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There are three definitions we discussed in class for linearity. | There are three definitions we discussed in class for linearity. | ||
| − | <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">'' | + | <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">''ax''<sub>1</sub>(''t'') + ''b''x''<sub>2</sub>(''t'')'' ''is ''ay''<sub>1</sub>(''t'') + ''b''y''<sub>2</sub>(''t''). </span> |
| − | </span> | + | |
| + | ''' | ||
<u>Definition 2</u> | <u>Definition 2</u> | ||
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<math> ax_1(t) + bx_2(t) \rightarrow \begin{bmatrix} system \end{bmatrix} \rightarrow ay_1(t) + by_2(t) </math> | <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 <span class="texhtml">''x''<sub>1</sub>(''t''),''x''<sub>2</sub>(''t'')</span> then we say the system is '''linear'''. | + | <span class="Apple-style-span" style="font-weight: normal;">for any <math>a,b\in </math> </span>''<span class="Apple-style-span" style="font-weight: normal;">all complex numbers</span>''<span class="Apple-style-span" style="font-weight: normal;">, any </span><span class="texhtml">''<span class="Apple-style-span" style="font-weight: normal;">x</span>''<sub><span class="Apple-style-span" style="font-weight: normal;">1</span></sub><span class="Apple-style-span" style="font-weight: normal;">(</span>''<span class="Apple-style-span" style="font-weight: normal;">t</span>''<span class="Apple-style-span" style="font-weight: normal;">),</span>''<span class="Apple-style-span" style="font-weight: normal;">x</span>''<sub><span class="Apple-style-span" style="font-weight: normal;">2</span></sub><span class="Apple-style-span" style="font-weight: normal;">(</span>''<span class="Apple-style-span" style="font-weight: normal;">t</span>''<span class="Apple-style-span" style="font-weight: normal;">)</span></span><span class="Apple-style-span" style="font-weight: normal;"> then we say the system is</span> '''linear'''. |
<u>Definition 3</u> | <u>Definition 3</u> | ||
| + | ''' | ||
| − | <u></u>A system is '''linear ''' | + | <u></u>A system is''''''linear'''''' |
Revision as of 06:26, 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 ax1(t) + bx2(t) is ay1(t) + by2(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 x1(t),x2(t) then we say the system is linear.
Definition 3
A system is'linear'
