GATE Electrical Engineering (EE) 2021 Solved Papers

Question : 13 of 65

Marks: +1, -0

If the input

x(t)

and output

y(t)

of a system are related as

y(t)=\max (0, x(t))

, then the system is:

Solution:

y(t)=\max (0, x(t))

y(t)=\{\table 0 , x(t)<0 ; x(t) , x(t) ≥ 0.

y(t)=x(t) ⋅(x(t))

For Linearity:
Additive + homogeneity
'Homogeneity'

y(t)=x(t)(x(t))

K y(t)=K x(t)(x(t))

'additive'

y_{1}(t)=x_{1}(t)(x(t))

...(1)

y_{2}(t) x_{2}(t)(x_{2}(t))

.....(2)

y_{1}(t)+y_{2}(t)-x_{1}(t)(x_{1}(t))+x_{2}(t)(x_{2}(t))

....(3)

y_{3}(t)=[x_{1}(t)+x_{2}(t)][x_{1}(t)+x_{2}(t)]

.....(4)

y_{3}(t) ≠ y_{1}(t)+y_{2}(t)

So, system is Non-linear.
For time variant or time invariant:

y(t)=x(t)(x(t))

y(t, t_{0})=x(t-t_{0})(x(t-t_{0})) ⋅s(1)

....(1)

y(t-t_{0})=x(t-t_{0})(x(t)-t_{0}) ⋅s(2)

.....(2)

y(t, t_{0})=y(t-t_{0})

So, system is time Invariant.

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