GATE EC 2019 Question Paper (09-Feb-2019) (Shift 1)

The RC circuit shown below has a variable resistance R (t) given by the following expression:

$R\left(t\right)=R_0\left(1-\frac{t}{T}\right)for0\le t<T$

Where R₀ = 1Ω, and C = 1 F. We are also given that T = 3 R₀C and the source voltage is Vₛ = 1 V. If the current at time t = 0 is 1 A, then the current I (t), in amperes, at time t = T/2 is _______ (rounded off to 2 decimal places).

Consider a unity feedback system, as in the figure shown, with an integral compensator $\frac{k}{s}$ and open-loop transfer function

$G\left(s\right)=\frac{1}{s^2+3s+2}$

where K > 0. The positive value of K for which there are exactly two poles of the unity feedback system on the jω axis is equal to ______ (rounded off to two decimal places).

Consider the homogeneous ordinary differential equation $x^2\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}+3y=0,$ x > 0 with y(x) as a general solution. Given that y(1) = 1 and y(2) = 14 the value of y(1.5), rounded off to two decimal places, is ______.

Let h[n] be a length-7 discrete-time finite impulse response filter, given by

h[0] = 4, h[1] = 3, h[2] = 2, h[3] = 1,

h[−1] = −3, h[−2] = −2, h[−3] = −1,

and h[n] is zero for |n| ≥ 4. A length-3 finite impulse response approximation g[n] of h[n] has to be obtained such that

$E\left(h,g\right)=\underset{-\pi }{\overset{\pi }{\int }}{\left|H\left(e^{j\omega }\right)-G\left(e^{j\omega }\right)\right|}^{2}d\omega$

is minimized, where H(e${}^{j\omega }$) and G(e${}^{j\omega }$) are the discrete-time Fourier transforms of h[n] and g[n], respectively. For the filter that minimizes E(h, g), the value of 10g[−1] + g[1], rounded off to 2 decimal places, is ________ .

Let a random process Y(t) be described as Y(t) = h(t) ∗ X(t) + Z(t), where X(t) is a white noise process with power spectral density S${}_X$ (f) = 5 W/Hz. The filter h(t) has a magnitude response given by |H(f)| = 0.5 for −5 ≤ f ≤ 5, and zero elsewhere. Z(t) is a stationary random process, uncorrelated with X(t), with power spectral density as shown in the figure. The power in Y(t), in watts, is equal to ________ W (rounded off to two decimal places).