; \mathrm{Hz}$. The bandwidth of the output would be

A medium of relative permittivity $\varepsilon_{r 2}=2$ forms an interface with free-space. A point source of electronagnetic energy is located in the medium at a depth of $1$ meter from the interface. Due to the total internal reflection, the transmitted beam has a circular cross-section over the interface. The area of the beam cross-section at the …

A medium is divided into regions I and II about $x=0$ plane, as shown in the figure below. An electromagnetic wave with electric field $\mathrm{E}_{1}=4 \hat{a}_{x}+3 \hat{a}_{y}+5 \hat{a}_{z}$ is incident normally on the interface from region-I. The electric field $E_{2}$ in region-II at the interface is

When a plane wave travelling in free-space is incident normally on a medium having $\varepsilon_{t}=4.0$, then fraction of power transmitted into the medium is given by

A rectangular waveguide having $\mathrm{TE}_{10}$ mode as dominant mode is having a cutoff frequency of $18-\mathrm{GHz}$ for the $\mathrm{TE}_{w,}$ mode. The inner broad-wall dimension of the rectangular waveguide is

A mast antenna consisting of a $50$ meter long vertical conductor operates over a perfectly conducting ground plane. It is base-fed at a frequency of $600-\mathrm{kHz}$. The radiation resistance of the antenna in ohms is