GATE EC question library

​​​​​The propagation delay of the $2 \times 1$ MUX shown in the circuit is $10 \mathrm{~ns}$. Consider the propagation delay of the inverter as $0 \mathrm{~ns}$. If $\text{S}$ is set to $1$ then the output $\text{Y}$ is $\_\_\_\_\_\_$.

​​​​​The sequence of states $\left(Q_{1} Q_{0}\right)$ of the given synchronous sequential circuit is - $\_\_\_\_\_\_\_$.

​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ is $\_\_\_\_\_\_$.

Consider two continuous time signals $x(t)$ and $y(t)$ as shown below If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is $\_\_\_\_\_\_$.

​​​​​A source transmits a symbol $s$, taken from $\{-4,0,4\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is given by $r=s+w$, where the noise $w$ is zero mean with variance 4 and is independent of $s$. Using $Q(x)=\frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} e^{\frac{-t^{2}}{2}} d t$, the …

​​​​​A full scale sinusoidal signal is applied to a $10$ -bit $\mathrm{ADC}$. The fundamental signal component in the $\mathrm{ADC}$ output has a normalized power of $1 \mathrm{~W}$, and the total noise and distortion normalized power is $10 \mu \mathrm{W}$. The effective number of bits (rounded off to the nearest integer) of the $\mathrm{ADC}$ …

​​​​​The information bit sequence $\{111010101\}$ is to be transmitted by encoding with Cyclic Redundancy Check $4$ (CRC-$4$) code, for which the generator polynomial is $C(x)=x^{4}+x+1$. The encoded sequence of bits is $\_\_\_\_\_\_\_$.

​​​​A continuous time signal $x(t)=2 \cos (8 \pi t+\pi / 3)$ is sampled at a rate of $15 \mathrm{~Hz}$. The sampled signal $x_{s}(t)$ when passed through an LTI system with impulse response $$h(t)=\left(\frac{\sin 2 \pi t}{\pi t}\right) \cos (38 \pi t-\pi / 2)$$ produces an output $x_{o}(t)$. The expression for $x_{o}(t)$ is $\_\_\_\_\_$.

​​​​​The opamps in the circuit shown are ideal, but have saturation voltages of $\pm 10 \mathrm{~V}$. Assume that the initial inductor current is $0 \mathrm{~A}$. The input voltage $\left(\mathrm{V}_{\mathrm{i}}\right)$ is a triangular signal with peak voltages of $\pm 2 \mathrm{~V}$ and time period of $8 \mu \mathrm{s}$. Which one of the following …

In the circuit below, the opamp is ideal. If the circuit is to show sustained oscillations, the respective values of $R_{1}$ and the corresponding frequency of oscillation are

In the circuit shown below, the transistors $M_{1}$ and $M_{2}$ are biased in saturation. Their small signal transconductances are $g_{m 1}$ and $g_{m 2}$ respectively. Neglect body effect, channel length modulation and intrinsic device capacitances. Assuming that capacitor $C_{1}$ is a short circuit for $\mathrm{AC}$ analysis, the exact …

Which of the following statements is/are true for a $\text{BJT}$ with respect to its DC current gain $\beta$ ?

Consider a system $S$ represented in state space as $$\frac{d x}{d t}=\left[\begin{array}{ll} 0 & -2 \\ 1 & -3 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] r, y=\left[\begin{array}{ll} 2 & -5 \end{array}\right] x .$$ Which of the state space representations given below has/have the same transfer function as that of …

Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)$ evaluates to the same value along any path $C$ that starts at $A$ and ends at $B$. Then which of the following …

​​​​​Consider the matrix $\left[\begin{array}{ll}1 & k \\ 2 & 1\end{array}\right]$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

​​​​​The radian frequency value(s) for which the discrete time sinusoidal signal $x[n]=A \cos (\Omega n+\pi / 3)$ has a period of $40$ is/are $\_\_\_\_\_\_$.

Let $X(t)=A \cos \left(2 \pi f_{0} t+\theta\right)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2,2]$ and $[0,2 \pi]$, respectively. $X(t)$ is fed to an 8-bit uniform mid-rise type quantizer. Given that the autocorrelation of $X(t)$ is …

A lossless transmission line with characteristic impedance $\mathrm{Z}_{0}=50 \Omega$ is terminated with an unknown load. The magnitude of the reflection co-efficient is $|\Gamma|=0.6$. As one moves towards the generator from the load, the maximum value of the input impedance magnitude looking towards the load (in $\Omega$ ) is $\_\_\_\_\_\_.$

The relationship between any N-length sequence $x[n]$ and its corresponding $\mathrm{N}$-point discrete Fourier transform $X[k]$ is defined as $$X[k]=\mathcal{F}\{x[n]\} .$$ Another sequence $y[n]$ is formed as below $$y[n]=\mathcal{F}\{\mathcal{F}\{\mathcal{F}\{\mathcal{F}\{x[n]\}\}\}\} .$$ For the sequence $x[n]=\{1,2,1,3\}$, the value of …

For the two port network shown below, the value of the $Y_{21}$ parameter (in Siemens) is $\_\_\_\_\_\_$.