Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 121

Solution

Lety(t)be the system response corresponding tox(t). Assume then that we scale the input by a

factorαso that the input isαx(t). The corresponding output is then

1

T

∫t

t−T

αx(τ)dτ+B=

α

T

∫t

t−T

x(τ)dτ+B

which is not equal to

αy(t)=

α

T

∫t

t−T

x(τ)dτ+αB

so the system is not linear. Notice that the difference is due to the term associated withB, which is

not affected at all by the scaling of the input. So to make the system linear we letB=0.

The constantBis the response due to zero input, and as such, the response can be seen as the sum

of a linear system and a zero-input response. This type of system is calledincrementally lineargiven

that if

S[x 1 (t)]=y 1 (t)−B

S[x 2 (t)]=y 2 (t)−B

then

S[x 1 (t)−x 2 (t)]=S[x 1 (t)]−S[x 2 (t)]=y 1 (t)−y 2 (t)

=

1

T

∫t

t−T

[x 1 (τ)−x 2 (τ)]dτ

That is, the difference of the responses to two inputs is linear. n

nExample 2.2

Whenever the explicit relation between the input and the output of a system is represented by a

nonlinear expression the system is nonlinear. Consider the following input–output relations that

show the corresponding systems are nonlinear:

(i)y(t)=|x(t)|

(ii)z(t)=cos(x(t))assuming|x(t)|≤ 1

(iii)v(t)=x^2 (t)

wherex(t)is the input andy(t),z(t), andv(t)are the outputs.