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<u></u><u>Definition 1</u> | <u></u><u>Definition 1</u> | ||
| − | <u></u>A system is called '''linear''' if for any constants <math>a,b\in </math> ''all complex numbers'' and for any input signals <span class="texhtml">''x''<sub>1</sub>(''t''),''x''<sub>2</sub>(''t'')</span> with response <span class="texhtml">''y''<sub>1</sub>(''t''),''y''<sub>2</sub>(''t'')</span>, respectively, the system's response to <span class="texhtml">''a''''x''<sub>1</sub>(''t'') + ''b''''x''<sub>2</sub>(''t'')</span> is <span class="texhtml">''a''''y''<sub>1</sub>(''t'') + ''b''''y''<sub>2</sub>(''t'')</span>. | + | <u></u>A system is called '''linear''' if for any constants <math>a,b\in </math> ''all complex numbers'' and for any input signals <span class="texhtml">''x''<sub>1</sub>(''t''),''x''<sub>2</sub>(''t'')</span> with response <span class="texhtml">''y''<sub>1</sub>(''t''),''y''<sub>2</sub>(''t'')</span>, respectively, the system's response to <span class="texhtml">''a''''x'''''<b><sub>1</sub>(''t'') + ''b'''</b>''x''<sub>2</sub>(''t'')</span> is <span class="texhtml">''a''''y'''''<b><sub>1</sub>(''t'') + ''b'''</b>''y''<sub>2</sub>(''t'')</span>. |
| − | <u>Definition 2</u> | + | <u>Definition 2</u> |
| − | If | + | If <math> x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} |
Revision as of 06:15, 6 May 2011
Linearity
There are three definitions we discussed in class for linearity.
Definition 1
A system is called linear if for any constants $ a,b\in $ all complex numbers and for any input signals x1(t),x2(t) with response y1(t),y2(t), respectively, the system's response to a'x1(t) + bx2(t) is a'y1(t) + by2(t).
Definition 2
If $ x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} $
