GATE EC 2017 Question Paper (05-Feb-2017) (Shift 1)

Let X(t) be a wide sense stationary random process with the power spectral density S ${}_X$ (f) as shown in Figure (a), where f is in Hertz (Hz). The random process X(t) is input to an ideal lowpass filter with the frequency response:

H≤ft( f ) = ≤ft\{ {*20c 1,&≤ft| f | ≤ 1/2Hz\\ 0,&≤ft| f | > 1/2Hz } .

This is as shown in Figure (b). The output of the lowpass filter is Y(t).



Let E be the expectation operator. Consider the following statements:

I. E(X(t)) = E(Y(t))

II. E(X ²(t)) = E(Y ²(t))

III. E(Y ²(t)) = 2

Select the correct option:

A continuous time signal x(t) = 4 cos(200πt) + 8 cos(400πt), where t is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response

h≤ft( t ) = ≤ft\{ {*20c {{{2≤ft( 300π t )}}π t}&t ≠ 0\\ 600,&t = 0 } .

Let y(t) be the output of this filter. The maximum value of |y(t)| is ________.

An optical fiber is kept along the ẑ direction. The refractive indices for the electric fields along the x̂ and ŷ directions in the fiber are nₓ = 1.5000 and n${}_y$ = 1.5001, respectively (nₓ ≠ n${}_y$ due to the imperfection in the fiber cross-section). The free space wavelength of a light wave propagating in the fiber is 1.5 μm. If the lightwave is circularly polarized at the input of the fiber, the minimum propagation distance after which it becomes linearly polarized, in centimeters, is ________.

The expression for an electric field in free space is $E=E_0\left(\hat{x}+\hat{y}+j2\hat{z}\right)e^{-j\left(\omega t-kx+ky\right)}$, where x,y,z represent the spatial coordinates, t represents time, and ω, k are constants. This electric field

A half wavelength dipole is kept in the x-y plane and oriented along 45° from the x-axis. Determine the direction of null in the radiation pattern for 0 ≤ Φ ≤ π. Here the angle θ (0 ≤ θ ≤ π) is measured from the z-axis, and the angle Φ (0 ≤ Φ ≤ 2π) is measured from the x-axis in the x-y plane.