Linear Systems and Nonlinear Systems

If the operator T satisfies the following two conditions, then T is called a
linear operator and the system represented by a linear operator T is called a linear system:

1. Additivity:

Given that Tx, = y, and Tx, = y,, then
T{x, +x2) =y, +Y,

for any signals x, and x2.

2. Homogeneity (or Scaling):

for any signals x and any scalar a.

Any system that does not satisfy T is classified as a
nonlinear system. Equation can be combined into a single condition as


T { ~+ wI 2 ) = ~ I Y+I a 2 Y z (1.68)

where a, and a, are arbitrary scalars. Equation (1.68) is known as the superposition
property. Examples of linear systems are the resistor and the capacitor.

Examples of nonlinear systems are
y =x 2 (1.69)
y = cos x (1.70)

Note that a consequence of the homogeneity (or scaling) property of linear
systems is that a zero input yields a zero output. This follows readily by setting a = 0 . This is another important property of linear systems.