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

Consider a binary, digital communication system which used pulses $\mathrm{g}\left(\mathrm{t}\right)$ and $-\mathrm{g}\left(\mathrm{t}\right)$ for transmitting bits over an AWGN channel. If the receiver uses a matched filter, which one of the following pulses will give the minimum probability of bit error?

Let X ∈ {0,1} and Y ∈ {0,1} be two independent binary random variables. If P(X = 0) = p and P(Y = 0) = q, then P(X + Y) ≥ 1 is equal to

An air – filled rectangular waveguide of internal dimensions $\mathrm{a}\mathrm{cm}\times \mathrm{b}\mathrm{cm}(\mathrm{a}>\mathrm{b})$ has a cutoff frequency of $6\mathrm{GHz}$ for the dominant $\mathrm{T}{\mathrm{E}}_{10}$ mode. For the same waveguide, if the frequency of the $\mathrm{T}{\mathrm{M}}_{11}$ mode is $15\mathrm{GHz}$, the cutoff frequency of the $\mathrm{T}{\mathrm{E}}_{01}$ mode in GHz is_________.

Two half – wave dipole antennas placed as shown in the figure are excited with sinusoidal varying current of frequency $3\mathrm{MHz}$ and phase shift of π/2 between them (the element at the origin leads in phase.) If the maximum radiated E – field at the point P in the x-y plane occurs at an azimuthal angle of 60°, the distance $\mathrm{d}$ (in meters) between the antennas is__________

The electric field of a plane wave propagation in a lossless non – magnetic medium is given by the following expression

$\mathrm{E}\left(\mathrm{z},\mathrm{t}\right)={\hat{\mathrm{a}}}_{\mathrm{x}}5\cos \left(2\pi \times {10}^9\mathrm{t}+\beta \mathrm{z}\right)+{\hat{\mathrm{a}}}_{\mathrm{y}}3\cos \left(2\pi \times {10}^9\mathrm{t}+\beta \mathrm{z}-\frac{\pi }{2}\right)$

The type of the polarization is