Difference between revisions of "HW2-C Vivek Ravi ECE301Fall2008mboutin" - Rhea

Revision as of 10:55, 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).$ {a*y_1(t)+b*y_2(t)}={a*tsin(t)+b*tcos(t)} $

And, (2).$ {H[{a*x_1(t)+b*x_2(t)}]}={t{asin(t)+bcos(t)}}={a*tsin(t)+b*tcos(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).$ {a*y_1(t)+b*y_2(t)}={a*{t}^6+b*{sin}^2(t)} $

(2).$ {H[{a*x_1(t)+b*x_2(t)}]}={[{a*{t}^3}+{b*sin(t)}]^2} $

When we observe (1) and (2) we notice that they are not equal. Thus the system is not linear.