Introduction to systems - Continuous and Discrete Time Signals and Systems - page 75

and

x 2 ( t ) → e x 2 ( t )

= y 2 ( t ) ,

giving

α x 1 ( t ) + β x 2 ( t ) → e α x 1 ( t ) +β x 2 ( t ) .

Since
e α x 1 ( t ) +β x 2 ( t ) = e α x 1 ( t ) e β x 2 ( t ) = [ y 1 ( t )] α + [ y 2 ( t )] β = α y 1 ( t ) + β y 2 ( t ) ,

the exponential amplifier represented by Eq. (2.34) is not a linear system.
(c) From (2.35), it follows that

x 1 ( t ) → 3 x 1 ( t ) = y 1 ( t )

and

x 2 ( t ) → 3 x 2 ( t ) = y 2 ( t ) ,

giving

α x 1 ( t ) + β x 2 ( t ) → 3 α x 1 ( t ) + β x 2 ( t ) = 3 α x 1 ( t ) + 3 β x 2 ( t )
= α y 1 ( t ) + β y 2 ( t ) .

Therefore, the amplifier of Eq. (2.35) is a linear system.
(d) From Eq. (2.36), we can write

x 1 ( t ) → 3 x 1 ( t ) + 5 = y 1 ( t )

and

x 2 ( t ) → 3 x 2 ( t ) + 5 = y 2 ( t ) ,

giving

α x 1 ( t ) + β x 2 ( t ) → 3[ α x 1 ( t ) + β x 2 ( t )] + 5 .

Since

3[ α x 1 ( t ) + β x 2 ( t )] + 5 = α y 1 ( t ) + β y 2 ( t ) − 5 ,

the amplifier with an additive bias as specified in Eq. (2.36) is not a linear
system.
An alternative approach to check if a system is non-linear is to apply the
zero-input, zero-output property. For system (b), if x ( t ) = 0, then y ( t ) = 1.
System (b) does not satisfy the zero-input, zero-output property, hence system
(b) is non-linear. Likewise, for system (d), if x ( t ) = 0 then y ( t ) = 5. Therefore,
system (d) is not a linear system.
If a system does not satisfy the zero-input, zero-output property, we can safely
classify the system as a non-linear system. On the other hand, if it satisfies
the zero-input, zero-output property, it can be linear or non-linear. Satisfying
the zero-input, zero-output property is not a sufficient condition to prove the