List of practice Questions

Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the $60^\circ N$ latitude circle, that contains the north pole in its interior is $\_\_\_\_$.

Consider a unity negative feedback control system with forward path gain: \[ G(s) = \frac{K}{(s+1)(s+2)(s+3)}. \] The impulse response of the closed-loop system decays faster than $e^{-t}$ if $\_\_\_\_$.

A satellite attitude control system, as shown below, has a plant with transfer function: $ G(s) = \frac{1}{s^2}, $ cascaded with a compensator: $ C(s) = \frac{K(s + \alpha)}{s + 4}, $ where $ K $ and $ \alpha $ are positive real constants. In order for the closed-loop system to have poles at $ -1 \pm j\sqrt{3} $, the value of $ \alpha $ must be $\_\_\_\_\_$.

A uniform plane wave with electric field: \[ \vec{E}(x) = A_y \hat{a}_y e^{-\frac{j2\pi x}{3}} \, {V/m}, \] is traveling in the air (relative permittivity, $\epsilon_r = 1$, and relative permeability, $\mu_r = 1$) in the $+x$ direction. It is incident normally on an ideal electric conductor (conductivity, $\sigma = \infty$) at $x = 0$. The position of the first null of the total magnetic field in the air (measured from $x = 0$, in meters) is $\_\_\_\_$.

A 4-bit priority encoder has inputs $ D_3, D_2, D_1, $ and $ D_0 $ in descending order of priority. The two-bit output $ AB $ is generated as 00, 01, 10, and 11 corresponding to inputs $ D_3, D_2, D_1, $ and $ D_0 $, respectively. The Boolean expression of the output bit $ B $ is $\_\_\_\_$.

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

The sequence of states ($Q_1 Q_0$) 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: \[ \oint_C f(z) \, dz \] 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: \[ Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{t^2}{2}} dt, \] the optimum symbol error probability is $\_\_\_\_$.

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

The information bit sequence $\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}$ 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 Hz. The sampled signal $x_s(t)$ when passed through an LTI system with impulse response: \[ h(t) = \frac{\sin(2\pi t)}{\pi t} \cos(38\pi t - \pi/2), \] produces an output $x_0(t)$. The expression for $x_0(t)$ is $\_\_\_\_$.

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

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 $g_{m1}$ and $g_{m2}$, respectively. Neglect body effect, channel length modulation, and intrinsic device capacitances. Assuming that capacitor $C_1$ is a short circuit for AC analysis, the exact magnitude of small signal voltage gain $\left|\frac{v_{{out}}}{v_{{in}}}\right|$ is $\_\_\_\_$.

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

Consider a system $S$ represented in state space as: \[ \frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix} x + \begin{bmatrix} 1 \\ 0 \end{bmatrix} r, \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix} x. \] Which of the state space representations given below has/have the same transfer function as that of $S$?

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: \[ \int_C \left(F_1 dx + F_2 dy + F_3 dz\right) \] evaluates to the same value along any path $C$ that starts at $A$ and ends at $B$. Then which of the following is/are true?

Consider the matrix: \[ \begin{bmatrix} 1 & k \\ 2 & 1 \end{bmatrix}, \] 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 $X(t) = A \cos(2\pi f_0 t + \theta)$ 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: \[ R_X(\tau) = \frac{2}{3} \cos(2\pi f_0 \tau), \] the signal-to-quantization noise ratio (in dB, rounded off to two decimal places) at the output of the quantizer is $\_\_\_\_$.

A lossless transmission line with characteristic impedance $Z_0 = 50 \, \Omega$ is terminated with an unknown load. The magnitude of the reflection coefficient 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 $N$-point discrete Fourier transform $X[k]$ is defined as: \[ X[k] = \mathcal{F}\{x[n]\}. \] Another sequence $y[n]$ is formed as: \[ y[n] = \mathcal{F}\{\mathcal{F}\{\mathcal{F}\{x[n]\}\}\}. \] For the sequence $x[n] = \{1, 2, 1, 3\}$, the value of $y[0]$ is $\_\_\_\_$.

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