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Linear System   Electrical and Electronics

  A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
Characterizing the complete input-output properties of a system by exhaustive measurement is usually impossible. When a system qualifies as a linear system, it is possible to use the responses to a small set of inputs to predict the response to any possible input. This can save the scientist enormous amounts of work, and makes it possible to characterize the system completely.
Now we come to one of the most important and revealing properties systems may have - Linearity. Basically, the principle of linearity is equivalent to the principle of superposition, i.e. a system can be said to be linear if, for any two input signals, their linear combination yields as output the same linear combination of the corresponding output signals.
For determining the linearity the system, we need to test whether it obeys certain rules that all linear systems obey. There are two rules which a system needs to obey in order to qualify for a linear system:
1.    Additivity:
A system is said to be additive if for any two input signals x1(t) and x2(t),
X1(t) -> S -> Y1 (t) ; X2(t) -> S -> Y2(t)
[X1(t) + X2(t)] -> S -> [Y1 (t) + Y2(t)]
 I.e. the output corresponding to the sum of any two inputs is the sum of the two outputs.
2.    Homogeneity:
It is also called scaling.
A system is said to be homogenous if, for any input signal X(t),
X1(t) -> S -> Y1 (t) ;  aX1(t) -> S -> aY1 (t)
 i.e. scaling any input signal scales the output signal by the same factor.
To say a system is linear is equivalent to saying the system obeys both Additivity and homogeneity.
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