| Line 7: | Line 7: | ||
<math>X2(t)=\ 2t^2</math> | <math>X2(t)=\ 2t^2</math> | ||
| − | assume the function <math>Y(t)=\ | + | assume the function <math>Y(t)=\ X(t)</math> |
| − | <math>Y1(t)=\ | + | <math>Y1(t)=\ 2t</math> |
| − | <math>Y2(t)=\ | + | <math>Y2(t)=\ 2t^2</math> |
| − | now for <math>aY1(t)+bY2(t)=\ | + | now for <math>aY1(t)+bY2(t)=\ a*2t+b*2t^2=[aX1(t)+bX2(t)]</math> |
| − | thus the given system<math>Y(t)=\ | + | thus the given system<math>Y(t)=\ X(t)</math> is linear |
==Examples of non linear system== | ==Examples of non linear system== | ||
| + | <math>X1(t)=\ t</math> | ||
| + | |||
| + | <math>X2(t)=\ t^2</math> | ||
| + | |||
| + | assume the function <math>Y(t)=\ sin[X(t)]</math> | ||
| + | |||
| + | <math>Y1(t)=\ sin(t)</math> | ||
| + | |||
| + | <math>Y2(t)=\ sin(t^2)</math> | ||
| + | |||
| + | now for <math>aY1(t)+bY2(t)=a*sin(t)+b*sin(t^2)\neq sin[aX1(t)+bX2(t)]</math> | ||
| + | |||
| + | thus the given system<math>Y(t)=\ sin[X(t)]</math> is '''not''' a linear system | ||
Latest revision as of 12:33, 11 September 2008
Linear system
A system is said to be linear if it satisfies the principle of superposition i.e if for an input A the system gives an output X and for an input B the system gives output then for an input ( a*A + b*B ) the system should yield the output as ( a*X + b*B ). Where a and b are any complex numbers.
Examples of linear system
$ X1(t)=\ 2t $
$ X2(t)=\ 2t^2 $
assume the function $ Y(t)=\ X(t) $
$ Y1(t)=\ 2t $
$ Y2(t)=\ 2t^2 $
now for $ aY1(t)+bY2(t)=\ a*2t+b*2t^2=[aX1(t)+bX2(t)] $
thus the given system$ Y(t)=\ X(t) $ is linear
Examples of non linear system
$ X1(t)=\ t $
$ X2(t)=\ t^2 $
assume the function $ Y(t)=\ sin[X(t)] $
$ Y1(t)=\ sin(t) $
$ Y2(t)=\ sin(t^2) $
now for $ aY1(t)+bY2(t)=a*sin(t)+b*sin(t^2)\neq sin[aX1(t)+bX2(t)] $
thus the given system$ Y(t)=\ sin[X(t)] $ is not a linear system
