Linear System

A system is said to be linear if superposition principle is applicable to the system.

Superposition Principle

The response resulting from several input signals can be computed as the sum of the responses resulting from each input signal acting alone. Let x1(t) and x2(t) be the two inputs. The corresponding responses/output are y1(t) and y2(t).

Then the system is linear i.e. follows the principle of superposition if

  1. The response to x1(t) + x2(t)  is y1(t) + y2(t). This is called Additivity property.
  2. The response to  ax1(t)  is y1(t), where a is any arbitrarily constant. This is called Scaling or homogeneity property.

For Continuous time system: ax1(t) + bx2(t) → ay1(t) + by2(t).

For Discreate time system: ax1(n) + bx2(n) → ay1(n) + by2(n).

When a system violates either the principle of superposition or the property of homogeneity, the system is said to be nonlinear.

Non-Linear System

A system is said to be non-linear if superposition principle or the property of homogeneity is not applicable to the system.

For a linear system, an input that is zero for all time results in an output that is zero all time. If x(t) → y(t), then homogeneity property tells us that

0 = 0* x(t) → 0* y(t) = 0

Linear and Non-Linear Systems

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