GATE EC question library

Let $X$ and $Y$ be two statistically independent random variables uniformly distributed in the ranges $(-1,1)$ and $(-2,1)$ respectively. Let $Z=X+Y$. Then the probability that $(Z \leq-2)$ is

Let $\mathrm{P}$ be linearity, $\mathrm{Q}$ be time-invariance, $\mathrm{R}$ be causality and $\mathrm{S}$ be stability. A discrete-time system has the input-output relationship, $$y(n)=\left\{\begin{array}{ll} x(n) & n \geq 1 \\ 0, & n=0 \\ x(n+1), & n \leq-1 \end{array}\right.$$ where $x(n)$ is the input and $y(n)$ is the output. …

Data for Q. 63-64 are given below. Solve the problems and choose the correct answers. The system under consideration is an RC low-pass filter (RC-LPF) with $R=1.0 \mathrm{k} \; \Omega \mathrm{k}$ and $\mathrm{C}=1.0 \; \mu \mathrm{F}$ Let $\mathrm{H}(f)$ denote the frequency response of the RC-LPF. Let $f_{1}$ be the highest frequency such that …

Data for Q. 63-64 are given below. Solve the problems and choose the correct answers. The system under consideration is an $R C$ low-pass filter (RC-LPF) with $R=1.0 \; \mathrm{k} \Omega \mathrm{k}$ and $\mathrm{C}=1.0 \; \mu \mathrm{F}$ Let $t_{g}(f)$ be the group delay function of the given RC-LPF and $f_{2}=100 \mathrm{~Hz}$. Then …

Data for Q. 65-66 are given below. Solve the problems and choose the correct answers. Let $X$ be the Gaussian random variable obtained by sampling the process at $t=t_{i}$ and let $$\mathrm{Q}(\alpha)=\int_{\alpha}^{\infty} \frac{1}{\sqrt{2 \pi}} e^{\frac{x^{2}}{2}} d y$$ The probability that $[x \leq 1]$ is

Let $Y$ and $Z$ be the random variables obtained by sampling $X(t)$ at $t=2$ and $t=4$ respectively. Let $W$ $=Y-Z$. The variance of $W$ is

Let $x(t)=2 \cos (800 \pi t)+\cos (1400 \pi t) . x(t)$ is sampled with the rectangular pulse train shown in the figure. The only spectral components (in $\mathrm{kHz}$ ) present in the sampled signal in the frequency range $2.5 \; \mathrm{kHz}$ to $3.5 \; \mathrm{kHz}$ are

The signal flow graph of a system is shown in the figure. The transfer function $\frac{C(s)}{R(s)}$ of the system is

The root locus of the system $\mathrm{G}(\mathrm{s}) \mathrm{H}(s)$ $=\frac{\mathrm{K}}{s(s+2)(s+3)}$ has the break-away point located at

The approximate Bode magnitude plot of a minimum-phase system is shown in the figure. The transfer function of the system is

A second-order system has the transfer function $\frac{C(s)}{R(s)}=\frac{4}{s^{2}+4 s+4}$. With $r(t)$ as the unit-step function, the response $c(t)$ of the system is represented by

The gain margin and the phase margin of a feedback system with $\mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s})=\frac{\mathrm{s}}{(\mathrm{s}+100)^{3}}$ are

The zero-input response of a system given by the state-space equation $\left[\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]$ and …

A DSB-SC signal is to be generated with a carrier frequency $f_{t}=1 \; \mathrm{MHz}$ using a non-linear device with the input-output characteristic $$v_{0}=a_{0} v_{i}+a_{1} v_{i}^{3}$$ where $a_{0}$ and $a_{1}$ are constants. The output of the non-linear device can be filtered by an appropriate band-pass filter. Let …

The data for Q. 75-76 are given below. Solve the problems and choose the correct answers. Let $m(t)=\cos \left[\left(4 \pi \times 10^{3}\right) t\right]$ be the message signal and $\left.c(t)=5 \cos \left[2 \pi \times 10^{b}\right) t\right]$ be the carrier $c(t)$ and $m(t)$ are used to generate an AM signal. The modulation index of the generated AM …

The data for Q. 75-76 are given below. Solve the problems and choose the correct answers. Let $m(t)=\cos \left[\left(4 \pi \times 10^{3}\right) t\right]$ be the message signal and $\left.c(t)=5 \cos \left[2 \pi \times 10^{b}\right) t\right]$ be the carrier $c(t)$ and $m(t)$ are used to generate an $\text{FM}$ signal. If the peak frequency deviation …

Choose the correct one from among the alternative $\text{A, B, C. D}$ after matching an item in Group $1$ with the most appropriate item in Group $2.$ …

A superheterodyne receiver is to operate in the frequency range $550 \; \mathrm{kHz}-1650 \; \mathrm{kHz}$, with the intermediate frequency of $450 \; \mathrm{kHz}$. Let $\mathrm{R}=\frac{\mathrm{C}_{\operatorname{max}}}{\mathrm{C}_{\text{min}}}$ denote the required capacitance ratio of the local oscillator and $\text{I}$ denote the image frequency …

A sinusoidal signal with peak-to-peak amplitude of $1.536 \mathrm{~V}$ is quantized into $128$ levels using a mid-rise uniform quantizer. The quantization-noise power is

If $E_{b^{\prime}}$ the energy per bit of a binary digital signal, is $10^{-6}$ watt-sec and the one-sided power spectral density of the white noise, $\mathrm{N}_{0}=10^{-5} \mathrm{~W} / \mathrm{Hz}$, then the output $\text{SNR}$ of the matched filter is