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A system is called linear if and only if: | A system is called linear if and only if: | ||
| − | <math>f(ax_1 + bx_2) = af(x_1) + bf(x_2)</math> | + | <math>f(ax_1 + bx_2) = af(x_1) + bf(x_2)\,</math> |
== Example of a linear system == | == Example of a linear system == | ||
System is: <math> f(x) = 23x \,</math> | System is: <math> f(x) = 23x \,</math> | ||
| + | |||
<math>X_1(t) = t^2 \,</math> | <math>X_1(t) = t^2 \,</math> | ||
| + | |||
<math>X_2(t) = 2t^2 \,</math> | <math>X_2(t) = 2t^2 \,</math> | ||
| + | |||
<math>f(aX_1 + bX_2) = af(X_1) + bf(X_2) \,</math> | <math>f(aX_1 + bX_2) = af(X_1) + bf(X_2) \,</math> | ||
| − | <math>f(at^2 + 2bt^2) = af(t^2) + bf( | + | |
| + | <math>f(at^2 + 2bt^2) = af(t^2) + bf(2t^2) \,</math> | ||
| + | |||
<math>f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \,</math> | <math>f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \,</math> | ||
| + | |||
<math>f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \,</math> | <math>f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \,</math> | ||
| + | |||
<math> f(x) = 23x \,</math> | <math> f(x) = 23x \,</math> | ||
| + | |||
| + | |||
| + | |||
== Example of a non-linear system == | == Example of a non-linear system == | ||
| + | System is: <math> f(x) = 23x + 1\,</math> | ||
| + | |||
| + | <math>X_1(t) = t^2 \,</math> | ||
| + | |||
| + | <math>X_2(t) = 2t^2 \,</math> | ||
| + | |||
| + | |||
| + | <math>f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \,</math> | ||
| + | |||
| + | <math>f(at^2 + 2bt^2) \neq af(t^2) + bf(2t^2) \,</math> | ||
| + | |||
| + | <math>f(at^2 + 2bt^2) \neq a(23t^2+1) + b(23*(2t^2)+1) \,</math> | ||
| + | |||
| + | <math>f(at^2 + 2bt^2) \neq 23 at^2 + 1 + 46 bt^2 + b \,</math> | ||
| + | |||
| + | <math> f(at^2 + 2bt^2) \neq 23 (at^2 + 2bt^2) + a + b \,</math> | ||
| + | |||
| + | <math> f(x) \neq 23x + 1 \,</math> | ||
| + | |||
| + | == Reference == | ||
| + | |||
| + | |||
| + | http://kiwi.ecn.purdue.edu/ECE301Fall2008mboutin/index.php/Concepts_and_Formulae | ||
Latest revision as of 16:48, 10 September 2008
Contents
Linearity
A system is called linear if and only if:
$ f(ax_1 + bx_2) = af(x_1) + bf(x_2)\, $
Example of a linear system
System is: $ f(x) = 23x \, $
$ X_1(t) = t^2 \, $
$ X_2(t) = 2t^2 \, $
$ f(aX_1 + bX_2) = af(X_1) + bf(X_2) \, $
$ f(at^2 + 2bt^2) = af(t^2) + bf(2t^2) \, $
$ f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \, $
$ f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \, $
$ f(x) = 23x \, $
Example of a non-linear system
System is: $ f(x) = 23x + 1\, $
$ X_1(t) = t^2 \, $
$ X_2(t) = 2t^2 \, $
$ f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \, $
$ f(at^2 + 2bt^2) \neq af(t^2) + bf(2t^2) \, $
$ f(at^2 + 2bt^2) \neq a(23t^2+1) + b(23*(2t^2)+1) \, $
$ f(at^2 + 2bt^2) \neq 23 at^2 + 1 + 46 bt^2 + b \, $
$ f(at^2 + 2bt^2) \neq 23 (at^2 + 2bt^2) + a + b \, $
$ f(x) \neq 23x + 1 \, $
Reference
http://kiwi.ecn.purdue.edu/ECE301Fall2008mboutin/index.php/Concepts_and_Formulae
