GATE EC 2015 Question Paper (01-Feb-2015) (Shift 1)

The complex envelope of the bandpass signal $x\left(t\right)=-\sqrt{2}\left(\frac{\sin \left(\frac{\pi t}{5}\right)}{\frac{\pi t}{5}}\right)\sin \left(\pi t-\frac{\pi }{4}\right),$ centered about f = 1/2 Hz, is

A random binary wave, y(t) is given by

$y\left(t\right)=\sum _{n=-\infty }^{\infty }{X}_{n}p\left(t-nT-\varphi \right)$

Where p(t) = u(t) – u(t - T), u(t) is the unit step function and ϕ is an independent random variable with uniform distribution in [0, T]. The sequence {Xₙ} consists of independent and identically distributed binary valued random variables with P{Xₙ = +1} = P{Xₙ = -1} = 0.5 for each n.

The value of the autocorrelation $R_{yy}\left(\frac{3T}{4}\right)\triangleq E\left[y\left(t\right)y\left(t-\frac{3T}{4}\right)\right]$ equals ___________.

Consider the 3 m long lossless air-filled transmission line shown in the figure. It has a characteristic impedance of 120π Ω, is terminated by a short circuit, and is excited with a frequency of 37.5 MHz. What is the nature of the input impedance (Zᵢₙ)?

A 200 m long transmission line having parameters shown in the figure is terminated into a load R${}_L$. The line is connected to a 400 V source having source resistance R${}_S$ through a switch, which is closed at t = 0. The transient response of the circuit at the input of the line (z = 0) is also drawn in the figure. The value of R${}_L$ (in Ω) is ___________.

A coaxial capacitor of inner radius 1 mm and outer radius 5 mm has a capacitance per unit length of 172 pF/m. If the ratio of outer radius to the inner radius is doubled, the capacitance per unit length (in pF/m) is ________.