Signals and Systems - Electrical Engineering

120 C H A P T E R 2: Continuous-Time Systems

When developing a mathematical model for a continuous-time system it is important to contrast the

accuracy of the model with its simplicity and practicality. The following are some of the characteristics

of the model being considered:

n Linearity

n Time invariance

n Causality

n Stability

The linearity between the input and the output, as well as the constancy of the system parameters,

simplify the mathematical model. Causality, or nonanticipatory behavior of the system, relates to

the cause–effect relationship between the input and the output. It is essential when the system is

working under real-time situations—that is, when there is limited time for the system to process

signals coming into the system. Stability is needed in practical systems. A stable system behaves well

under reasonable inputs. Unstable systems are useless.

2.3.1 Linearity

A system represented bySis said to belinearif for inputsx(t)andv(t), and any constantsαandβ,

superposition holds—that is,

S[αx(t)+βv(t)]=S[αx(t)]+S[βv(t)]

=αS[x(t)]+βS[v(t)] (2.2)

When checking the linearity of a system we first need to check the scaling—that is, if the output

y(t)=S[x(t)] for some inputx(t)is known, then for a scaled inputαx(t)the output should beαy(t)=

αS[x(t)]. If this condition is not satisfied, the system is nonlinear. If the condition is satisfied, you

would then test the additivity or that the response to the sum of weighted inputs,S[αx(t)+βv(t)], is

the sum of the corresponding responsesαS[x(t)]+βS[v(t)].

The scaling property of linear systems indicates that whenever the input of a linear system is zero

the output is zero. Thus, if the output corresponding to an inputx(t)isy(t), then the output

corresponding toαx(t)isαy(t); and if, in particular,α=0, then both input and output are zero.

nExample 2.1

Consider a biased averager—that is, the outputy(t)of such a system is given by

y(t)=

1

T

∫t

t−T

x(τ)dτ+B

for an inputx(t). The system finds the average over an intervalTand adds a constant valueB. Is

this system linear? If not, is there a way to make it linear? Explain.