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= a*x1[n]+b*x2[n] -> [sys] -> 2*z[n]^3 = 2*(a*x1[n] + b*x2[n])^3 | = a*x1[n]+b*x2[n] -> [sys] -> 2*z[n]^3 = 2*(a*x1[n] + b*x2[n])^3 | ||
x2[n] -> (X)*b +++ | x2[n] -> (X)*b +++ | ||
| + | |||
| + | |||
| + | However, since 2*a*x1[n]^3 + 2*b*x2[n] != 2(a*x1[n] + b*x2[n])^3 = 8*a^3*x1[n]^3 + 8*b^3*x2[n]^3 | ||
| + | the system is not linear because the two inflexive operations are not equal to each other. | ||
Revision as of 07:12, 12 September 2008
Linearity
So a system is linear if its inputs x1(t), x2(t) or (x1[n], x2[n] for Discrete Time signals) yield outputs y1(t), y2(t) such as the response: a*x1(t)+b*x2(t) => a*y1(t)+b*y2(t).
Example: Linear
Example: Non-Linear
One-way
y[n] = 2*x[n]^3
x1[n] -> [sys] -> y1[n]=2*x1[n]^3 -> (X)*a +++
= a*2*x1[n]^3+2*b*x2[n]^3
x2[n] -> [sys] -> y2[n]=2*x2[n]^3 -> (X)*b +++
Reverse-way
x1[n] -> (X)*a +++
= a*x1[n]+b*x2[n] -> [sys] -> 2*z[n]^3 = 2*(a*x1[n] + b*x2[n])^3
x2[n] -> (X)*b +++
However, since 2*a*x1[n]^3 + 2*b*x2[n] != 2(a*x1[n] + b*x2[n])^3 = 8*a^3*x1[n]^3 + 8*b^3*x2[n]^3
the system is not linear because the two inflexive operations are not equal to each other.
