(New page: 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)}) |
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| + | == Definition == | ||
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A system is Linear if it is both additive and homogeneous | A system is Linear if it is both additive and homogeneous | ||
That is, | That is, | ||
T{a x1 (n) + b x2 (n)} = a T{x1 (n)} + b T{x2 (n)} | T{a x1 (n) + b x2 (n)} = a T{x1 (n)} + b T{x2 (n)} | ||
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| + | == Linearity check == | ||
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| + | Let us check if the following signal is linear. | ||
| + | y[n]= cos[nQ]*x[n] | ||
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| + | '''First we check if its additive''' | ||
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| + | y[x1[n]]=cos(nQ)* x1[n] | ||
| + | y[x2[n]]=cos(nQ)* x2[n] | ||
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| + | Therefore, | ||
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| + | y[x1[n]+x2[n]]= cos(nQ)x1[n] + cos(nQ)*x2[n] | ||
| + | = cos(nQ)[x1[n]+ x2[n]]..................(1) | ||
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| + | Also, | ||
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| + | y[x1[n]+x2[n]] = cos[nQ][x1[n]+x2[n]]...............(2) | ||
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| + | From (1) and (2) we see that y[n] is additive | ||
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| + | '''Now we check it it is homogeneous''' | ||
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| + | y[c*x[n]] = cos[nQ]* [c*x[n]] = c*cos[nQ]*x[n].......................(1) | ||
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| + | c*y[x[n]] = c*cos[nQ]*x[n]...............................(2) | ||
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| + | From (1) and (2) we see that it is also homogeneous | ||
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| + | '''Hence we can say that the above function os linear.''' | ||
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| + | == Non linearity Check == | ||
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| + | Let us check if the following signal is linear | ||
| + | y[n] = <math>x{[n]^2}</math> | ||
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| + | '''First we check if it is additive''' | ||
| + | |||
| + | *y[x1[n]]= <math>x1{[n]^2}</math> | ||
| + | *y[x2[n]]= <math>x2{[n]^2}</math> | ||
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| + | Therefore, | ||
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| + | y[x1[n]]+y[x2[n]] = <math>x1{[n]^2} + x2{[n]^2}</math>...........(1) | ||
| + | |||
| + | Also, | ||
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| + | y[x1[n]+x2[n]]= <math>[[x1[n]+ x2[n_ECE301Fall2008mboutin]]^{2}</math> = <math>x1[n]^2 + 2 x1[n] x2[n] + x2[n]^2</math>.......(2) | ||
| + | |||
| + | From (1) and (2) we see that the above system is not additive | ||
| + | |||
| + | '''Hence it is not linear''' | ||
Latest revision as of 09:42, 12 September 2008
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
