HW 2.c Hetong Li ECE301Fall2008mboutin - Rhea

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.