GATE EC question library

The impulse response $h|n|$ of a linear time invariant system is given as $$h[n]=\left\{\begin{array}{cc} -2 \sqrt{2} & n=1,-1 \\ 4 \sqrt{2} & n=2,-2 \\ 0, & \text{ otherwise } \end{array}\right.$$ If the input to the above system is the sequence $\mathrm{e}^{j \pi n / 4}$, then the output is

Let $x(t)$ and $y(t)$ (with Fourier transforms $X(f)$ and $Y(f)$ respectively) be related as shown in the given figure. Then $Y(f)$ is

A system has poles at $0.01 \mathrm{~Hz}, 1 \mathrm{~Hz}$ and $80 \mathrm{~Hz}$; zeros at $5 \mathrm{~Hz}, 100 \mathrm{~Hz}$ and $200 \mathrm{~Hz}$. The approximate phase of the system response at $20 \mathrm{~Hz}$ is

Consider the signal flow graph shown in the figure below. The gain $\frac{x_{5}}{x_{1}}$ is

If $A=\left[\begin{array}{cc}-2 & 2 \\ 1 & -3\end{array}\right]$, then $\sin A t$ is

The open-loop transfer function of a unity feedback system is $G(s)=\frac{K}{s\left(s^{2}+s+2\right)(s+3)}$ The range of $\mathrm{K}$ for which the system is stable is

For the polynomial $P(s)=s^{5}+s^{4}+2 s^{3}+2 s^{2}+3 s+15$, the number of roots which lie in the right half of the s-plane is

The state variable equations of a system are: $\begin{array}{l} 1. \; x_{1}=-3 x_{1}-x_{2}+u \\ 2. \; x_{2}=2 x_{1} \\ \quad y=x_{1}+u \end{array}$ The system is

Given $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, the state transition matrix $e^{A t}$ is given by

Consider the signal $x(t)$ shown in the figure. Let $h(t)$ denote the impulse response of the filter matched to $x(t)$, with $h(t)$ being non-zero only in the interval 0 to $4 \; \mathrm{sec}$. The slope of $h(t)$ in the interval $3 < t < \text{sec}.$ is

A $1 \mathrm{~mW}$ video signal having a bandwidth of $100 \; \mathrm{MHz}$ is transmitted to a receiver through a cable that has $40 \mathrm{~dB}$ loss. If the effective one-sided noise spectral density at the receiver is $10^{-20}$ Watt $/ \mathrm{Hz}$, then the signal-to-noise ratio at the receiver is

A $100 \; \mathrm{MHz}$ carrier of $1 \mathrm{~V}$ amplitude and a $1 \; \mathrm{MHz}$ modulating signal of $1 \mathrm{~V}$ amplitude are fed to a balanced modulator. The output of the modulator is passed through an ideal high-pass filter with cut-off frequency of $100 \; \mathrm{MHz}$. The output of the filter is added with $100 \; \mathrm{MHz}$ …

Two sinusoidal signals of same amplitude and frequencies $10 \; \mathrm{kHz}$ and $10.1 \; \mathrm{kHz}$ are added together. The combined signal is given to an ideal frequency detector. The output of the detector is

Consider a binary digital communication system with equally likely $0$'s and $1$'s. When binary 0 is transmitted the voltage at the detector input can lie between the levels $-0.25 \mathrm{~V}$ and $+0.25 \mathrm{~V}$ with equal probability: when binary $1$ is transmitted, the voltage at the detector can have any value between $0$ and …

A random variable $\mathrm{X}$ with uniform density in the interval $0$ to $1$ is quantized as follows: $\begin{array}{ll} \qquad \text{If 0 < X < 0.3,} & \text{x}_{4} = 0 \\ \qquad \text{If 0.3 < X < 1,} & \text{x}_{4} = 0.7 \end{array}$ where $\text{x}_{4}$ is the quantized value of $\text{X}$. The root-mean square value of …

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

Three analog signals, having bandwidths $1200 \mathrm{~Hz}$, $600 \mathrm{~Hz}$ and $600 \mathrm{~Hz}$, are sampled at their respective Nyquist rates, encoded with $12$ bit words, and time division multiplexed. The bit rate for the multiplexed signal is

Consider a system shown in the figure. Let $X(f)$ and $Y(f)$ denote the Fourier transforms of $X(t)$ and $y(t)$ respectively. The ideal HPF has the cutoff frequency $10 \mathrm{kHz}$. The positive frequencies where $Y(f)$ has spectral peaks are

A parallel plate air-filled capacitor has plate area of $10^{-1} \mathrm{~m}^{2}$ and plate separation of $10^{-3} \mathrm{~m}$. It is connected to a $0.5 \mathrm{~V}, 3.6 \; \mathrm{GHz}$ source. The magnitude of the displacement current is $\left(\varepsilon_{0}=1 / 36 \pi \times 10^{-9} \mathrm{~F} / \mathrm{m}\right)$

A source produces binary data at the rate of $\text{10 kbps.}$ The binary symbols are represented as shown in the figure given below. The source output is transmitted using two modulation schemes, namely Bìnary PSK (BPSK) and Quadrature PSK (QPSK). Let $\mathrm{B}_{1}$ and $\mathrm{B}_{2}$ be the bandwidth requirements of BPSK and QPSK respectively. …