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81
GATE ECE 2023 | Question: 4
The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is $-\pi(1+j)$ $\pi(1+j)$ $\pi(1-j)$ $-\pi(1-j)$
The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$,...
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GATE ECE 2023 | Question: 21
The signal-to-noise ratio $\mathrm{SNR}$ of an $\mathrm{ADC}$ with a full-scale sinusoidal input is given to be $61.96 \mathrm{~dB}$. The resolution of the $\mathrm{ADC}$ is_____________bits (rounded off to the nearest integer).
The signal-to-noise ratio $\mathrm{SNR}$ of an $\mathrm{ADC}$ with a full-scale sinusoidal input is given to be $61.96 \mathrm{~dB}$. The resolution of the $\mathrm{ADC}$...
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GATE ECE 2023 | Question: 5
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{v_k \mid 1 \leq k \leq n\right\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where ... $\left\{P^{-1} v_k \mid 1 \leq k \leq n\right\}$
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{v_k \mid 1 \leq k \leq n\right\}$, respectivel...
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GATE ECE 2023 | Question: 7
For an intrinsic semiconductor at temperature $\text{T}=0 \; \text{K}$, which of the following statement is true? All energy states in the valence band are filled with electrons and all energy states in the conduction band are empty ... the valence and conduction band are filled with holes. All energy states in the valence and conduction band are filled with electrons.
For an intrinsic semiconductor at temperature $\text{T}=0 \; \text{K}$, which of the following statement is true?All energy states in the valence band are filled with ele...
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GATE ECE 2023 | Question: 14
The synchronous sequential circuit shown below works at a clock frequency $1 \; \mathrm{GHz}$. The throughput, in $\text{Mbits/s,}$ and the latency, in $\text{ns,}$ respectively, are $1000,3$ $333.33,1$ $2000,3$ $333.33,3$
The synchronous sequential circuit shown below works at a clock frequency $1 \; \mathrm{GHz}$. The throughput, in $\text{Mbits/s,}$ and the latency, in $\text{ns,}$ respe...
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GATE ECE 2023 | Question: 50
A transparent dielectric coating is applied to glass $\left(\varepsilon_r=4, \mu_r=1\right)$ to eliminate the reflection of red light $\left(\lambda_0=0.75 \; \mu \mathrm{m}\right)$. The minimum thickness of the dielectric coating, in $\mu \mathrm{m}$, that can be used is_____________(rounded off to two decimal places).
A transparent dielectric coating is applied to glass $\left(\varepsilon_r=4, \mu_r=1\right)$ to eliminate the reflection of red light $\left(\lambda_0=0.75 \; \mu \mathrm...
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GATE ECE 2023 | Question: 9
For a $\text{MOS}$ capacitor, $\text{V}_{\mathrm{fb}}$ and $\text{V}_{\mathrm{t}}$ are the flat-band voltage and the threshold voltage, respectively. The variation of the depletion width $\left(\mathrm{W}_{\mathrm{dep}}\right)$ for varying gate voltage $\left(\text{V}_g\right)$ is best represented by
For a $\text{MOS}$ capacitor, $\text{V}_{\mathrm{fb}}$ and $\text{V}_{\mathrm{t}}$ are the flat-band voltage and the threshold voltage, respectively. The variation of the...
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GATE ECE 2023 | Question: 16
Consider a system with input $x(t)$ and output $y(t)=x\left(e^{t}\right)$. The system is Causal and time invariant. Non-causal and time varying. Causal and time varying. Non-causal and time invariant.
Consider a system with input $x(t)$ and output $y(t)=x\left(e^{t}\right)$. The system isCausal and time invariant.Non-causal and time varying.Causal and time varying.Non-...
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GATE ECE 2023 | Question: 30
In the circuit shown below, $D_1$ and $D_2$ are silicon diodes with cut-in voltage of $0.7 \mathrm{~V}$. $\mathrm{V}_{\mathrm{IN}}$ and $\mathrm{V}_{\mathrm{OUT}}$ are input and output voltages in volts. The transfer characteristic is
In the circuit shown below, $D_1$ and $D_2$ are silicon diodes with cut-in voltage of $0.7 \mathrm{~V}$. $\mathrm{V}_{\mathrm{IN}}$ and $\mathrm{V}_{\mathrm{OUT}}$ are in...
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GATE ECE 2023 | Question: 26
A random variable $\mathrm{X}$, distributed normally as $\mathrm{N(0,1)}$ undergoes the transformation $\mathrm{Y}=\mathrm{h}(\mathrm{X})$, given in the figure. The form of the probability density function of $\mathrm{Y}$ is (In the options given below, $a, b, c$ are non-zero ... $a \delta(y+2)+b \delta(y)+c \delta(y-2)+g(y)$ $a \delta(y+2)+b \delta(y-2)+g(y)$
A random variable $\mathrm{X}$, distributed normally as $\mathrm{N(0,1)}$ undergoes the transformation$\mathrm{Y}=\mathrm{h}(\mathrm{X})$, given in the figure. The form o...
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GATE ECE 2023 | Question: 17
Let $m(t)$ be a strictly band-limited signal with bandwidth $B$ and energy $E$. Assuming $\omega_{0}=10 B$, the energy in the $m(t) \cos \omega_{0} t$ is $\frac{E}{4}$ $\frac{E}{2}$ $E$ $2E$
Let $m(t)$ be a strictly band-limited signal with bandwidth $B$ and energy $E$. Assuming $\omega_{0}=10 B$, the energy in the $m(t) \cos \omega_{0} t$ is$\frac{E}{4}$$\f...
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GATE ECE 2023 | Question: 8
A series $\text{RLC}$ circuit has a quality factor $\text{Q}$ of $1000$ at a center frequency of $10^{6} \; \mathrm{rad} / \mathrm{s}$. The possible values of $\text{R, L}$ and $\text{C}$ are $R=1 \; \Omega, L=1 \; \mu H$ and $C=1 \; \mu F$ ... $C=1 \; \mu F$ $R=0.001 \; \Omega, L=1 \; \mu H$ and $C=1 \; \mu F$
A series $\text{RLC}$ circuit has a quality factor $\text{Q}$ of $1000$ at a center frequency of $10^{6} \; \mathrm{rad} / \mathrm{s}$. The possible values of $\text{R,...
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GATE ECE 2023 | Question: 27
The value of the line integral $\int_{P}^{Q}\left(z^{2} d x+3 y^{2} d y+2 x z \; d z\right)$ along the straight line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is $20$ $24$ $29$ $-5$
The value of the line integral $\int_{P}^{Q}\left(z^{2} d x+3 y^{2} d y+2 x z \; d z\right)$ along the straight line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is$20$$2...
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GATE ECE 2023 | Question: 18
The Fourier transform $X(\omega)$ of $x(t)=e^{-t^{2}}$ is Note: $\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi}$ $\sqrt{\pi} e^{\frac{\omega^{2}}{2}}$ $\frac{e^{-\frac{\omega^{2}}{4}}}{2 \sqrt{\pi}}$ $\sqrt{\pi} e^{-\frac{\omega^{2}}{4}}$ $\sqrt{\pi} e^{-\frac{\omega^{2}}{2}}$
The Fourier transform $X(\omega)$ of $x(t)=e^{-t^{2}}$ isNote: $\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi}$$\sqrt{\pi} e^{\frac{\omega^{2}}{2}}$$\frac{e^{-\frac{\o...
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GATE ECE 2023 | Question: 20
For a real signal, which of the following is/are valid power spectral density/densities? $\text{S}_X(\omega)=\frac{2}{9+\omega^2}$ $\text{S}_X(\omega)=e^{-\omega^2} \cos ^2 \omega$
For a real signal, which of the following is/are valid power spectral density/densities?$\text{S}_X(\omega)=\frac{2}{9+\omega^2}$$\text{S}_X(\omega)=e^{-\omega^2} \cos ^2...
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GATE ECE 2023 | Question: 15
The open loop transfer function of a unity negative feedback system is $G(s)=\frac{k}{s\left(1+s T_1\right)\left(1+s T_2\right)}$, where $k, T_1$ and $T_2$ are positive constants. The phase cross-over frequency, in $\text{rad/s,}$ is $\frac{1}{\sqrt{T_1 T_2}}$ $\frac{1}{T_1 T_2}$ $\frac{1}{T_1 \sqrt{T_2}}$ $\frac{1}{T_2 \sqrt{T_1}}$
The open loop transfer function of a unity negative feedback system is $G(s)=\frac{k}{s\left(1+s T_1\right)\left(1+s T_2\right)}$, where $k, T_1$ and $T_2$ are positive c...
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GATE ECE 2023 | Question: 19
In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics (i) Continuous, aperiodic (a) Continuous, aperiodic (ii) Continuous, periodic (b) Continuous, periodic (iii) Discrete, aperiodic (c) Discrete, aperiodic (iv) ...
In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics (i) Continuous, aperiodic(a) Continuous, aperiod...
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GATE ECE 2023 | Question: 22
In the circuit shown below, the current $i$ flowing through $200 \; \Omega$ resistor is___________ $\mathrm{mA}$ (rounded off to two decimal places).
In the circuit shown below, the current $i$ flowing through $200 \; \Omega$ resistor is___________ $\mathrm{mA}$ (rounded off to two decimal places).
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GATE ECE 2023 | Question: 29
The $\frac{V_{\text {OUT }}}{V_{\text {IN }}}$ of the circuit shown below is $-\frac{\mathrm{R}_4}{\mathrm{R}_3}$ $\frac{\mathrm{R}_4}{\mathrm{R}_3}$ $1+\frac{\text{R}_4}{\text{R}_3}$ $1-\frac{\mathrm{R}_4}{\mathrm{R}_3}$
The $\frac{V_{\text {OUT }}}{V_{\text {IN }}}$ of the circuit shown below is$-\frac{\mathrm{R}_4}{\mathrm{R}_3}$$\frac{\mathrm{R}_4}{\mathrm{R}_3}$$1+\frac{\text{R}_4}{\t...
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GATE ECE 2023 | Question: 54
In the circuit below, the voltage $V_L$ is _________ $\mathrm{V}$ (rounded off to two decimal places).
In the circuit below, the voltage $V_L$ is _________ $\mathrm{V}$ (rounded off to two decimal places).
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GATE ECE 2023 | Question: 31
A closed loop system is shown in the figure where $\mathrm{k}>0$ and $\alpha>0$. The steady state error due to a ramp input $\left(\mathrm{R}(\mathrm{s})=\alpha / \mathrm{s}^2\right)$ is given by $\frac{2 \alpha}{\mathrm{k}}$ $\frac{\alpha}{\mathrm{k}}$ $\frac{\alpha}{2 \mathrm{k}}$ $\frac{\alpha}{4 \mathrm{k}}$
A closed loop system is shown in the figure where $\mathrm{k}>0$ and $\alpha>0$. The steady state error due to a ramp input $\left(\mathrm{R}(\mathrm{s})=\alpha / \mathrm...
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GATE ECE 2023 | Question: 25
In the circuit shown below, switch $\mathrm{S}$ was closed for a long time. If the switch is opened at $t=0$, the maximum magnitude of the voltage $\mathrm{V}_{\mathrm{R}}, \text{in volts,}$ is_____________ (rounded off to the nearest integer).
In the circuit shown below, switch $\mathrm{S}$ was closed for a long time. If the switch isopened at $t=0$, the maximum magnitude of the voltage $\mathrm{V}_{\mathrm{R}}...
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GATE ECE 2023 | Question: 46
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} \mathrm{~cm}^{-3}$. The ratio of electron to hole mobility for equal hole and electron drift current is given as ________ (rounded off to two decimal places).
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} \mathrm{~cm}^{-...
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GATE ECE 2023 | Question: 55
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 $\text{"yes/no"}$ questions which are assumed to be truthfully answered. The average number of questions when asked in the most ... $\frac{1}{16}$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{128}$
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 $\text{"yes/no"}$ questions ...
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GATE ECE 2023 | Question: 43
The electric field of a plane electromagnetic wave is $\boldsymbol{E}=\boldsymbol{a}_{x} C_{1 x} \cos (\omega t-\beta z)+\boldsymbol{a}_{y} C_{1 y} \cos (\omega t-\beta z+\theta) \quad \mathrm{V} / \mathrm{m}$. Which of the following combination(s) will give rise to a left handed ... $C_{1 x}=1, C_{1 y}=2, \theta=3 \pi / 2$ $C_{1 x}=2, C_{1 y}=1, \theta=3 \pi / 4$
The electric field of a plane electromagnetic wave is$\boldsymbol{E}=\boldsymbol{a}_{x} C_{1 x} \cos (\omega t-\beta z)+\boldsymbol{a}_{y} C_{1 y} \cos (\omega t-\beta z+...
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GATE ECE 2023 | Question: 40
Let $\mathrm{x}_1(\mathrm{t})$ and $\mathrm{x}_2(\mathrm{t})$ be two band-limited signals having bandwidth $B=4 \pi \times 10^3 \; \mathrm{rad} / \mathrm{s}$ each. In the figure below, the Nyquist sampling frequency, in $\mathrm{rad} / \mathrm{s}$, required to sample $y(\mathrm{t})$, is $20 \pi \times 10^3$ $40 \pi \times 10^3$ $8 \pi \times 10^3$ $32 \pi \times 10^3$
Let $\mathrm{x}_1(\mathrm{t})$ and $\mathrm{x}_2(\mathrm{t})$ be two band-limited signals having bandwidth $B=4 \pi \times 10^3 \; \mathrm{rad} / \mathrm{s}$ each. In the...
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GATE ECE 2007 | Question: 9
The electron and hole concentrations in an intrinsic semiconductor are $n_i$ per $\mathrm{cm}^3$ at $300 \mathrm{~K}$. Now, if acceptor impurities are introduced with a concentration of $N_A$ per $\mathrm{cm}^3$ (where $N_A \gg n_i$ ), the electron concentration per $\mathrm{cm}^3$ at $300 \mathrm{~K}$ will be $n_{i}$ $n_i+N_A$ $N_A-n_{i}$ $\frac{n_i^2}{N_A}$
The electron and hole concentrations in an intrinsic semiconductor are $n_i$ per $\mathrm{cm}^3$ at $300 \mathrm{~K}$. Now, if acceptor impurities are introduced with a c...
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GATE ECE 2023 | Question: 41
The $S$-parameters of a two-port network is given as $ [S]=\left[\begin{array}{ll} S_{11} & S_{12} \\ S_{21} & S_{22} \end{array}\right] $ with reference to $Z_0$. Two lossless transmission line sections of electrical lengths $\theta_1=\beta l_1$ ...
The $S$-parameters of a two-port network is given as$$[S]=\left[\begin{array}{ll}S_{11} & S_{12} \\S_{21} & S_{22}\end{array}\right]$$with reference to $Z_0$. Two lossles...
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GATE ECE 2023 | Question: 39
Let $x(t)=10 \cos (10.5 W t)$ be passed through an LTI system having impulse response $h(t)=\pi\left(\frac{\sin W t}{\pi t}\right)^{2} \cos 10 Wt$. The output of the system is $\left(\frac{15 W}{4}\right) \cos (10.5 W t)$ $\left(\frac{15 W}{2}\right) \cos (10.5 W t)$ $\left(\frac{15 W}{8}\right) \cos (10.5 W t)$ $(15 W) \cos (10.5 W t)$
Let $x(t)=10 \cos (10.5 W t)$ be passed through an LTI system having impulse response $h(t)=\pi\left(\frac{\sin W t}{\pi t}\right)^{2} \cos 10 Wt$. The output of the sys...
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GATE ECE 2023 | Question: 38
Let an input $x[n]$ having discrete-time Fourier transform $X\left(e^{j \Omega}\right)=1-e^{-j \Omega}+2 e^{-3 j \Omega}$ be passed through an LTI system. The frequency response of the LTI system is $H\left(e^{j \Omega}\right)=1-\frac{1}{2} e^{-j 2 \Omega}$. The ... $\delta[n]+\delta[n-1]+\frac{1}{2} \delta[n-2]+\frac{5}{2} \delta[n-3]+\delta[n-5]$
Let an input $x[n]$ having discrete-time Fourier transform$X\left(e^{j \Omega}\right)=1-e^{-j \Omega}+2 e^{-3 j \Omega}$ be passed through an LTI system. The frequency re...
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GATE ECE 2023 | Question: 33
The state equation of a second order system is $\dot{\boldsymbol{x}}(t)=\mathrm{A} \boldsymbol{x}(t), \; \boldsymbol{x}(0)$ is the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of $\mathrm{A}$ and $v_1$ and $v_2$ are ... $\sum_{i=1}^2 \alpha_i e^{4 \lambda_i \mathrm{t}} \boldsymbol{v}_i$
The state equation of a second order system is$\dot{\boldsymbol{x}}(t)=\mathrm{A} \boldsymbol{x}(t), \; \boldsymbol{x}(0)$ is the initial condition.Suppose $\lambda_1$ an...
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GATE ECE 2023 | Question: 34
The switch $\mathrm{S}_{1}$ was closed and $\mathrm{S}_{2}$ was open for a long time. At $t=0$, switch $\mathrm{S}_{1}$ is opened and $\mathrm{S}_{2}$ is closed, simultaneously. The value of $\mathrm{i}_{\mathrm{c}}\left(0^{+}\right)$, in amperes, is $1$ $-1$ $0.2$ $0.8$
The switch $\mathrm{S}_{1}$ was closed and $\mathrm{S}_{2}$ was open for a long time. At $t=0$, switch $\mathrm{S}_{1}$ is opened and $\mathrm{S}_{2}$ is closed, simultan...
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GATE ECE 2023 | Question: 51
In a semiconductor device, the Fermi-energy level is $0.35 \; \mathrm{eV}$ above the valence band energy. The effective density of states in the valence band at $T=300 \mathrm{~K}$ is $1 \times 10^{19} \mathrm{~cm}^{-3}$. The thermal equilibrium hole ... $\mathrm{kT}$ at $300 \mathrm{~K}$ is $0.026 \; \mathrm{eV}$.
In a semiconductor device, the Fermi-energy level is $0.35 \; \mathrm{eV}$ above the valence band energy. The effective density of states in the valence band at $T=300 \m...
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GATE ECE 2023 | Question: 37
Consider a discrete-time periodic signal with period $N=5$. Let the discrete-time Fourier series $\text{(DTFS}$) representation be $x[n]=\sum_{k=0}^4 a_k e^{\frac{j k 2 \pi n}{5}}$, where $a_0=1, a_1=$ $3 j, a_2=2 j, a_3=-2 j$ and $a_4=-3 j$. The value of the sum $\sum_{n=0}^4 x[n] \sin \frac{4 \pi n}{5}$ is $-10$ $10$ $-2$ $2$
Consider a discrete-time periodic signal with period $N=5$. Let the discrete-time Fourier series $\text{(DTFS}$) representation be $x[n]=\sum_{k=0}^4 a_k e^{\frac{j k 2 \...
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GATE ECE 2023 | Question: 42
The standing wave ratio on a $50 \; \Omega$ lossless transmission line terminated in an unknown load impedance is found to be $2.0$. The distance between successive voltage minima is $30 \mathrm{~cm}$ and the first minimum is located at $10 \mathrm{~cm}$ ... $R_m=100 \; \Omega, l_m=5 \mathrm{~cm}$ $R_m=25 \; \Omega, l_m=5 \mathrm{~cm}$
The standing wave ratio on a $50 \; \Omega$ lossless transmission line terminated in an unknown load impedance is found to be $2.0$. The distance between successive volta...
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GATE ECE 2023 | Question: 32
In the following block diagram, $\mathrm{R}(\mathrm{s})$ and $\mathrm{D}(\mathrm{s})$ are two inputs. The output $\mathrm{Y}(\mathrm{s})$ is expressed as $Y(s)=G_1(s) R(s)+G_2(s) D(s)$. $\mathrm{G}_1(\mathrm{~s})$ and $\mathrm{G}_2(\mathrm{~s})$ ...
In the following block diagram, $\mathrm{R}(\mathrm{s})$ and $\mathrm{D}(\mathrm{s})$ are two inputs. The output $\mathrm{Y}(\mathrm{s})$ is expressed as $Y(s)=G_1(s) R(s...
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GATE ECE 2023 | Question: 49
Let $X(t)$ be a white Gaussian noise with power spectral density $\frac{1}{2} \mathrm{~W} / \mathrm{Hz}$. If $X(t)$ is input to an LTI system with impulse response $e^{-t} u(t)$. The average power of the system output is _____________ $\mathrm{W}$ (rounded off to two decimal places).
Let $X(t)$ be a white Gaussian noise with power spectral density $\frac{1}{2} \mathrm{~W} / \mathrm{Hz}$. If $X(t)$ is input to an LTI system with impulse response $e^{-t...
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GATE ECE 2023 | Question: 36
The $\text{h}$-parameters of a two port network are shown below. The condition for the maximum small signal voltage gain $\frac{\mathrm{V}_{\text {out }}}{\mathrm{V}_{\mathrm{s}}}$ is $\mathrm{h}_{11}=0, \mathrm{~h}_{12}=0, \mathrm{~h}_{21}=$ ... $\mathrm{h}_{11}=0, \mathrm{~h}_{12}=0, \mathrm{~h}_{21}=$ very high and $\mathrm{h}_{22}=$ very high
The $\text{h}$-parameters of a two port network are shown below. The condition for the maximum small signal voltage gain $\frac{\mathrm{V}_{\text {out }}}{\mathrm{V}_{\ma...
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GATE ECE 2023 | Question: 35
Let a frequency modulated $\text{(FM)}$ signal $x(t)=A \cos \left(\omega_{c} t+k_{f} \int_{-\infty}^{t} m(\lambda) d \lambda\right)$, where $m(t)$ is a message signal of bandwidth $\text{W.}$ ... to recover $x(t)$ from $y(t)$ is $B_{T}+W$ $\frac{3}{2} B_{T}$ $2 B_{T}+W$ $\frac{5}{2} B_{T}$
Let a frequency modulated $\text{(FM)}$ signal$x(t)=A \cos \left(\omega_{c} t+k_{f} \int_{-\infty}^{t} m(\lambda) d \lambda\right)$, where $m(t)$ is a message signal of b...
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GATE ECE 2023 | Question: 52
A sample and hold circuit is implemented using a resistive switch and a capacitor with a time constant of $1 \; \mu \mathrm{s}$. The time for the sampling switch to stay closed to charge a capacitor adequately to a full scale voltage of $1 \mathrm{~V}$ with $12$-bit accuracy is ___________ $\mu \mathrm{s}$ (rounded off to two decimal places).
A sample and hold circuit is implemented using a resistive switch and a capacitor with a time constant of $1 \; \mu \mathrm{s}$. The time for the sampling switch to stay ...
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