List of top Networks, Signals and Systems Questions on z-transform asked in GATE EC

In the feedback control system shown in the figure below, \[ G(s) = \frac{6}{s(s+1)(s+2)}. \]\(R(s)\), \(Y(s)\), and \(E(s)\) are the Laplace transforms of \(r(t)\), \(y(t)\), and \(e(t)\), respectively. If the input \(r(t)\) is a unit step function, then:

 

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: \[ \vec{F_1} = A(\hat{i}y + \hat{j}x) \quad {and} \quad \vec{F_2} = A(\hat{i}y - \hat{j}x)? \]
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:

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

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: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0? \]
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 \(\_\_\_\_\).
A non-degenerate n-type semiconductor has \(5\%\) neutral dopant atoms. Its Fermi level is located at \(0.25 \, {eV}\) below the conduction band (\(E_C\)) and the donor energy level (\(E_D\)) has a degeneracy of \(2\). Assuming the thermal voltage to be \(20 \, {mV}\), the difference between \(E_C\) and \(E_D\) (in \({eV}\), rounded off to two decimal places) is \(\_\_\_\_\).
A machine has a 32-bit architecture with 1-word long instructions. It has 24 registers and supports an instruction set of size 40. Each instruction has five distinct fields, namely opcode, two source register identifiers, one destination register identifier, and an immediate value. Assuming that the immediate operand is an unsigned integer, its maximum value is___.
In the given circuit, the current Ix (in mA) is
the given circuit, the current Ix (in mA)
As shown in the circuit, the initial voltage across the capacitor is 10 V, with the switch being open. The switch is then closed at t 0. The total energy dissipated = in the ideal Zener diode (Vz=5V) after the switch is____. the initial voltage across the capacitor
The relationship between any N-length sequence x[n] and its corresponding N-point discrete Fourier transform X[k] is defined as
X[k] = F(x[n]}. 
Another sequence y[n] is formed as below 
y[n] = F F{F{F{F{x[n]}}}}. 
For the sequence x[n] = {1, 2, 1, 3}, the value of Y[0] is___.

In the given circuit, the current \(I_x\) (in mA) is \(\_\_\_\_\). 

As shown in the circuit, the initial voltage across the capacitor is \(10 \, {V}\), with the switch being open. The switch is then closed at \(t = 0\). The total energy dissipated in the ideal Zener diode \((V_Z = 5 \, {V})\) after the switch is closed (in mJ, rounded off to three decimal places) is \(\_\_\_\_\). 

A machine has a 32-bit architecture with 1-word long instructions. It has 24 registers and supports an instruction set of size 40. Each instruction has five distinct fields, namely opcode, two source register identifiers, one destination register identifier, and an immediate value. Assuming that the immediate operand is an unsigned integer, its maximum value 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 \(\_\_\_\_\).
A non-degenerate n-type semiconductor has \(5\%\) neutral dopant atoms. Its Fermi level is located at \(0.25 \, {eV}\) below the conduction band (\(E_C\)) and the donor energy level (\(E_D\)) has a degeneracy of \(2\). Assuming the thermal voltage to be \(20 \, {mV}\), the difference between \(E_C\) and \(E_D\) (in \({eV}\), rounded off to two decimal places) is \(\_\_\_\_\).