Contents
Linearity
Background
Language Definition
A system is considered linear if 2 separate inputs, multiplied by 2 different constants, can produce 2 separate outputs multiplied by those same constants.
Mathematical Definition
A system is called linear if: For any inputs $ x_1(t) $ and $ x_2(t) $ yielding outputs of $ y_1(t) $ and $ y_2(t) $,
$ ax_1(t)+bx_2(t)=ay_1(t)+by_2(t)\,\! $
Example of Linear system
The easiest way to determine linearity is using standard definition, going from the y's to the x's, then from the x's to the y's, and checking the results to make sure that they are the same:
Lets take the system $ y(t)=8x(t) $ ,
x's going through the system yield:
$ ay_1(t)+by_2(t)=8ax_1(t)+8bx_2(t)\,\! $
y's going through the system yield:
$ ax_1(t)+bx_2(t)=8Z(t)\,\! $ where $ \,\!Z(t)=ay_1(t)+by_2(t) $ so
$ \,\!8Z(t)=8(ax_1(t)+bx_2(t))=8ax_1(t)+8bx_2(t) $ so both sides are satisfied.
Example of Non-Linear system
Lets take the non-linear equation $ y(t)=x(t)^3 $ , and use the same method
x's going through the system yield:
$ \,\!ay_1(t)+by_2(t)=ax_1(t)^3+bx_2(t)^3 $
y's going through the system yield:
$ \,\!ax_1(t)+bx_2(t)=Z(t)^3 $ where $ \,\!Z(t)=ay_1(t)+by_2(t) $ but
$ 8Z(t)=\,\!ax_1(t)^3+bx_2(t)^3\ne (ax_1(t)+bx_2(t))^3 $
so both sides are not satisfied.
