GATE 2023 Electronics & Communication

Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is

The state equation of a second order system is $$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition. Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and...

A sample and hold circuit is implemented using a resistive switch and a capacitor with a time constant of 1 $$\mu$$s. The time for the sampling switch to stay closed to charge a ca...

A series $$RLC$$ circuit has a quality factor Q of 1000 at a center frequency of $$10^6$$ rad/s. The possible values of R, L and C are

Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the...

Consider a narrow band signal, propagating in a lossless dielectric medium $$(\varepsilon_r=4,\mu_r=1)$$, with phase velocity $$v_p$$ and group velocity $$v_g$$. Which of the follo...

For a real signal, which of the following is/are valid power spectral density/densities?

Let a frequency modulated (FM) signal $$x(t) = A\cos ({\omega _c}t + {k_f}\int_{ - \infty }^t {m(\lambda )d\lambda )} $$, where $$m(t)$$ is a message signal of bandwidth W. It is p...

The following circuit(s) representing a lumped element equivalent of an infinitesimal sectioin of a transmission line is/are

Let X(t) be a white Gaussian noise with power spectral density $$\frac{1}{2}$$W/Hz. If X(t) is input to an LTI system with impulse response $$e^{-t}u(t)$$. The average power of the...

A transparent dielectric coating is applied to glass ($$\varepsilon _r=4,\mu_r=1$$) to eliminate the reflection of red light ($$\lambda_0=0.75~\mu\mathrm{m}$$). The minimum thickne...

The frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of "yes/no" questions which are assumed to be...

Let $$x(t) = 100\cos (10.5Wt)$$ be passed through an LTI system having impulse response $$h(t) = \pi {\left( {{{\sin Wt} \over {\pi t}}} \right)^2}\cos 10Wt$$. The output of the sy...

Let $${w^4} = 16j$$. Which of the following cannot be a value of $$w$$?

In a semiconductor, if the Fermi energy level lies in the conduction band, then the semiconductor is known as

For an intrinsic semiconductor at temperature $$T=0K$$, which of the following statement is true?

For a MOS capacitor, $$\mathrm{V_{fb}}$$ and $$\mathrm{V_{t}}$$ are the flat-band voltage and the threshold voltage, respectively. The variation of the depletion width ($$\mathrm{W...

Consider a system with input $$x(t)$$ and output $$y(t) = x({e^t})$$. The system is

Let $$m(t)$$ be a strictly band-limited signal with bandwidth B and energy E. Assuming $${\omega _0} = 10B$$, the energy in the signal $$m(t)\cos {\omega _0}t$$ is

The Fourier transform $$x(\omega )$$ of $$x(t) = {e^{ - {t^2}}}$$ is Note : $$\int\limits_{ - \infty }^\infty {{e^{ - {y^2}}}dy = \sqrt \pi } $$

In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics (i) Continuous, aperiodic (a) Continuous, aperiodic (ii) Con...

Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be $$x[n] = \sum\limits_{k = 0}^4 {{a_k}{e^{{{jk2\pi m} \over...

Let an input $$x[n]$$ having discrete time Fourier transform $$x({e^{j\Omega }}) = 1 - {e^{ - j\Omega }} + 2{e^{ - 3j\Omega }}$$ be passed through an LTI system. The frequency resp...

In an extrinsic semiconductor, the hole concentration is given to be 1.5$$n_i$$ where $$n_i$$ is the intrinsic carrier concentration of 1 $$\times$$ 10$$^{10}$$ $$cm^{-3}$$. The ra...

In a semiconductor device, the Fermi-energy level is 0.35 eV above the valence band energy. The effective density of states in the valence band at T = 300 K is 1 $$\times$$ 10$$^{1...

The value of the contour integral, $$\oint\limits_C {\left( {{{z + 2} \over {{z^2} + 2z + 2}}} \right)dz} $$, where the contour C is $$\left\{ {z:\left| {z + 1 - {3 \over 2}j} \rig...

The rate of increase, of a scalar field $$f(x,y,z) = xyz$$, in the direction $$v = (2,1,2)$$ at a point (0,2,1) is

Let $${v_1} = \left[ {\matrix{ 1 \cr 2 \cr 0 \cr } } \right]$$ and $${v_2} = \left[ {\matrix{ 2 \cr 1 \cr 3 \cr } } \right]$$ be two vectors. The value of the coefficient $$\alpha$...

A 100 cm $$\times$$ 32 cm rectangular sheet is folded 5 times. Each time the sheet is folded, the long edge aligns with its opposite side. Eventually, the folded sheet is a rectang...

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strateg...

Which one of the following options can be inferred from the given passage alone? When I was a kid, I was partial to stories about other worlds and interplanetary travel. I used to...

Forestland is a planet inhabited by different kinds of creatures. Among other creatures, it is populated by animals all of whom are ferocious. There are also creatures that have cl...

In a class of 100 students, (i) there are 30 students who neither like romantic movies nor comedy movies, (ii) the number of students who like romantic movies is twice the number o...

What is the smallest number with distinct digits whose digits add up to 45?

A cascade of common-source amplifiers in a unity gain feedback configuration oscillates when

"I cannot support this proposal. My __________ will not permit it."

The electric field of a plane electromagnetic wave is $$E = {a_x}{C_{1x}}\cos (\omega t - \beta z) + {a_y}{C_{1y}}\cos (\omega t - \beta z + \theta )$$ V/m. Which of the following...

Courts : __________ : : Parliament : Legislature (By word meaning)

The open loop transfer function of a unity negative feedback system is $$G(s) = {k \over {s(1 + s{T_1})(1 + s{T_2})}}$$, where $$k,T_1$$ and $$T_2$$ are positive constants. The pha...

The signal-to-noise ratio (SNR) of an ADC with a full-scale sinusoidal input is given to be 61.96 dB. The resolution of the ADC is __________ bits (rounded off to the nearest integ...

The value of the line integral $$\int_P^Q {({z^2}dx + 3{y^2}dy + 2xz\,dz)} $$ along the straight line joining the points $$P(1,1,2)$$ and $$Q(2,3,1)$$ is

"I cannot support this proposal. My ________ will not permit it."

Courts : ________ :: Parliament : Legislature (By word meaning)

What is the smallest number with distinct digits whose digits add up to 45?

In a class of 100 students, (i) there are 30 students who neither like romantic movies nor comedy movies, (ii) the number of students who like romantic movies is twice the number o...

How many rectangles are present in the given figure?

Forestland is a planet inhabited by different kinds of creatures. Among other creatures, it is populated by animals all of whom are ferocious. There are also creatures that have cl...

Which one of the following options represents the given graph?

Which one of the following options can be inferred from the given passage alone? When I was a kid, I was partial to stories about other worlds and interplanetary travel. I used to...

Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strateg...

A 100 cm × 32 cm rectangular sheet is folded 5 times. Each time the sheet is folded, the long edge aligns with its opposite side. Eventually, the folded sheet is a rectangle of dim...

Let $v_1 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}$ be two vectors. The value of the coefficient $\alpha$ in the expression...

The rate of increase, of a scalar field $f(x, y, z) = xyz$, in the direction $v = (2,1,2)$ at a point $(0,2,1)$ is

Let w⁴ = 16j. Which of the following cannot be a value of w?

The value of the contour integral, $\oint_C \left(\frac{z+2}{z^2+2z+2}\right) dz$, where the contour C is $\{z: |z + 1 - \frac{3}{2}j| = 1\}$, taken in the counter clockwise direct...

Let the sets of eigenvalues and eigenvectors of a matrix B be {λk | 1 ≤ k ≤ n} and {vk | 1 ≤ k ≤ n}, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvec...

In a semiconductor, if the Fermi energy level lies in the conduction band, then the semiconductor is known as

For an intrinsic semiconductor at temperature T = 0 K, which of the following statement is true?

A series RLC circuit has a quality factor Q of 1000 at a center frequency of 10^6 rad/s. The possible values of R, L and C are

For a MOS capacitor, Vfb and Vt are the flat-band voltage and the threshold voltage, respectively. The variation of the depletion width (Wdep) for varying gate voltage (Vg) is best...

Consider a narrow band signal, propagating in a lossless dielectric medium ($\epsilon_r = 4, \mu_r = 1$), with phase velocity $v_p$ and group velocity $v_g$. Which of the following...

In the circuit shown below, V₁ and V₂ are bias voltages. Based on input and output impedances, the circuit behaves as a

A cascade of common-source amplifiers in a unity gain feedback configuration oscillates when

In the circuit shown below, P and Q are the inputs. The logical function realized by the circuit shown below is

The synchronous sequential circuit shown below works at a clock frequency of 1 GHz. The throughput, in Mbits/s, and the latency, in ns, respectively, are

The open loop transfer function of a unity negative feedback system is G(s) = k / (s(1+sT₁)(1+sT₂)), where k, T₁ and T₂ are positive constants. The phase cross-over frequency, in r...

Consider a system with input x(t) and output y(t) = x(eᵗ). The system is

Let m(t) be a strictly band-limited signal with bandwidth B and energy E. Assuming ω₀ = 10B, the energy in the signal m(t) cos ω₀t is

The Fourier transform X(ω) of x(t) = e⁻ᵗ² is Note: ∫₋∞^∞ e⁻ʸ² dy = √π

In the table shown below, match the signal type with its spectral characteristics. Signal type (i) Continuous, aperiodic (ii) Continuous, periodic (iii) Discrete, aperiodic (iv) Di...

For a real signal, which of the following is/are valid power spectral density/densities?

The signal-to-noise ratio (SNR) of an ADC with a full-scale sinusoidal input is given to be 61.96 dB. The resolution of the ADC is _________ bits (rounded off to the nearest intege...

In the circuit shown below, the current i flowing through 200 Ω resistor is _________ mA (rounded off to two decimal places).

For the two port network shown below, the [Y]-parameters is given as $[Y] = \frac{1}{100} \begin{bmatrix} 2 & -1 \\ -1 & 4/3 \end{bmatrix} S$ The value of load impedance ZL, in Ω,...

For the circuit shown below, the propagation delay of each NAND gate is 1 ns. The critical path delay, in ns, is ________ (rounded off to the nearest integer).

In the circuit shown below, switch S was closed for a long time. If the switch is opened at t = 0, the maximum magnitude of the voltage VR, in volts, is ________ (rounded off to th...

A random variable X, distributed normally as N(0,1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density function of Y is (In the option...

The value of the line integral $\int_P^Q(z^2 dx + 3y^2 dy + 2xz dz)$ along the straight line joining the points P (1,1,2) and Q (2,3,1) is

Let $\mathbf{x}$ be an $n \times 1$ real column vector with length $l = \sqrt{\mathbf{x}^T\mathbf{x}}$. The trace of the matrix $P = \mathbf{x}\mathbf{x}^T$ is

The $\frac{V_{OUT}}{V_{IN}}$ of the circuit shown below is

In the circuit shown below, D₁ and D₂ are silicon diodes with cut-in voltage of 0.7 V. Vᵢₙ and Vₒᵤₜ are input and output voltages in volts. The transfer characteristic is

A closed loop system is shown in the figure where k > 0 and α > 0. The steady state error due to a ramp input (R(s) = α/s²) is given by

In the following block diagram, R(s) and D(s) are two inputs. The output Y(s) is expressed as Y(s) = G₁(s)R(s) + G₂(s)D(s). G₁(s) and G₂(s) are given by

The state equation of a second order system is $\dot{x}(t) = Ax(t)$, $x(0)$ is the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of A and $v_1...

The switch S₁ was closed and S₂ was open for a long time. At t = 0, switch S₁ is opened and S2 is closed, simultaneously. The value of i_c(0+), in amperes, is

Let a frequency modulated (FM) signal x(t) = A cos(ω_c t + k_f ∫_{-∞}^t m(λ)dλ), where m(t) is a message signal of bandwidth W. It is passed through a non-linear system with output...

The h-parameters of a two port network are shown below. The condition for the maximum small signal voltage gain $\frac{V_{out}}{V_s}$ is

Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be x[n] = $\sum_{k=0}^{4} a_k e^{\frac{jk2\pi n}{5}}$, where...

Let an input x[n] having discrete time Fourier transform $X(e^{j\Omega}) = 1 - e^{-j\Omega} + 2e^{-3j\Omega}$ be passed through an LTI system. The frequency response of the LTI sys...

Let x(t) = 10 cos(10.5Wt) be passed through an LTI system having impulse response h(t) = π (sin Wt / πt)^2 cos 10Wt. The output of the system is

Let x1(t) and x2(t) be two band-limited signals having bandwidth B = 4π × 10³ rad/s each. In the figure below, the Nyquist sampling frequency, in rad/s, required to sample y(t), is

The S-parameters of a two port network is given as $[S] = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix}$ with reference to Z₀. Two lossless transmission line sec...

The standing wave ratio on a 50 Ω lossless transmission line terminated in an unknown load impedance is found to be 2.0. The distance between successive voltage minima is 30 cm and...

The electric field of a plane electromagnetic wave is $E = a_xC_{1x} \cos(\omega t – \beta z) + a_yC_{1y} \cos(\omega t – \beta z + \theta)$ V/m. Which of the following combination...

The following circuit(s) representing a lumped element equivalent of an infinitesimal section of a transmission line is/are

The value of the integral $\iint_R xy \, dx \, dy$ over the region R, given in the figure, is ________ (rounded off to the nearest integer).

In an extrinsic semiconductor, the hole concentration is given to be $1.5n_i$ where $n_i$ is the intrinsic carrier concentration of $1 \times 10^{10} \, cm^{-3}$. The ratio of elec...

The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is $G(s) = \frac{k(s+z)^a}{s^b (s+p)^c}$, where k, z, p, a,...

Let x1(t) = u(t + 1.5) - u(t - 1.5) and x2(t) is shown in the figure below. For y(t) = x1 (t) * x2 (t), the $\int_{-\infty}^{\infty}y(t)dt$ is ________ (rounded off to the nearest...

Let X(t) be a white Gaussian noise with power spectral density $\frac{1}{2}$ W/Hz. If X(t) is input to an LTI system with impulse response $e^{-t}u(t)$. The average power of the sy...

A transparent dielectric coating is applied to glass ($\epsilon_r = 4, \mu_r = 1$) to eliminate the reflection of red light ($\lambda_0 = 0.75 \mu m$). The minimum thickness of the...

In a semiconductor device, the Fermi-energy level is 0.35 eV above the valence band energy. The effective density of states in the valence band at T = 300 K is $1 \times 10^{19}cm^...

A sample and hold circuit is implemented using a resistive switch and a capacitor with a time constant of 1 $\mu s$. The time for the sampling switch to stay closed to charge a cap...

In a given sequential circuit, initial states are Q1 = 1 and Q2 = 0. For a clock frequency of 1 MHz, the frequency of signal Q2 in kHz, is ________ (rounded off to the nearest inte...

In the circuit below, the voltage VL is ________ V (rounded off to two decimal places).

The frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of "yes/no" questions which are assumed to be...