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A system is defined by its input relationship y(t) = 2x(t + 2) + 2 where y(t) and x(t) are the output and the input of the system, respectively. The system is

A system is defined by its input relationship y(t) = 2x(t + 2) + 2 where y(t) and x(t) are the output and the input of the system, respectively. The system is Correct Answer non-linear and non-causal

Concept:

Causal systems:

  • A system is said to be causal if its output depends upon present and past inputs, and does not depend upon future input.
    Example: y(t) = x(t − 1) + x(t)
    Here, x(t) is purely present value dependent function and x(t - 1) is purely past value dependent function.
     
Non-causal system:
  • A system is said to be causal if its output depends upon the present, past and, future input.
    y(t) = x(t) + x(t + 2)
    Here, x(t) is purely present value dependent function and x(t + 2) is purely future value dependent function.
     
Linear System:
  • The system which follows the principle of superposition principle (Law of additivity and homogeneity) is known as a linear system.
     
Non-Linear System:
  • The system which does not follow the principle of superposition principle (Law of additivity and homogeneity) is known as a Non-linear system.
     

Calculation:

Given,

y(t) = 2x(t + 2) + 2 

Checking Casuality:

 2x(t + 2) is the future input, hence it is Non-casual system.

Checking Linearity:

ky(t) = 2kx(t + 2) + 2k 

And, kx(t) = 2kx(t + 2)

Hence, ky(t) ≠ kx(t)

Hence, system is Non-linear.

Additional Information

Principle of Superposition:
A system will follow the principle of superposition if they follow the Law of Additivity as well as the Law of Homogeneity.

 Law of Additivity:
Consider two systems in which input is x1(t), x2(t), and output is y1(t), y2(t).

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[ alt="F1 Jai.P 15-03-21 Savita D2" src="//storage.googleapis.com/tb-img/production/21/03/F1__Jai.P_15-03-21_Savita_D2.png" style="width: 228px; height: 42px;">

It follow the law of Additivity if,
x1(t) + x2(t) = y1(t) + y2(t)

Law of Homogeneity:
Considerd a systtem has input x(t) and output y(t).

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It follows the law of Homogeneity, if
kx(t) = ky(t)

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