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= <math>\frac{1}{2}*x1(t) + \frac{1}{2}*x2(t))</math> | = <math>\frac{1}{2}*x1(t) + \frac{1}{2}*x2(t))</math> | ||
| + | |||
| + | Using the defition of linearity above, | ||
| + | a = <math>\frac{1}{2}</math> | ||
| + | b = <math>\frac{1}{2}</math> | ||
| + | x1(t) = <math>e^{2tj}</math> | ||
| + | x2(t) = <math>e^{-2tj}</math> | ||
Revision as of 08:21, 18 September 2008
Basics of Linearity
Definition of Linearity: For any constants a and b (that are complext numbers), and inputs x1(t) and x2(t) which yield outputs y1(t) and y2(t),
$ a * x1(t) + b * x2(t) ---> Sys ---> a * y1(t) + b * y2(t) $
We are given a linear system that behaves as follows,
$ e^{2jt} --> Sys --> t*e^{-2jt} $
and asked to find the response to find the response to cos(2t).
Solution:
Using the properties of cosine we can convert cos(2t) to an exponential function.
cos(2t) = $ \frac{e^{2tj}+e^{-2tj}}{2} $
= $ \frac{1}{2}*(x1(t) + x2(t)) $
= $ \frac{1}{2}*x1(t) + \frac{1}{2}*x2(t)) $
Using the defition of linearity above, a = $ \frac{1}{2} $ b = $ \frac{1}{2} $ x1(t) = $ e^{2tj} $ x2(t) = $ e^{-2tj} $
