Difference between revisions of "HW 2.c Hetong Li ECE301Fall2008mboutin" - Rhea

Latest revision as of 16:02, 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.

@@ Line 2: / Line 2: @@
 == Linear System ==
-For a function y=f(x), if its derivitive is a constant, then we say it is a 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 ==
-<pre>
+A)LINEAR
-For function y=2t+1, its derivitive y'=2
+  Let:
-y' is a constant
+     x1(t)=t, x2(t)=2t;
-Thus y=2t+1 is a linear system.
+     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
-For function y=sin(t), y'=cos(t)
+     a*x1(t)+b*x2(t)=at+2bt->system->Output=(at+2bt)^2 ----------(4)
-y' is not a constant
-Thus y=sin(t) is a non-linear system.
-</pre>
+     (3)!=(4),so non-linear.

Difference between revisions of "HW 2.c Hetong Li ECE301Fall2008mboutin" - Rhea

Latest revision as of 16:02, 12 September 2008

Linear System

Example

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