(New page: == Linearity == Because engineers talk in symbols instead of words...the following describes linearity. *A/*B = multiply by any complex number x(t) --> system --> *A ...) |
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*A/*B = multiply by any complex number | *A/*B = multiply by any complex number | ||
| − | + | <pre> | |
x(t) --> system --> *A | x(t) --> system --> *A | ||
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y(t) --> *B | y(t) --> *B | ||
| − | + | </pre> | |
If z(t) == w(t) then the system is linear. | If z(t) == w(t) then the system is linear. | ||
In linearity, the values are scalable and follow the rules of superposition. | In linearity, the values are scalable and follow the rules of superposition. | ||
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| + | ==An example of a non-linear system== | ||
| + | |||
| + | <math>y(t)=t^2*sqrt(x(t))</math> | ||
| + | |||
| + | x1(t)-->system--> *A =x1'(t) | ||
| + | x2(t)-->system--> *B =x2'(t) | ||
| + | |||
| + | w(t)=x1'(t)+x2'(t)= <math>A*t^2*sqrt(x1(t))+B*t^2*sqrt(x2(t))</math> | ||
| + | |||
| + | |||
| + | x3(t)--> *A -->system--> = x3'(t) | ||
| + | x4(t)--> *B -->system--> = x4'(t) | ||
| + | |||
| + | z(t)=x3'(t) + x4'(t)= <math>t^2*sqrt(A*x1(t))+t^2*sqrt(B*x2(t))</math> | ||
| + | |||
| + | Since z(t) is not equal to w(t) (because one equation has root A and root B and the other just has A and B) the system is non-linear. | ||
| + | |||
| + | ==An example of a linear system== | ||
| + | |||
| + | <math>y(t)=t^2*x(t)</math> | ||
| + | |||
| + | x1(t)-->system--> *A =x1'(t) | ||
| + | x2(t)-->system--> *B =x2'(t) | ||
| + | |||
| + | w(t)=x1'(t)+x2'(t)= <math>A*t^2*x1(t)+B*t^2*x2(t)</math> | ||
| + | |||
| + | |||
| + | x3(t)--> *A -->system--> = x3'(t) | ||
| + | x4(t)--> *B -->system--> = x4'(t) | ||
| + | |||
| + | z(t)=x3'(t) + x4'(t)= <math>t^2*A*x1(t)+t^2*B*x2(t)</math> | ||
| + | |||
| + | Since z(t) is equal to w(t) the system is linear. | ||
Latest revision as of 13:20, 10 September 2008
Linearity
Because engineers talk in symbols instead of words...the following describes linearity.
- A/*B = multiply by any complex number
x(t) --> system --> *A
|
+ --> w(t)
|
y(t) --> system --> *B
x(t) --> *A
|
+ --> system --> z(t)
|
y(t) --> *B
If z(t) == w(t) then the system is linear.
In linearity, the values are scalable and follow the rules of superposition.
An example of a non-linear system
$ y(t)=t^2*sqrt(x(t)) $
x1(t)-->system--> *A =x1'(t) x2(t)-->system--> *B =x2'(t)
w(t)=x1'(t)+x2'(t)= $ A*t^2*sqrt(x1(t))+B*t^2*sqrt(x2(t)) $
x3(t)--> *A -->system--> = x3'(t)
x4(t)--> *B -->system--> = x4'(t)
z(t)=x3'(t) + x4'(t)= $ t^2*sqrt(A*x1(t))+t^2*sqrt(B*x2(t)) $
Since z(t) is not equal to w(t) (because one equation has root A and root B and the other just has A and B) the system is non-linear.
An example of a linear system
$ y(t)=t^2*x(t) $
x1(t)-->system--> *A =x1'(t) x2(t)-->system--> *B =x2'(t)
w(t)=x1'(t)+x2'(t)= $ A*t^2*x1(t)+B*t^2*x2(t) $
x3(t)--> *A -->system--> = x3'(t)
x4(t)--> *B -->system--> = x4'(t)
z(t)=x3'(t) + x4'(t)= $ t^2*A*x1(t)+t^2*B*x2(t) $
Since z(t) is equal to w(t) the system is linear.
