Difference between revisions of "HW2.C Naman Anand Verma ECE301Fall2008mboutin" - Rhea

Latest revision as of 17:13, 12 September 2008

Part C: Linearity

A linear system is as follows:

When two separate signals x(t) and y (t) enter two systems individually and their outputs are separately multiplied by constants a and b and the added .The resulting signal after addition can be called z(t).

Now if we multiply x(t) and y(t) by a and b respectively and then add them and let them enter the system.Let us call the resulting output w(t).

If w(t) and z(t) are same then the system is called a Linear system

If w(t) and z(t) are different ,then the system is called a Non Linear system.

Example of a Linear system:

x1 (t) --->SYSTEM---->Y1 (t) multiply by  a ---> aY1(t)


x2 (t) --->SYSTEM---->Y2 (t) multiply by  b ---> bY2(t)


aY1(t) + bY2(t) --------> Z (t)


Now if

x1 (t) multiply by  a ---> aX1(t)


x2 (t) multiply by  b ---> bX2(t)


aX1(t) + bX2(t)---------->SYSTEM ---->W(t)

When W(t) = Z(t)

It is a linear system.





Example of a Non linear system :


y[n] = x[n]^2


x1 (n) --->SYSTEM---->Y1 (n) multiply by  a ---> ax1[n]^2          - eq 1

x2 (n) --->SYSTEM---->Y2 (n) multiply by  b ---> bx2[n]^2          - eq 2

adding equation 1 and 2 we get

ax1[n]^2 + bx2[n]^2                 - eq 5




x1 (n) multiply by  a ---> ax1[n]      -eq 3

x2 (n) multiply by  b ---> bx2[n]      -eq 4

adding eq 1 and eq2 

ax1[n] + bx2[n]-------> SYSTEM----> (ax1[n] + bx2[n]) ^2         - eq 6

Since both outputs (eq 5 and eq 6) are differeny

The system is non linear.