GATE EC 2016 Question Paper (31-Jan-2016) (Shift 1)

In the feedback system shown below $\mathrm{G}(\mathrm{s})=\frac{1}{\left(\mathrm{s}+1\right)\left(\mathrm{s}+2\right)\left(\mathrm{s}+3\right)}$:



The positive value of 𝑘 for which the gain margin of the loop is exactly 0 dB and the phase margin of the loop is exactly zero degrees is ________.

The asymptotic Bode phase plot of $\mathrm{G}(\mathrm{s})=\frac{\mathrm{K}}{\left(\mathrm{s}+0.1\right)\left(\mathrm{s}+10\right)\left(\mathrm{s}+{\mathrm{p}}_1\right)}$, with K and 𝑝₁ both positive, is shown below.



The value of 𝑝₁ is ________.

An information source generates a binary sequence {𝛼${}_{\text{n}}$ }. 𝛼${}_{\text{n}}$ can take one of the two possible values −1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence {𝛽${}_{\text{n}}$ }, as ${\text{𝛽}}_{\text{n}}={\text{𝛼}}_{\text{n}}+\text{k}{\text{𝛼}}_{\text{n}-3}$ . The sequence {𝛽${}_{\text{n}}$ } is used to modulate a pulse $\mathrm{X}(\mathrm{t})$ to generate the baseband signal

$\mathrm{X}\left(\mathrm{t}\right)=\sum _{\mathrm{n}=-\infty }^{\infty }{\beta }_{\mathrm{n}}\mathrm{g}\left(\mathrm{t}-\mathrm{nT}\right),$ where ≤ft( ) = ≤ft\{ {*20c 1,&{0 ≤ ≤ }\\ 0,&{.} } .

If there is a null at 𝑓 = $\frac{1}{3\mathrm{T}}$ in the power spectral density of $\mathrm{X}(\mathrm{t})$, then 𝑘 is ________

An ideal band-pass channel 500 Hz - 2000 Hz is deployed for communication. A modem is designed to transmit bits at the rate of 4800 bits/s using 16-QAM. The roll-off factor of a pulse with a raised cosine spectrum that utilizes the entire frequency band is ________

Consider a random process $\mathrm{X}(\mathrm{t})=3V(\mathrm{t})-8$, where (𝑡) is a zero mean stationary random process with autocorrelation ${\mathrm{R}}_{\mathrm{V}}(\tau )=4{\mathrm{e}}^{-5|\tau |}$. The power in $\mathrm{X}(\mathrm{t})$ is ________