Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 9

voltage as output, since v C ¼ð 1 = C Þ R t 1 i C ð t Þ dt : On the other hand, a squarer
is a memoryless system, since y(t) = [x(t)] 2 .
5. Stable and unstable systems: if a system takes a bounded input signal (i.e.,
|x(t)| \?) and always produces a bounded output signal y(t), the system is said
to be bounded-input, bounded-output (BIBO) stable.
6. Linear and non-linear systems: a system which produces an operation T on a
signal is called homogeneous if it satisfies the scaling property:

T½ c x ð t Þ¼ c T½ x ð t Þ;

where c is an arbitrary constant. A system is referred to as additive if it satisfies
the additivity condition:

T½ x 1 ð t Þþ x 2 ð t Þ¼T½ x 1 ð t ÞþT½ x 2 ð t Þ:

A linear system satisfies the superposition property, which is the combination
of scaling (homogeneity) and additivity:

T½ a x 1 ð t Þþ b x 2 ð t Þ¼ a T½ x 1 ð t Þþ b T½ x 2 ð t Þ;

where a and b are arbitrary constants.

Example 1 The system represented mathematically by y(T) = x(T) ? 2 is not
linear. To see this more clearly, assume that x(t) = a x 1 (t) ? b x 2 (t), where a and
b are constants. Then it follows that:

T½ a x 1 ð t Þþ b x 2 ð t Þ¼T½ x ð t Þ¼ x ð t Þþ 2 ¼ a x 1 ð t Þþ b x 2 ð t Þþ 2 ;

whereas

a T½ x 1 ð t Þþ b T½ x 2 ð t Þ¼ a ½ x 1 ð t Þþ 2 þ b ½ x 2 ð t Þþ 2
¼ a x 1 ð t Þþ b x 2 ð t Þþ 2a þ 2b :

Example 2 The system represented mathematically by y(t) = ln[x(t)] is non-
linear since ln[c x(t)] = c ln[x(t)].

Example 3 The system y(t) = dx(t)/dt is linear since it can be shown to satisfy
both homogeneity and additivity.

1.1.9 Linear Time-Invariant Systems

Linear time-invariant (LTI) systems are of fundamental importance in practical
analysis firstly because they are relatively simple to analyze and secondly because
they provide reasonable approximations to many real-world systems. LTI sys-
tems exhibit both the linearity and time-invariance properties described

above.