Definition of Linearity

A system is linear if for any inputs $ \,x_1(t), x_2(t)\, $ yielding outputs $ \,y_1(t), y_2(t)\, $, respectively, the response to

$ \,ax_1(t)+bx_2(t)\, $ is

$ \,ay_1(t)+by_2(t)\, $, where $ \,a,b\in \mathbb{C}, a\not= 0 ,b\not= 0\, $.

Example of a Linear System

The following system is linear:

$ \,s(t)=2x(t+3)\, $


Proof:

We have two functions: $ \,x_1(t), x_2(t)\, $.

After applying the functions to the system $ \,s(t)\, $, we get:

$ \,y_1(t)=2x_1(t+3)\, $

$ \,y_2(t)=2x_2(t+3)\, $

Thus,

$ \,ay_1(t)+by_2(t)=\, $

$ \,a(2x_1(t+3))+b(2x_2(t+3))=\, $

$ \,2ax_1(t+3)+2bx_2(t+3)\, $


Now, apply $ \,ax_1(t)+bx_2(t)\, $ to the system $ \,s(t)\, $:

$ \,2(ax_1(t+3)+bx_2(t+3))=\, $

$ \,2ax_1(t+3)+2bx_2(t+3)\, $


Since the two results are equal

$ \,2ax_1(t+3)+2bx_2(t+3)=2ax_1(t+3)+2bx_2(t+3)\, $

the system is linear.

Example of a Non-Linear System

The following system is non-linear:

$ \,s(t)=2x(t)+3\, $


Proof:

We have two functions: $ \,x_1(t), x_2(t)\, $.

After applying the functions to the system $ \,s(t)\, $, we get:

$ \,y_1(t)=2x_1(t)+3\, $

$ \,y_2(t)=2x_2(t)+3\, $

Thus,

$ \,ay_1(t)+by_2(t)=\, $

$ \,a(2x_1(t)+3)+b(2x_2(t)+3)=\, $

$ \,2ax_1(t)+3a+2bx_2(t)+3b\, $


Now, apply $ \,ax_1(t)+bx_2(t)\, $ to the system $ \,s(t)\, $:

$ \,2(ax_1(t)+bx_2(t))+3=\, $

$ \,2ax_1(t)+2bx_2(t)+3\, $


Since the two results are not equal

$ \,2ax_1(t)+3a+2bx_2(t)+3b\not= 2ax_1(t)+2bx_2(t)+3\, $

the system is non-linear.

Alumni Liaison

Questions/answers with a recent ECE grad

Ryne Rayburn