GATE EC 2010 Question Paper (14-Feb-2010) (Shift 1)

A continuous time LTI system is described by

$\frac{d^{2}y(t)}{d{t}^2}+4\frac{dy(t)}{dt}+3y(t)=2\frac{dx(t)}{dt}+4x(t)$

Assuming zero initial conditions, the response y(t) of the above system for the input x(t) = e${}^{-2t}$ u(t) is given by

The transfer function of a discrete-time LTI system is given by

$H(z)=\frac{2-\frac{3}{4}{z}^{-1}}{1-\frac{3}{4}{z}^{-1}+\frac{1}{8}{z}^{-2}}$

Consider the following statements:

S₁: The system is stable and casual for ROC |z| > 1/2

S₂: The system is stable but not casual for ROC |z| < 1/4

S₃: The system is neither stable nor casual for ROC 1/4 < |z| < 1/2

Which one of the following statements is valid?

The Nyquist sampling rate for the signal $x\left(t\right)=\frac{\sin \left(500\pi t\right)}{\pi t}\times \frac{\sin \left(700\pi t\right)}{\pi t}$ is

A unity negative feedback closed loop system has a plant with the transfer function $G(s)=\frac{1}{s^2+2s+2}$ and a controller Gₑ(s) in the feedforward path. For a unit step input, the transfer function of the controller that gives minimum steady slate error is

X(t) is a stationary process with the power spectral density Sₓ(f) > 0 for all f. The Process is passed through a system shown below.



Let S${}_{\gamma }$ (f) be the power spectral density of Y(t). Which one of the following statements is correct ?