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== Linear System == | == Linear System == | ||
| − | + | A system is called "Linear" | |
| − | + | if for any constants a,b and for any inputs x1(t),x2(t),(x1[n],x2[n]) | |
| + | yielding output y1(t),y2(t),respectively, the respond to a*x1(t)+b*x2(t) is a*y1(t)+b*y2(t) | ||
== Example == | == Example == | ||
| + | A)LINEAR | ||
| + | Let: | ||
| + | x1(t)=t, x2(t)=2t; | ||
| + | System: y(t)=3*x(t) | ||
| + | Thus, y1(t)=3t,y2(t)=6t | ||
| + | So say a,b are any non-zero constant | ||
| + | a*x1(t)->system->3at | ||
| + | + --->Output= 3at+6bt -----(1) | ||
| + | b*x2(t)->system->6bt | ||
| + | |||
| + | |||
| + | a*x1(t)+b*x2(t)=at+2bt->system->Output=3*(at+2bt)= 3at+6bt ----------(2) | ||
| + | |||
| + | (1)=(2),so linear. | ||
| + | |||
| + | B)NON-LINEAR | ||
| + | Let: | ||
| + | x1(t)=t, x2(t)=2t; | ||
| + | System: y(t)=x(t)^2 | ||
| + | Thus, y1(t)=t^2,y2(t)=4t^2 | ||
| + | |||
| + | So say a,b are any non-zero constant | ||
| + | a*x1(t)->system->a*t^2 | ||
| + | + --->Output= a*t^2+4b*t^2 -----(3) | ||
| + | b*x2(t)->system->b*4t^2 | ||
| + | |||
| + | a*x1(t)+b*x2(t)=at+2bt->system->Output=(at+2bt)^2 ----------(4) | ||
| − | + | (3)!=(4),so non-linear. | |
| − | + | ||
| − | + | ||
Latest revision as of 17:02, 12 September 2008
Linear System
A system is called "Linear" if for any constants a,b and for any inputs x1(t),x2(t),(x1[n],x2[n]) yielding output y1(t),y2(t),respectively, the respond to a*x1(t)+b*x2(t) is a*y1(t)+b*y2(t)
Example
A)LINEAR
Let:
x1(t)=t, x2(t)=2t;
System: y(t)=3*x(t)
Thus, y1(t)=3t,y2(t)=6t
So say a,b are any non-zero constant
a*x1(t)->system->3at
+ --->Output= 3at+6bt -----(1)
b*x2(t)->system->6bt
a*x1(t)+b*x2(t)=at+2bt->system->Output=3*(at+2bt)= 3at+6bt ----------(2)
(1)=(2),so linear.
B)NON-LINEAR
Let:
x1(t)=t, x2(t)=2t;
System: y(t)=x(t)^2
Thus, y1(t)=t^2,y2(t)=4t^2
So say a,b are any non-zero constant
a*x1(t)->system->a*t^2
+ --->Output= a*t^2+4b*t^2 -----(3)
b*x2(t)->system->b*4t^2
a*x1(t)+b*x2(t)=at+2bt->system->Output=(at+2bt)^2 ----------(4)
(3)!=(4),so non-linear.
