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

The state transition matrix ϕ (t) of a system

≤ft[ {*20c {x_1}\\ {x_2} } ] = ≤ft[ {*20c 0&1\\ 0&0 } ]≤ft[ {*20c x_1\\ x_2 } ]

is

Consider a transfer function $G_p\left(s\right)=\frac{p{s}^2+3ps-2}{s^2+\left(3+p\right)s+\left(2-p\right)}$ with p a positive real parameter. The maximum value of p until which Gₚ remains stable is _______

The characteristic equation of a unity negative feedback system is 1 + KG(𝑠) = 0. The open-loop transfer function (𝑠) has one pole at 0 and two poles at -1. The root locus of the system for varying K is shown in the figure.



The constant damping ratio line, for ξ = 0.5, intersects the root locus at point A. The distance from the origin to point A is given as 0.5. The value of 𝐾 at point A is ________ .

Consider a communication scheme where the binary-valued signal X satisfies {𝑋 = +1} = 0.75 and {𝑋 = −1} = 0.25. The received signal Y = X + Z, where Z is a Gaussian random variable with zero mean and variance 𝜎². The received signal Y is fed to the threshold detector. The output of the threshold detector X̂ is:

X = ≤ft\{ {*20c + 1, Y > \\ - 1, Y ≤ } .

The achieve a minimum probability of error P{X̂ ≠ X}, the threshold τ should be

Consider the Z-channel given in the figure. The input is 0 or 1 with equal probability.



If the output is 0, the probability that the input is also 0 equals _________