List of practice Questions

A digital communication system transmits through a noiseless bandlimited channel $[-W, W]$. The received signal $z(t)$ at the output of the receiving filter is given by: \[ z(t) = \sum_n b[n] x(t - nT), \] where $b[n]$ are the symbols and $x(t)$ is the overall system response to a single symbol. The received signal is sampled at $t = mT$. The Fourier transform of $x(t)$ is $X(f)$. The Nyquist condition that $X(f)$ must satisfy for zero intersymbol interference at the receiver is:

Consider a lossless transmission line terminated with a short circuit as shown in the figure below. As one moves towards the generator from the load, the normalized impedances $z_{inA}$, $z_{inB}$, $z_{inC}$, and $z_{inD}$ (indicated in the figure) are: \begin{center} \includegraphics[width=8cm]{15.png} \end{center}

Let $\hat{i}$ and $\hat{j}$ be the unit vectors along $x$ and $y$ axes, respectively, and let $A$ be a positive constant. Which one of the following statements is true for the vector fields: \[ \vec{F_1} = A(\hat{i}y + \hat{j}x) \quad {and} \quad \vec{F_2} = A(\hat{i}y - \hat{j}x)? \]

In the circuit below, assume that the long channel NMOS transistor is biased in saturation. The small signal transconductance of the transistor is $g_m$. Neglect body effect, channel length modulation, and intrinsic device capacitances. The small signal input impedance $Z_{in}(j\omega)$ is:

For the closed-loop amplifier circuit shown below, the magnitude of open-loop low-frequency small signal voltage gain is 40. All the transistors are biased in saturation. The current source $I_{SS}$ is ideal. Neglect body effect, channel length modulation, and intrinsic device capacitances. The closed-loop low-frequency small signal voltage gain $\frac{v_{out}}{v_{in}}$ (rounded off to three decimal places) is: \begin{center} \includegraphics[width=6cm]{18.png} \end{center}

For the Boolean function: \[ F(A, B, C, D) = \Sigma m(0, 2, 5, 7, 8, 10, 12, 13, 14, 15), \] the essential prime implicants are:

A white Gaussian noise $w(t)$ with zero mean and power spectral density $\frac{N_0}{2}$, when applied to a first-order RC low-pass filter produces an output $n(t)$. At a particular time $t = t_k$, the variance of the random variable $n(t_k)$ is:

A causal and stable LTI system with impulse response $h(t)$ produces an output $y(t)$ for an input signal $x(t)$. A signal $x(0.5t)$ is applied to another causal and stable LTI system with impulse response $h(0.5t)$. The resulting output is:

For non-degenerately doped n-type silicon, which one of the following plots represents the temperature ($T$) dependence of free electron concentration ($n$)?

In the circuit shown, the $n:1$ step-down transformer and the diodes are ideal. The diodes have no voltage drop in forward-biased condition. If the input voltage (in Volts) is $V_s(t) = 10\sin\omega t$ and the average value of load voltage $V_L(t)$ (in Volts) is $2.5/\pi$, the value of $n$ is $\_\_\_\_$.

For a causal discrete-time LTI system with transfer function: \[ H(z) = \frac{2z^2 + 3}{(z + \frac{1}{3})(z - \frac{1}{3})}, \] which of the following statements is/are true?

Let $\rho(x, y, z, t)$ and $\mathbf{u}(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho}{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$, and $\mathbf{\hat{n}}$ be the outward unit normal of $S$. Which of the following equations is/are equivalent to: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0? \]

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 $\_\_\_\_$.