HW2.C Jayanth Athreya ECE301Fall2008mboutin - Rhea

now If

      X(t)-----> System---->z1(t)$ \times $a---->a.z1(t)
                                                    

      Y(t)-----> System---->z2(t)$ \times $b---->b.z2(t)

                       a.z1(t)+bz2(t)----->Z(t)       equation 1

and

     X(t)$ \times $a----->w1(t).a

     Y(t)$ \times $b----->w2(t).b

now

              w1(t).a+w2(t).b------>System----->W(t)       equation 2

IF eq 1 = eq 2 the system is linear.

      a,b are complex numbers.

Example of a linear System. Y[n]=X[n-1].

  Proof: X1[n]--->system--->Y1[n]=X1[n-1]--->a--->a.X1[n-1]

         X2[n]--->system--->Y2[n]=X2[n-1]--->b--->b.X2[n-1]

         Now   a.X1[n-1] +  b.X2[n-1]= Z(n)

         And

         X1[n]---->a-------->a.X1[n]
         X2[n]---->b-------->b.X2[n]

   {a.X1[n]+b.X2[n]}----->System------>W[n-1] = a.X1[n-1] +  b.X2[n-1] 

                 As the 2 results match the System is Linear

Example Of a non-linear System Y[t]=X[t]$ ^2 $

now If

X1(t)----> System---->Y1(t) =X1[t]$ ^2 $ $ \times $a---->a.X1[t]$ ^2 $
                                                    

X2(t)----> System---->Y2(t) =X2[t]$ ^2 $ $ \times $b---->b.X2[t]$ ^2 $

     a.X1[t]$ ^2 $ +  b.X2[t]$ ^2 $= Z(t)               equation 1

and

     X1(t)$ \times $a----->X1(t).a

     X2(t)$ \times $b----->X2(t).b

now

              {X1(t).a+X2(t).b}------>System----->{X1(t).a+X2(t).b}$ ^2 $      equation 2

IF eq 1 not equal to eq 2 the system is not linear.

      a,b are complex numbers.