GATE EC question library

Consider a causal second-order system with the transfer function $$G(s)=\dfrac{1}{1+2s+s^{2}}$$ with a unit-step $R(s)=\dfrac{1}{s}$ as an input. Let $C(s)$ be the corresponding output. The time taken by the system output $c(t)$ to reach $94\%$ of its steady-state value $\underset{t\rightarrow \infty}{\lim}\:c(t),$ rounded off to two decimal …

The block diagram of a system is illustrated in the figure shown, where $X(s)$ is the input and $Y(s)$ is the output. The transfer function $H(s)=\dfrac{Y(s)}{X(s)}$ is

Let the state-space representation of an LTI system be $x(t)=A x(t)+B u(t), y(t)=Cx(t)+du(t)$ where $A,B,C$ are matrices, $d$ is a scalar, $u(t)$ is the input to the system, and $y(t)$ is its output. Let $B=[0\quad0\quad1]^{T}$ and $d=0$ .Which one of the following options for $A$ and $C$ will ensure that the transfer function of this LTI …

A single bit, equally likely to be $0$ and $1$, is to be sent across an additive white Gaussian noise (AWGN) channel with power spectral density $N_{0}/2.$ Binary signaling with $0 \mapsto p(t),$ and $1 \mapsto q(t),$ is used for the transmission, along with an optimal receiver that minimizes the bit-error probability. Let …

The quantum efficiency $(\eta)$ and responsivity $(R)$ at wavelength $\lambda \:(\text{in}\: \mu m)$ in a p-i-n photodetector are related by

Two identical copper wires $W1$ and $W2$ placed in parallel as shown in the figure, carry currents $I$ and $2I$, respectively, in opposite directions. If the two wires are separated by a distance of $4r$, then the magnitude of the magnetic field $\overrightarrow{B}$ between the wires at a distance $r$ from $W1$ is

The dispersion equation of a waveguide, which relates the wavenumber $k$ to the frequency $\omega$ is $$k(\omega)= (1/c) \sqrt{\omega^{2}-\omega_{\circ}^{2}}$$ where the speed of light $c= 2 \times 10^{8}\: m/s$ and $\omega_{\circ}$ is a constant . If the group velocity is $2 \times 10^{8}\: m/s$, then the phase velocity is

In the circuit shown, the breakdown voltage and the maximum current of the Zener diode are $20\:V$ and $60\:mA$, respectively. The values of $R_{1}$ and $R_{L}$ are $200\: \Omega$ and $1\:k\Omega,$ respectively. What is the range of $V_{i}$ that will maintain the Zener diode in the ‘on’ state?

The state transition diagram for the circuit shown is

In the circuits shown the threshold voltage of each $\text{nMOS}$ transistor is $0.6\:V.$ Ignoring the effect of channel length modulation and body bias. the values of $\text{Vout}1$ and $\text{Vout} 2,$ respectively, in volts, are

The $\text{RC}$ circuit shown below has a variable resistance $R(t)$ given by the following expression: $$R(t)=R_{0}\left(1-\frac{t}{T}\right) \text{for} \:\: 0 \leq t < T$$ where $R_{0}=1\: \Omega,$ and $C=1\:F.$ We are also given that $T=3\: R_{0}C$ and the source voltage is $V_{s}=1\: V.$ If the current at time $t=0$ is $1\: A,$ then …

Consider a unity feedback system, as in the figure shown, with an integral compensator $\dfrac{K}{s}$ and open-loop transfer function $$G(s)=\dfrac{1}{s^{2}+3s+2}$$ where $K>0.$ The positive value of $K$ for which there are exactly two poles of the unity feedback system on the $j\omega$ axis is equal to ________ (rounded off to two decimal …

Consider the homogenous ordinary differential equation $$x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+3y=0, \quad x>0$$ with $y(x)$ as a general solution. Given that $$y(1)=1 \quad \text{and} \quad y(2)=14$$ the value of $y(1.5),$ rounded off to two decimal places, is________.

Let $h[n]$ be a length – $7$ discrete-time finite impulse response filter, given by $$h[0]=4, \quad h[1]=3,\quad h[2]=2,\quad h[3]=1,$$ $$\quad h[-1]=-3, \quad h[-2]=-2, \quad h[-3]=-1,$$ and $h[n]$ is zero for $|n|\geq4.$ A length – $3$ finite impulse response approximation $g[n]$ of $h[n]$ has to be obtained such that …

Let a random process $Y(t)$ be described as $Y(t)=h(t) \ast X(t)+Z(t),$ where $X(t)$ is a white noise process with power spectral density $S_{x}(f)=5$W/Hz. The filter $h(t)$ has a magnitude response given by $\mid H(f) \mid =0.5$ for $-5\leq f\leq5,$ and zero elsewhere. $Z(t)$ is a stationary random process, uncorrelated with $X(t)$, with …

A voice signal $m(t)$ is in the frequency range $5\:kHz$ to $15\:kHz$. The signal is amplitude-modulated to generated an AM signal $f(t)=A\left(1+m(t)\right)\cos 2\pi f_{c}t,$ where $f_{c}=600\: kHz.$ The AM signal $f(t)$ is to be digitized and archived. This is done by first sampling $f(t)$ at $1.2$ times the Nyquist frequency, and then …

A random variable $X$ takes values $-1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is independent of $X,$ and is uniformly distributed over the interval $[-2,2].$ The receiver makes a …

A Germanium sample of dimensions $1\: cm \times 1\: cm$ is illuminated with a $20\:mW,$ $600\: nm$ laser light source as shown in the figure. The illuminated sample surface has a $100\: nm$ of loss-less Silicon dioxide layer that reflects one-fourth of the incident light. From the remaining light, one-third of the power is reflected from the Silicon …

In an ideal $pn$ junction with an ideality factor of $1$ at $T=300\:K,$ the magnitude of the reverse-bias voltage required to reach $75\%$ of its reverse saturation current, rounded off to $2$ decimal places, is ______ $mV.$ $$[k=1.38 \times 10^{-23} JK^{-1}, h=6.625 \times 10^{-34} J-s, q=1.602 \times 10^{-19}C]$$

Consider a long-channel MOSFET with a channel length $1\:\mu m$ and width $10\: \mu m.$ The device parameters are acceptor concentration $N_{A}=5 \times 10^{16}\: cm^{-3},$ electron mobility $\mu_{n}=800\: cm^{2}/V-s,$ oxide capacitance/area $C_{ox}=3.45 \times 10^{-7}\:F/cm^{2},$ threshold voltage $V_{T}=0.7\:V.$ The drain saturation current …