GATE EC question library

An N-type semiconductor having uniform doping is biased as shown in the figure. If $E_C$ is the lowest energy level of the conduction band, $E_V$ is the highest energy level of the valance band and $E_F$ is the Fermi level, which one of the following represents the energy band diagram for the biased N-type semiconductor?

Consider the common-collector amplifier in the figure (bias circuitry ensures that the transistor operates in forward active region, but has been omitted for simplicity). Let $I_C$ be the collector current, $V_{BE}$ be the base-emitter voltage and $V_T$ be the thermal voltage. Also, $g_m$ and $r_0$ are the small-signal transconductance and output …

A BJT in a common-base configuration is used to amplify a signal received by a $50 \Omega$ antenna. Assume $kT/q=25 \: mV$. The value of the collector bias current (in mA) required to match the input impedance of the amplifier to the impedance of the antenna is __________.

For the common collector amplifier shown in the figure, the BJT has high $\beta$, negligible $V_{CE(sat)}$, and $V_{BE}=0.7 \: V$. The maximum undistorted peak-to-peak output voltage $v_o$ (in Volts) is __________

An $8$- to $1$ multiplexer is used to implement a logical function $Y$ as shown in the figure. The output $Y$ is given by

A $16$ – bit ripple carry adder is realized using $16$ identical full adders (FA) as shown in the figure. The carry-propagation delay of each FA is $12$ ns and the sum-propagation delay of each FA is $15$ ns. The worst case delay (in ns) of this $16$-bit adder will be _______

An $8085$ microprocessor executes $“STA 1234H”$ with starting address location $1FFEH$ (STA copies the contents of the Accumulator to the $16$- bit address location). While the instruction is fetched and executed, the sequence of values written at the address pins $A_{15} – A_8$ is

A stable linear time variant (LTI) system has a transfer function $H(s) = \frac{1}{s^2+s-6}$. To make this system casual it needs to be cascaded with another LTI system having a transfer function $H_1(s)$. A correct choice for $H_1(s)$ among the following options is

A casual LTI system has zero initial conditions and impulse response $h(t)$. Its input $x(t)$ and output $y(t)$ are related through the linear constant-coefficient differential equation $$\frac{d^2y(t)}{dt^2} + a \frac{dy(t)}{dt}+a^2y(t)=x(t).$$ Let another signal $g(t)$ be defined as $$g(t)=a^2 \int_0^t h(\tau) d \tau +\frac{dh(t)}{dt}+ah(t).$$ If …

The $N$-point DFT $X$ of a sequence $x[n]$, $0 \leq n \leq N-1$ is given by $$X[k] = \frac{1}{\sqrt{N}} \Sigma_{n=0}^{N-1} x[n] e^{-j \frac{2\pi}{N}nk}, \: \: \: 0 \leq k \leq N-1.$$ Denote this relation as $X=DFT(x)$. For $N=4$, which one of the following sequences satisfies $DFT(DFT(x))=x$?

The state transition matrix $\phi(t)$ of a system $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ is

Consider a transfer function $G_p(s) = \frac{ps^2+3ps-2}{s^2+(3+p)s+(2-p)}$ with $p$ a positive real parameter. The maximum value of $p$ until which $G_p$ remains stable is ___________.

The characteristic equation of a unity negative feedback system is $1+KG(s)=0$. The open loop transfer function $G(s)$ has one pole at $0$ and two poles at $-1$. The root locus of the system for varying $K$ is shown in the figure. The constant damping ratio line, for $\xi = 0.5$, intersects the root locus at point A. The distance from the origin …

Consider a communication scheme where the binary valued signal $X$ satisfies $P\{X=+1\}=0.75$ and $P\{X=-1 \}=0.25$. The received signal $Y=X+Z$, where $Z$ is a Gaussian random variable with zero mean and variance $\sigma ^2$. The received signal $Y$ is fed to the threshold detector. The output of the threshold detector $\hat{X}$ is: …

Consider the $Z$-channel given in the figure. The input is $0$ or $1$ with equal probability. If the output is $0$, the probability that the input is also $0$ equals ___________

An M-level PSK modulation scheme is used to transmit independent binary digits over a band-pass channel with bandwidth $100$ kHz. The bit rate is $200$ kbps and the system characteristic is a raised-cosine spectrum with $100 \%$ excess bandwidth. The minimum value of M is __________

Consider a discrete-time channel $Y=X +Z$, where the additive noise $Z$ is signal-dependent. In particular, given the transmitted symbol $X \in \{-a , +a\}$ at any instant, the noise sample $Z$ is chosen independently from a Gaussian distribution with mean $\beta X$ and unit variance. Assume a threshold detector with zero threshold at the receiver. …

The electric field (assumed to be one-dimensional) between two points $A$ and $B$ is shown. Let $\psi _A$ and $\psi _B$ be the electrostatic potentials at $A$ and $B$, respectively. The value of $\psi _B – \psi _A$ in Volts is ____________

Gven $\overrightarrow{F} = z \hat{a}_x + x \hat{a}_y + y \hat{a}_z$. If $S$ represents the portion of the sphere $x^2 +y^2+z^2=1$ for $z \geq 0$, then $\int _s \nabla \times \overrightarrow{F} \cdot \overrightarrow{ds}$ is __________.

If $\overrightarrow{E} = -(2y^3 – 3yz^2)\hat{x} – (6xy^2-3xz^2) \hat{y}+(6xyz) \hat{z}$ is the electric field in a source free region, a valid expression for the electrostatic potential is