GATE EC 2015 Question Paper (31-Jan-2015) (Shift 2)

The state variable representation of a system is given as

x = ≤ft[ {*20c 0&1\\ 0& - 1 } ];≤ft( 0 ) = ≤ft[ {*20c 1\\ 0 } ]\\ = ≤ft[ {*20c 0&1 } ]

The response y(t) is

The output of a standard second-order system for a unit step input is given as

$\mathrm{y}\left(\mathrm{t}\right)=1-\frac{2}{\sqrt{3}}{\mathrm{e}}^{-\mathrm{t}}\cos \left(\sqrt{3\mathrm{t}}-\frac{\pi }{6}\right)$

The transfer function is the system is

The transfer function of a mass – spring – damper system is given by

$\mathrm{G}\left(\mathrm{s}\right)=\frac{1}{\mathrm{M}{\mathrm{s}}^2+\mathrm{Bs}+\mathrm{K}}$

The frequency response data for the system are given in the following table.The unit step response of the system approaches a steady state value of ________.

A zero mean white Gaussian noise having power spectral density $\frac{{\mathrm{N}}_0}{2}$ is passed through an LTI filter whose impulse response $\mathrm{h}\left(\mathrm{t}\right)$ is shown in the figure. The variance of the filtered noise at $\mathrm{t}=4$ is

${\left\{{\mathrm{X}}_{\mathrm{n}}\right\}}_{\mathrm{n}=-\infty }^{\mathrm{n}=\infty }$ is an independent and identically distributed (i.i.d.) random process with Xₙequally likely to be $+1$ or $-1$. ${\left\{{\mathrm{Y}}_{\mathrm{n}}\right\}}_{\mathrm{n}=-\infty }^{\mathrm{n}=\infty }$ is another random process obtained as ${\mathrm{Y}}_{\mathrm{n}}={\mathrm{X}}_{\mathrm{n}}+0.5{\mathrm{X}}_{\mathrm{n}-1}$. The autocorrelation function of ${\left\{{\mathrm{Y}}_{\mathrm{n}}\right\}}_{\mathrm{n}=-\infty }^{\mathrm{n}=\infty }$, denoted by ${\mathrm{R}}_{\mathrm{y}}\left[\mathrm{k}\right],$ is_______