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.
