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| − | [[ | + | ==Linearity== |
| + | '''What is a linear system?''' | ||
| + | A linear system is a mathematical model of a system based on the use of a linear operator. A system is called "linear" if for any constants a,b<math>{\in}</math>complex number and for any inputs x1(t) and x2(t) yielding output y1(t),y2(t) respectively the response to a.x1(t)+b.x2(t) is a.y1(t)+b.y2(t). | ||
| + | A more mathematical description would be, | ||
| + | given two valid inputs | ||
| + | |||
| + | <math>{x_1(t)}</math> | ||
| + | |||
| + | <math>{x_2(t)}</math> | ||
| + | |||
| + | and their respective outputs | ||
| + | |||
| + | <math>({y_1(t)}=h*{x_1(t)}</math> | ||
| + | |||
| + | <math>{y_2(t)}=h*{x_2(t)}</math> | ||
| + | then a linear system must satisfy | ||
| + | |||
| + | <math>{a*y_1(t)}+{b*y_2(t)}=H*[{a*x_1(t)+b*y_1(t)}]</math> | ||
| + | |||
| + | ==Example for a linear system== | ||
| + | Consider, | ||
| + | <math>{x_1(t)=sin(t)}</math> | ||
| + | |||
| + | |||
| + | <math>{x_2(t)=cos(t)}</math> | ||
| + | |||
| + | Let, | ||
| + | <math>{y_1(t)=tsin(t)}</math> | ||
| + | |||
| + | |||
| + | <math>y_2(t)=tcos(t)</math> | ||
| + | |||
| + | Now, | ||
| + | |||
| + | (1).<math>{ay_1(t)+by_2(t)}={atsin(t)+btcos(t)}</math> | ||
| + | |||
| + | And, | ||
| + | (2).<math>{H[{ax_1(t)+bx_2(t)}]}={t{asin(t)+bcos(t)}}={atsin(t)+btcos(t)}</math> | ||
| + | |||
| + | Thus since (1) and (2) are the same the system is linear. | ||
| + | |||
| + | ==Example for non linear system== | ||
| + | <math>{x_1(t)=t^3}</math> | ||
| + | |||
| + | <math>{x_2(t)=sin(t)}</math> | ||
| + | |||
| + | <math>{y_1(t)={[x_1(t)]^2}}</math> | ||
| + | |||
| + | <math>{y_2(t)={[x_2(t)]^2}}</math> | ||
| + | |||
| + | Therefore, | ||
| + | |||
| + | (1).<math>{ay_1(t)+by_2(t)}={a{t}^6+b{sin}^2(t)}</math> | ||
| + | |||
| + | |||
| + | (2).<math>{H[{ax_1(t)+bx_2(t)}]}={[{a{t}^3}+{bsin(t)}]^2}</math> | ||
| + | |||
| + | When we observe (1) and (2) we notice that they are not equal. Thus the system is not linear. | ||
Latest revision as of 09:56, 12 September 2008
Linearity
What is a linear system? A linear system is a mathematical model of a system based on the use of a linear operator. A system is called "linear" if for any constants a,b$ {\in} $complex number and for any inputs x1(t) and x2(t) yielding output y1(t),y2(t) respectively the response to a.x1(t)+b.x2(t) is a.y1(t)+b.y2(t). A more mathematical description would be, given two valid inputs
$ {x_1(t)} $
$ {x_2(t)} $
and their respective outputs
$ ({y_1(t)}=h*{x_1(t)} $
$ {y_2(t)}=h*{x_2(t)} $ then a linear system must satisfy
$ {a*y_1(t)}+{b*y_2(t)}=H*[{a*x_1(t)+b*y_1(t)}] $
Example for a linear system
Consider, $ {x_1(t)=sin(t)} $
$ {x_2(t)=cos(t)} $
Let,
$ {y_1(t)=tsin(t)} $
$ y_2(t)=tcos(t) $
Now,
(1).$ {ay_1(t)+by_2(t)}={atsin(t)+btcos(t)} $
And, (2).$ {H[{ax_1(t)+bx_2(t)}]}={t{asin(t)+bcos(t)}}={atsin(t)+btcos(t)} $
Thus since (1) and (2) are the same the system is linear.
Example for non linear system
$ {x_1(t)=t^3} $
$ {x_2(t)=sin(t)} $
$ {y_1(t)={[x_1(t)]^2}} $
$ {y_2(t)={[x_2(t)]^2}} $
Therefore,
(1).$ {ay_1(t)+by_2(t)}={a{t}^6+b{sin}^2(t)} $
(2).$ {H[{ax_1(t)+bx_2(t)}]}={[{a{t}^3}+{bsin(t)}]^2} $
When we observe (1) and (2) we notice that they are not equal. Thus the system is not linear.
