GATE EC question library

In the circuit shown, assume that the diodes $D1$ and $D2$ are ideal. The average value of voltage $V_{ab}\: (\text{in Volts}),$ across terminals $‘a’$ and $‘b’$ is _________.

Suppose $x[n]$ is an absolutely summable discrete-time signal. Its $z$-transform is a rational function with two poles and two zeroes. The poles are at $z = \pm 2j.$ Which one of the following statements is TRUE for the signal $x[n]$?

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input $x[n ],$ the response $y[n ]$ is

Let $\widetilde{x}[n] = 1 + \cos\left(\dfrac{\pi n}{8}\right)$ be a periodic signal with period $16.$ Its DFS coefficients are defined by $a_{k} = \displaystyle{}\dfrac{1}{16}\sum_{n=0}^{15}\widetilde{x}[n] \text{exp}\big(-j\dfrac{\pi}{8} kn\big)$ for all $k .$ The value of the coefficient ܽ$a_{31}$ is _______.

Consider a continuous-time signal defined as $$x(t)=\left(\dfrac{\sin(\pi t/2)}{(\pi t /2)}\right)\ast \sum _{n=-\infty}^{\infty}\delta(t-10n)$$ where $’\ast’$ denotes the convolution operation and $t$ is in seconds. The Nyquist sampling rate $\text{(in samples/sec)}$ for $x(t)$ is _______.

The position control of a DC servo-motor is given in the figure. The values of the parameters are $K_{T}=1 \: N-m/A, R_{a}=1\Omega, L_{a} = 0.1H,J=5kg-m^{2},B=1N-m/(rad/sec)$ and $K_{b} = 1V/(rad/sec) .$ The steady-state position response (in radians) due to unit impulse disturbance torque $T_{d}$ is _______.

For the system shown in the figure, $s=-2.75$ lies on the root locus if $K$ is _______.

The characteristic equation of an LTI system is given by $F(s) = s^{5} + 2s^{4} + 3s^{3} + 6s^{2} – 4s – 8 = 0.$ The number of roots that lie strictly in the left half $s$-plane is _________.

Two sequences $x_{1}[n]$ and $x_{2}[n]$ have the same energy. Suppose $x_{1}[n]=\alpha\: 0.5 ^{n}\:u[n],$ where $\alpha$ is a positive real number and $u[n]$is the unit step sequence. Assume ݊$$x_{2}[n]= \begin{cases} \sqrt{1.5}& \text{for } n = 0,1 \\ 0 &\text{otherwise.} \end{cases}$$ Then the value of $\alpha$ is _________.

The variance of the random variable $X$ with probability density function $f(x)=\dfrac{1}{2}\mid x \mid e^{- \mid x \mid}$ is __________.

The complex envelope of the bandpass signal $x(t)=-\sqrt{2}\left(\dfrac{\sin (\pi t/5)}{\pi t/5}\right)\sin (\pi t – \dfrac{\pi}{4}),$ centered about $f=\dfrac{1}{2}\:Hz,$ is

A random binary wave $y(t)$ is given by $$y(t) = \sum_{n = -\infty}^{\infty}X_{n}\:p(t-nT-\phi)$$ where $p(t)=u(t)-u(t-T),u(t)$ is the unit step function and $\phi$ is an independent random variable with uniform distribution in $[0,T].$The sequence $\{X_{n}\}$ consist of independent and identically distributed binary valued random variables with …

Consider the $3\:m$ long lossless air-filled transmission line shown in the figure. It has a characteristic impedance of $120\pi\: \Omega,$ is terminated by a short circuit, and is excited with a frequency of $37.5\: MHz.$ What is the nature of the input impedance $(Z_{in})?$

A $200\: m$ long transmission line having parameters shown in the figure is terminated into a load ܴ$R_{L}$. The line is connected to a $400\: V$ source having source resistance $R_{S}$ through a switch, which is closed at $t = 0.$ The transient response of the circuit at the input of the line $(z = 0)$ is also drawn in the figure. The value of …

A coaxial capacitor of inner radius $1\: mm$ and outer radius $5\: mm$ has a capacitance per unit length of $172\: pF/m.$ If the ratio of outer radius to inner radius is doubled, the capacitance per unit length $\text{(in}\: pF/m)$ is ________.