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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'''''<b><sub>1</sub>(''t'') + ''b''' | + | <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'''''x''<sub>2</sub>(''t'')'''''</span>'''''is <span class="texhtml" />''a''''y'''''<b><sub>1</sub>(</b>'''''t'') + ''b''''''''y''<sub>2</sub>(''t''). |
<u>Definition 2</u> | <u>Definition 2</u> | ||
| − | If <math> x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} | + | If |
| + | |||
| + | <math> x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_1(t) </math> | ||
| + | |||
| + | <math> x_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_2(t) </math> | ||
| + | |||
| + | then | ||
| + | |||
| + | <math> ax_1(t) + bx_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow ay_1(t) + by_2(t) </math> | ||
| + | |||
| + | for any <math>a,b\in </math> ''all complex numbers'', any <math>x_1(t), x_2(t)</math> then we say the system is linear. | ||
Revision as of 07:22, 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)</span>is <span class="texhtml" />a'y<b>1(t) + b'''y2(t).
Definition 2
If
$ x_1(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_1(t) $
$ x_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow y_2(t) $
then
$ ax_1(t) + bx_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow ay_1(t) + by_2(t) $
for any $ a,b\in $ all complex numbers, any $ x_1(t), x_2(t) $ then we say the system is linear.
