Definition
A system is Linear if it is both additive and homogeneous That is,
T{a x1 (n) + b x2 (n)} = a T{x1 (n)} + b T{x2 (n)}
Linearity check
Let us check if the following signal is linear.
y[n]= cos[nQ]*x[n]
First we check if its additive
y[x1[n]]=cos(nQ)* x1[n] y[x2[n]]=cos(nQ)* x2[n]
Therefore,
y[x1[n]+x2[n]]= cos(nQ)x1[n] + cos(nQ)*x2[n] = cos(nQ)[x1[n]+ x2[n]]..................(1)
Also,
y[x1[n]+x2[n]] = cos[nQ][x1[n]+x2[n]]...............(2)
From (1) and (2) we see that y[n] is additive
Now we check it it is homogeneous
y[c*x[n]] = cos[nQ]* [c*x[n]] = c*cos[nQ]*x[n].......................(1)
c*y[x[n]] = c*cos[nQ]*x[n]...............................(2)
From (1) and (2) we see that it is also homogeneous
Hence we can say that the above function os linear.
Non linearity Check
Let us check if the following signal is linear
y[n] = $ x{[n]^2} $
First we check if it is additive
- y[x1[n]]= $ x1{[n]^2} $
- y[x2[n]]= $ x2{[n]^2} $
Therefore,
y[x1[n]]+y[x2[n]] = $ x1{[n]^2} + x2{[n]^2} $...........(1)
Also,
y[x1[n]+x2[n]]= $ [[x1[n]+ x2[n_ECE301Fall2008mboutin]]^{2} $ = $ x1[n]^2 + 2 x1[n] x2[n] + x2[n]^2 $.......(2)
From (1) and (2) we see that the above system is not additive
Hence it is not linear
