| Line 90: | Line 90: | ||
| − | + | X1(t)<math>\times</math>'''a'''----->X1(t).a | |
| − | + | X2(t)<math>\times</math>'''b'''----->X2(t).b | |
'''now ''' | '''now ''' | ||
| − | + | {X1(t).a+X2(t).b}------>'''System'''----->{X1(t).a+X2(t).b}<math>^2</math> equation 2 | |
| − | IF eq 1 | + | IF eq 1 '''not equal to''' eq 2 the '''system is not linear'''. |
'''a,b''' are complex numbers. | '''a,b''' are complex numbers. | ||
Latest revision as of 12:51, 12 September 2008
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.
