(New page: == What Is a Linear System== A linear system has to satisfy these contions: <br> If the inputs x1(t),x2(t),(x1[n],x2[n]) multiplied/divided by any constant a,b, then the output y1(t),y2(t)...) |
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A linear system has to satisfy these contions: <br> | A linear system has to satisfy these contions: <br> | ||
If the inputs x1(t),x2(t),(x1[n],x2[n]) multiplied/divided by any constant a,b, then the output y1(t),y2(t),(y1[n],y2[n]) will yield a*x1(t)+b*x2(t) --> a*y1(t)+b*y2(t) and respectively | If the inputs x1(t),x2(t),(x1[n],x2[n]) multiplied/divided by any constant a,b, then the output y1(t),y2(t),(y1[n],y2[n]) will yield a*x1(t)+b*x2(t) --> a*y1(t)+b*y2(t) and respectively | ||
| + | |||
| + | ==Example== | ||
| + | '''Linear''' | ||
| + | |||
| + | Given: | ||
| + | |||
| + | x1(t)=t, x2(t)=5 | ||
| + | |||
| + | System: y(t)=2*x(t) | ||
| + | |||
| + | Thus, y1(t)=2t,y2(t)=10t<br> | ||
| + | So say a,b are any non-zero constant<br> | ||
| + | a*x1(t)-> system ->2at<br> | ||
| + | b*x2(t)-> system ->10bt<br> | ||
| + | Therefore, the output is 2at + 10bt <-- answer A | ||
| + | |||
| + | |||
| + | a*x1(t)+b*x2(t)=at+5bt-> system ->Output=2*(at+2bt)= 2at + 10bt <-- answer B | ||
| + | |||
| + | answer A = answer B, proves that it's a linear system | ||
| + | |||
| + | '''Non Linear''' | ||
| + | |||
| + | Given: | ||
| + | |||
| + | x1(t)=t, x2(t)=5t | ||
| + | |||
| + | System: y(t)=x(t)^2 | ||
| + | |||
| + | Thus, y1(t)=t^2,y2(t)=(5t)^2<br> | ||
| + | So say a,b are any non-zero constant<br> | ||
| + | a*x1(t)-> system ->(at)^2<br> | ||
| + | b*x2(t)-> system ->(5bt)^2<br> | ||
| + | Therefore, the output is (at)^2 + (5bt)^2 <-- answer A | ||
| + | |||
| + | |||
| + | a*x1(t)+b*x2(t)=at+5bt-> system ->Output=(at+25t)^2 <-- answer B | ||
| + | |||
| + | answer A != answer B, proves that it's a non linear system | ||
Latest revision as of 18:46, 12 September 2008
What Is a Linear System
A linear system has to satisfy these contions:
If the inputs x1(t),x2(t),(x1[n],x2[n]) multiplied/divided by any constant a,b, then the output y1(t),y2(t),(y1[n],y2[n]) will yield a*x1(t)+b*x2(t) --> a*y1(t)+b*y2(t) and respectively
Example
Linear
Given:
x1(t)=t, x2(t)=5
System: y(t)=2*x(t)
Thus, y1(t)=2t,y2(t)=10t
So say a,b are any non-zero constant
a*x1(t)-> system ->2at
b*x2(t)-> system ->10bt
Therefore, the output is 2at + 10bt <-- answer A
a*x1(t)+b*x2(t)=at+5bt-> system ->Output=2*(at+2bt)= 2at + 10bt <-- answer B
answer A = answer B, proves that it's a linear system
Non Linear
Given:
x1(t)=t, x2(t)=5t
System: y(t)=x(t)^2
Thus, y1(t)=t^2,y2(t)=(5t)^2
So say a,b are any non-zero constant
a*x1(t)-> system ->(at)^2
b*x2(t)-> system ->(5bt)^2
Therefore, the output is (at)^2 + (5bt)^2 <-- answer A
a*x1(t)+b*x2(t)=at+5bt-> system ->Output=(at+25t)^2 <-- answer B
answer A != answer B, proves that it's a non linear system
