(New page: ==Diagrammatical Explanations== ==Mathematical Explanations== As some people find the mathematical explanations simpler to understand and/or work with, they will be presented here: ===Co...) |
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==References== | ==References== | ||
ECE301 lectures by Mimi Boutin, Purdue University, Fall 2008 | ECE301 lectures by Mimi Boutin, Purdue University, Fall 2008 | ||
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http://en.wikipedia.org/wiki/Linearity | http://en.wikipedia.org/wiki/Linearity | ||
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http://en.wikipedia.org/wiki/Time_invariant | http://en.wikipedia.org/wiki/Time_invariant | ||
Revision as of 13:22, 10 September 2008
Contents
Diagrammatical Explanations
Mathematical Explanations
As some people find the mathematical explanations simpler to understand and/or work with, they will be presented here:
Concepts
Linearity: The function ("The system") f is linear iff $ \forall x_1(t), x_2(t) \text{ and } \forall a,b \in \mathbb{C}, f(ax_1 + bx_2) = af(x_1) + bf(x_2) $
Time Invariant: Define $ S_{t_0} $ as the shifting operator $ S_{t_0}(x(t))=x(t-t_0). $ (In other words, $ S_{t_0} $ introduces a time delay of $ t_0 $ onto the function/signal x(t).) A function ("system") f is considered time-invariant iff $ f(S_{t_0}(x))=S_{t_0}(f(x))\ \forall x(t), t_0. $
Translations
- 'The system' <==> 'The function f'
- 'x --> [system] --> y' <==> 'y = f(x)' ('x --> [f] --> f(x)')
References
ECE301 lectures by Mimi Boutin, Purdue University, Fall 2008
