GATE Electronics & Communication

Consider the differential equation $\dot{w} = Aw$, with $w(t = 0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$. If $w(t) = e^t\vec{u}_x + e^{-2t}\vec{u}_y$ be the solution to the equati...

A surface is given by z² = 2x² – y² and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P} = \hat{i} + \sqrt{2} \hat{k}$. Which of the following...

The Laplace Transform of the signal $x(t) = u(t - 2) * (tu(t))$ is given by which of the following expressions? [" * " represents convolution operator]

Consider carrier transport in a Zener diode in the breakdown region. Which is the dominant transport mechanism for current flow in this case?

Two analog signals $x_1(t)$ and $x_2(t)$ ($t$ in second), are sampled at a rate $F_s = 40$ Hz, where $x_1(t) = \cos(20\pi t)$, $t \ge 0$, and $x_2(t) = \cos(100\pi t)$, $t \ge 0$....

The response of a discrete time system y[n] obeys the following relation: y[n] = (5/6)y[n-1] - (1/6)y[n – 2] + x[n]. The input to the system is x[n] = δ[n] – (1/3)δ[n – 1]. Which o...

The ideal OP-AMP circuit shown in the Figure produces output voltage V_o = x when the Switch, S, is open. Which of the options represents the output voltage when S is closed?

Consider the circuit shown in the Figure with $V_i = 3$ V and $V_{CC} = 12$ V. Assume $V_{BE} = 0.7$ V and $\beta_{dc} = 99$ for the BJT. Which of the following options is the corr...

A control system is shown in the Figure. Which option represents the correct transfer function of the system?

In the circuit shown in the Figure, A and B are logic inputs and Y is the logic output. Which of the following logic operations is realized by the circuit?

Consider the Friis' transmission equation $P_R = (P_T G_T G_R \lambda^2)/(4\pi D)^2$, where $P_R$ and $P_T$ are the received and the transmitted powers, respectively. $G_T$ and $G_...

Consider a discrete memoryless source with an alphabet of four source symbols. s(t) is a multi-level (-1, 0, +1, +2) signal representing a long sequence of random symbols from the...

The relation between the input current (I) and the output voltage (V) of a circuit is governed by the equation: $C \frac{dV}{dt} = I(t) - m(t)$. The circuit is excited by $I(t) = q...

The electric field of a monochromatic plane wave travelling in a lossless isotropic and homogenous medium is given by $\vec{E}(z,t) = E_0[\hat{x}cos(\omega t - kz) + \hat{y}sin(\om...

Consider a p-n junction diode when it is forward biased with 2 V. Which of the following is/are the correct magnitude(s) of the energy difference between quasi Fermi-levels, $E_{fn...

Consider the matrix M = $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 3 & 0 \\ -1 & a & b \end{bmatrix}$. Which of the following options is/ are TRUE if det(M) $\neq$ 0?

A binary ripple counter is designed to count (0)10 to (64)10. Which of the following is/are the number of flip-flops required to design the counter?

Which option(s) represents/ represent the dielectric loss tangent of a substrate?

Figure shows the output characteristics of two different Bipolar Junction Transistors (BJT), BJT 1 with magnitude of Early voltage |V_A1|, and BJT 2 with magnitude of Early voltage...

The output voltage V_o (in Volt) for the network given in the Figure is ____. (rounded off to two decimal places)

Consider the circuit shown in the Figure, where the input v_i(t) is in Volt. The average power (in mW) dissipated in the load resistance of 1 kΩ at the resonant frequency is ____....

A wireless digital transmission scheme is using 16-QAM over an additive white Gaussian noise channel and a maximum-likelihood receiver. Consider the information bit rate from sourc...

The cutoff frequency (in GHz) for the dominant TE10 mode of an air-filled rectangular waveguide of inner dimension 0.28 inch × 0.14 inch is ____. (rounded off to two decimal places...

For a lossless passive two-port network, |S11| and |S21| intersect at -3 dB. For a lossy passive two-port network, |S11| and |S21| intersect at – 4 dB.. The percentage of power dis...

The negative edge triggered JK flip-flop in the Figure has J and K inputs tied to Logic High and a square wave of 10 cycles/second is applied to its clock (C) input. The frequency...

Consider the two series, $S_A$ and $S_B$, where $S_A = \sum_{n=1}^{\infty} \frac{n^2}{2^n}$ $S_B = 1 + \frac{1}{2} + \frac{1}{8} + \frac{1}{16} + \frac{1}{64} + \frac{1}{128} + \fr...

The continuous time signal x(t) is real, periodic with period T and satisfies the Dirichlet conditions. The Fourier series representation of x(t) = $\sum_{-\infty}^{\infty} a_n e^{...

Let X, N, Y and Z be random variables. The variables X and N are independent of each other. X is uniformly distributed between -1 and 1; N follows Normal distribution with zero mea...

In the given circuit, $L = 1 \mu H$ and $C = 1 \mu F$. The phasor diagram for $I_C$ and $I_L$ is also shown. Assume that the phase $(\theta_1 + \theta_2)$ is $90^\circ$ at a freque...

For the control system shown in the Figure, the transfer function of a plant, $G(s) = \frac{1}{(s+1)(s+2)}$ is connected in cascade with a compensator $C(s) = K (s + \alpha)$, wher...

The state and output equations for a control system are: $\dot{x} = \begin{bmatrix} -4 & -1.5 \\ 4 & 0 \end{bmatrix}x + \begin{bmatrix} 2 \\ 0 \end{bmatrix}u$ $y = \begin{bmatrix}...

The address of the first location of a 256 kilo byte (KB) memory is (2500)H. Choose the correct address of the last location of the memory.

Consider a real signal x(t), -∞ < t <∞, such that x(t) = 0 for t <0, x(t) = 2 for 0 ≤ t < 1 and x(t) = 0 for t≥ 1. Let E[x(t)] = ∫_∞^[x(t)]^2dt. Which of the following options corr...

A QPSK modulated signal from an additive white Gaussian noise (AWGN) channel is received with an Eb/No = 8.4 dB at the input of a coherent QPSK demodulator. A maximum-likelihood re...

What is the 10's complement of (47)10?

Consider a real baseband signal x(t) = e^{-2t}, for t (in seconds) ≥ 0. If 99% of energy of x(t) lies within B Hz, then which of the following options is TRUE for the value of B?

Consider the discrete time system (S) with input x[n] and output y[n] as shown in the Figure. The two sub-systems represented by their impulse responses $h_1[n]$ and $h_2[n]$ are l...

A Boolean function, $f(x,y,z)$ with $x$ as MSB and $z$ as LSB is realized by 4:1 multiplexer (MUX) with select lines, $S_1$ and $S_0$ ($S_1$ is MSB, $S_0$ is LSB) and inputs, $I_0,...

A shift-left Shift Register (SR) and a D flip-flop are connected to a synchronized clock as shown in the Figure. Assume that the SR and D flip-flops are initially cleared and the X...

A small signal source, Vi(t) = A cos(10^5t) + B sin(10^7t) is applied to a BJT circuit as shown in the Figure. Assume zero source resistance, VBE = 0.7 V, βdc = 99, Early voltage =...

Consider the two-port network as shown in the Figure. Which of the following options provides the correct set of values of A, B, C and D parameters?

A complex load (in Ω) is represented as ΓL = 0.5∠30° on the Smith chart. A co-axial cable with a characteristic impedance of 50 Ω is connected to the load. The new input impedance...

A circuit using an ideal OP-AMP is shown in the Figure. Which of the following options gives the correct value of the current Iₓ?

Consider an LED based on a direct bandgap semiconductor material with energy bandgap 1.3 eV. Given: Plank's constant, h = 6.63 × 10^-34 J s and speed of light in free space is 3 ×...

Let the relevant bandwidth (B) of a digital communication system be 1 MHz and $kT = -174 \text{ dBm/Hz}$, where k is Boltzmann's constant and 'T' is equivalent noise temperature of...

Consider the four-variable Boolean function, f(w, x, y, z) = ∑m(0,2,5,7,8,10,13,14,15) with 'w' as MSB and 'z' as LSB. Which of the following expressions is/are the valid form(s) o...

Consider a real, narrowband signal $x(t) = A(t)\cos[2\pi f_c t + \theta(t)]$ where the maximum frequency components of $A(t)$ and $\theta(t)$ are $f_M$ and $f_c (= 1000 f_M)$, resp...

Let $x_1(t) = \cos(2\pi nt)$ and $x_2(t) = 2\sin(4\pi nt)$ represent two sinusoids for a positive integer $n$ and $-\infty < t < \infty$. Which of the following statements about $x...

Consider the unity negative feedback control system shown in the Figure. The value of gain K (>0) at which the given system will remain marginally stable is __. (Answer in integer)

An n-channel MOSFET is connected as shown in the Figure. Assume VTH = 1 V, VDD = 5 V, and μCox (W/L) = 2 mA V^-2 and neglect channel length modulation effects. The gate voltage (VG...

Consider the square region R in the X-Y plane as shown with the dark shading in the Figure. The value of $\iint_R (x^2 + y^2 - 1)dxdy$ is ______. (rounded off to two decimal places...

Consider an ideal OP-AMP circuit as shown in the Figure. The resistances $R_1 = R_2 = R_3 = R_4 = 50 \text{ k}\Omega$. The magnitude of the closed loop gain is ______. (rounded off...

The average bit error rate at the input of a (7, 4, 1) Hamming decoder is 0.10. The probability that the decoder will fail to decode a received word correctly is ____. (rounded off...

Consider that the concentration of electrons in a semiconductor bar varies linearly from 2 × 10^17 cm^−3 at x = 1 µm to 1× 10^16 cm^−3 at x = 4 µm along the x-direction. Assume tha...

Consider the ideal diodes D1 and D2 as shown in the Figure with cut-in voltage Vγ = 0 Volt and vi(t) is in Volt. The maximum voltage (Volt) of the output vo(t) is ____. (rounded of...

The 12 musical notes are given as C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. Frequency of each note is $\sqrt[12]{2}$ times the frequency of the previous note. If the frequency o...

The following figures show three curves generated using an iterative algorithm. The total length of the curve generated after 'Iteration n' is: Note: The figures shown are represen...

Which one of the following plots represents $f(x) = -\frac{|x|}{x}$, where x is a non-zero real number? Note: The figures shown are representative.

Consider the matrix A below: A = $\begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}$ For which of the following combinati...

Consider a frequency-modulated (FM) signal $f(t) = A_c \cos(2\pi f_c t + 3 \sin(2\pi f_1 t) + 4 \sin(6\pi f_1 t))$, where $A_c$ and $f_c$ are, respectively, the amplitude and frequ...

Consider an additive white Gaussian noise (AWGN) channel with bandwidth W and noise power spectral density $\frac{N_0}{2}$. Let $P_{av}$ denote the average transmit power constrain...

The Nyquist plot of a system is given in the figure below. Let $\omega_P, \omega_Q, \omega_R$, and $\omega_S$ be the positive frequencies at the points P, Q, R, and S, respectively...

Consider the discrete-time system below with input x[n] and output y[n]. In the figure, h₁[n] and h₂[n] denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also,...

Consider a continuous-time, real-valued signal f(t) whose Fourier transform F(ω) = ∫_(-∞)^∞ f(t) exp(-j ωt) dt exists. Which one of the following statements is always TRUE?

Consider a part of an electrical network as shown below. Some node voltages, and the current flowing through the 3Ω resistor are as indicated. The voltage (in Volts) at node X is _...

Let $i_C$, $i_L$, and $i_R$ be the currents flowing through the capacitor, inductor, and resistor, respectively, in the circuit given below. The AC admittances are given in Siemens...

A simplified small-signal equivalent circuit of a BJT-based amplifier is given below. The small-signal voltage gain Vo/Vs (in V/V) is _________.

The ideal BJT in the circuit given below is biased in the active region with a β of 100. If I_B is 10 µA, then V_CE (in Volts, rounded off to two decimal places) is ________.

A 3-input majority logic gate has inputs X, Y, and Z. The output F of the gate is logic '1' if two or more of the inputs are logic '1'. The output F is logic '0' if two or more of...

A full adder and an XOR gate are used to design a digital circuit with inputs X, Y, and Z, and output F, as shown below. The input Z is connected to the carry-in input of the full...

Consider the unity-negative-feedback system shown in Figure (i) below, where gain K ≥ 0. The root locus of this system is shown in Figure (ii) below. For what value(s) of K will th...

Let x[n] be a discrete-time signal whose Z-transform is X(z). Which of the following statements is/are TRUE?

Consider a message signal $m(t)$ which is bandlimited to $[-W, W]$, where $W$ is in Hz. Consider the following two modulation schemes for the message signal: * Double sideband-supp...

Which of the following statements is/are TRUE with respect to an ideal opamp?

Which of the following statements is/are TRUE with respect to ideal MOSFET-based DC-coupled single-stage amplifiers having finite load resistors?

Which of the following can be used as an n-type dopant for silicon? Select the correct option(s).

The function y(t) satisfies t²y''(t) - 2ty'(t) + 2y(t) = 0, where y'(t) and y''(t) denote the first and second derivatives of y(t), respectively. Given y'(0) = 1 and y'(1) = −1, th...

The generator matrix of a (6,3) binary linear block code is given by G = [1 0 0 1 0 1; 0 1 0 0 1 1; 0 0 1 1 1 0]. The minimum Hamming distance d_min between codewords equals ______...

All the components in the bandpass filter given below are ideal. The lower -3 dB frequency of the filter is 1 MHz. The upper -3 dB frequency (in MHz, rounded off to the nearest int...

A 4-bit weighted-resistor DAC with inputs b3, b2, b1, and b0 (MSB to LSB) is designed using an ideal opamp, as shown below. The switches are closed when the corresponding input bit...