(New page: ==Linearity== A system is linear if the responses to inputs multiplied by constants are the original responses multiplied by the same constants. <br> <br> In symbols: <br> '''If'''<br> <...) |
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This fulfills the requirements that <math> a*x(t)\,</math> yields <math> a*y(t)\,</math> | This fulfills the requirements that <math> a*x(t)\,</math> yields <math> a*y(t)\,</math> | ||
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| + | In the same sense, you can prove that <math> y(t) = x(t)^2\,</math> is '''not''' linear. | ||
| + | Multiplying <math> x(t)\,</math> by <math> a\,</math> would yield <math> y(t) = a^2*t^2\,</math> | ||
Latest revision as of 07:20, 12 September 2008
Linearity
A system is linear if the responses to inputs multiplied by constants are the original responses multiplied by the same constants.
In symbols:
If
$ y1(t)+y2(t)\, $ are the responses to inputs $ x1(t)+x2(t)\, $
Then
$ a*x1(t)+b*x2(t)\, $ should yield $ a*y1(t)+b*y2(t)\, $
Example
A very simple linear system is $ y(t) = t*x(t)\, $
If $ x(t) = t^2\, $, then $ y(t) = t^3\, $
Multiplying $ x(t)\, $ by a constant $ a\, $ yields $ y(t) = t*x(t) = t*a*t^2 = a*t^3\, $
This fulfills the requirements that $ a*x(t)\, $ yields $ a*y(t)\, $
In the same sense, you can prove that $ y(t) = x(t)^2\, $ is not linear.
Multiplying $ x(t)\, $ by $ a\, $ would yield $ y(t) = a^2*t^2\, $
