Mathematical explanation of Linearity and Time-invariance ECE301Fall2008mboutin - Rhea

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 \rightarrow \text{system} \rightarrow y \Leftrightarrow y = f(x) \left ( x \rightarrow \text{f} \rightarrow f(x) \right ) $

References

ECE301 lectures by Mimi Boutin, Purdue University, Fall 2008

http://en.wikipedia.org/wiki/Linearity

http://en.wikipedia.org/wiki/Time_invariant