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*'''Linearity:''' The function ("The system") f is '''linear''' iff <math>\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)</math> | *'''Linearity:''' The function ("The system") f is '''linear''' iff <math>\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)</math> | ||
| − | *'''Time Invariant:''' Define <math>S_{t_0}</math> as the shifting operator <math>S_{t_0}(x(t))=x(t-t_0).</math> (In other words, <math>S_{t_0}</math> introduces a time delay of <math>t_0</math> onto the function/signal x(t).) A function ("system") f is considered time-invariant iff <math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \forall x(t), t_0.</math> | + | *'''Time Invariant:''' Define <math>S_{t_0}</math> as the shifting operator <math>S_{t_0}(x(t))=x(t-t_0).</math> (In other words, <math>S_{t_0}</math> introduces a time delay of <math>t_0</math> onto the function/signal x(t).) A function ("system") f is considered '''time-invariant''' iff <math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \forall x(t), t_0.</math> |
Revision as of 13:23, 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 between Diagrammatical and Mathematical Explanations
- '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
