GATE EC 2014 Question Paper (16-Feb-2014) (Shift 2)

The output (Y) of the circuit shown in the figure is

A Fourier transform pair is given by:

${\left(\frac{2}{3}\right)}^{n}u\left[n+3\right]\stackrel{FT}{\iff }\frac{A{e}^{j6\pi f}}{1-\left(\frac{2}{3}\right)e^{-j2\pi f}}$

where u[n] denotes the unit step sequence. The values of A is _______.

A real-valued signal 𝑥(𝑡) limited to the frequency band $\left|f\right|\le \frac{W}{2}$ is passed through a linear time-invariant system whose frequency response is

H≤ft( f ) = ≤ft\{ {*20c {e^ - j4π f, ≤ft| f | ≤ W/2}\\ 0, ≤ft| f | > W/2 } .

The output of the system is

The sequence x[𝑛] = 0.5ⁿ u[𝑛], where 𝑢[𝑛] is the unit step sequence, is convolved with itself to obtain 𝑦[𝑛]. Then $\sum _{n=-\infty }^{+\infty }y\left[n\right]$ is ______

In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4${}^{th}$ order all-pole system –