Page 29 - Schaum's Outline of Theory and Problems of Signals and Systems

18 SIGNALS AND SYSTEMS [CHAP. 1

E. Linear Systems and Nonlinear Systems:

If the operator T in Eq. (1.60) 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 Eq. (1.66) and/or Eq. (1.67) is classified as a
nonlinear system. Equations (1.66) and ( 1.67) can be combined into a single condition as
+
T{~I 2 ) = ~IYI a2Yz (1.68)
+ w
where a, and a, are arbitrary scalars. Equation (1.68) is known as the superposition
property. Examples of linear systems are the resistor [Eq. (1.6111 and the capacitor [Eq.
( 1.62)]. 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 [Eq. (1.6711 of linear
systems is that a zero input yields a zero output. This follows readily by setting a = 0 in Eq.
(1.67). This is another important property of linear systems.

F. Time-Invariant and Time-Varying Systems:
A system is called rime-inuariant if a time shift (delay or advance) in the input signal
causes the same time shift in the output signal. Thus, for a continuous-time system, the
system is time-invariant if

for any real value of T. For a discrete-time system, the system is time-invariant (or
shift-incariant ) if
~{x[n -k]) =y[n -k] (1.72)
for any integer k. A system which does not satisfy Eq. (1.71) (continuous-time system) or
Eq. (1.72) (discrete-time system) is called a time-varying system. To check a system for
time-invariance, we can compare the shifted output with the output produced by the
shifted input (Probs. 1.33 to 1.39).

G. Linear Time-Invariant Systems
If the system is linear and also time-invariant, then it is called a linear rime-invariant
(LTI) system.