(New page: == LINEARITY == Linearity, in my definition, means that superposition always works. In other words, summation of inputs yield summation of outputs. == Example of Linearity and its proof ...) |
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<math>x2(t) \to System \to y2(t)=x2(2t) \to Scalar multiplication(*b) \to bx2(2t) </math> | <math>x2(t) \to System \to y2(t)=x2(2t) \to Scalar multiplication(*b) \to bx2(2t) </math> | ||
| − | <math>ax1(2t) and bx2(2t) \to SUM \to | + | <math>ax1(2t) and bx2(2t) \to SUM \to ax1(2t)+bx2(2t)</math> |
| Line 20: | Line 20: | ||
<math>x2(t) \to Scalar multiplication(*b) \to bx2(t)</math> | <math>x2(t) \to Scalar multiplication(*b) \to bx2(t)</math> | ||
| − | <math>ax1(t) and bx2(t) \to SUM \to \to System \to | + | <math>ax1(t) and bx2(t) \to SUM \to \to System \to ax1(2t)+bx2(2t)</math> |
Those two yielded the same outputs thus it is linear. | Those two yielded the same outputs thus it is linear. | ||
| − | |||
== Example of non-linearity and its proof == | == Example of non-linearity and its proof == | ||
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'''Proof:''' | '''Proof:''' | ||
| − | <math> | + | <math>x1(t) \to System \to y1(t)=e^{x1(t)} \to Scalar multiplication(*a) \to ae^{x1(t)} </math> |
| − | <math>\ | + | <math>x2(t) \to System \to y2(t)=e^{x2(t)}\to Scalar multiplication(*b) \to be^{x2(t)} </math> |
| + | <math>ae^{x1(t)} and be^{x2(t)} \to SUM \to ae^{x1(t)}+be^{x2(t)}</math> | ||
| − | |||
| − | <math>\ | + | <math>x1(t) \to Scalar multiplication(*a) \to ax1(t)</math> |
| + | |||
| + | <math>x2(t) \to Scalar multiplication(*b) \to bx2(t)</math> | ||
| + | <math>ax1(t) and bx2(t) \to SUM \to \to System \to e^{ax1(2t)+bx2(2t)}=e^{ax1(2t)}e^{bx2(2t)}</math> | ||
| − | + | Those two yielded different outputs, thus it is not linear. | |
Latest revision as of 17:53, 12 September 2008
LINEARITY
Linearity, in my definition, means that superposition always works. In other words, summation of inputs yield summation of outputs.
Example of Linearity and its proof
$ \,y(t)=x(2t)\, $
Proof:
$ x1(t) \to System \to y1(t)=x1(2t) \to Scalar multiplication(*a) \to ax1(2t) $
$ x2(t) \to System \to y2(t)=x2(2t) \to Scalar multiplication(*b) \to bx2(2t) $
$ ax1(2t) and bx2(2t) \to SUM \to ax1(2t)+bx2(2t) $
$ x1(t) \to Scalar multiplication(*a) \to ax1(t) $
$ x2(t) \to Scalar multiplication(*b) \to bx2(t) $
$ ax1(t) and bx2(t) \to SUM \to \to System \to ax1(2t)+bx2(2t) $
Those two yielded the same outputs thus it is linear.
Example of non-linearity and its proof
$ \,y(t)=e^{x(t)}\, $
Proof:
$ x1(t) \to System \to y1(t)=e^{x1(t)} \to Scalar multiplication(*a) \to ae^{x1(t)} $
$ x2(t) \to System \to y2(t)=e^{x2(t)}\to Scalar multiplication(*b) \to be^{x2(t)} $
$ ae^{x1(t)} and be^{x2(t)} \to SUM \to ae^{x1(t)}+be^{x2(t)} $
$ x1(t) \to Scalar multiplication(*a) \to ax1(t) $
$ x2(t) \to Scalar multiplication(*b) \to bx2(t) $
$ ax1(t) and bx2(t) \to SUM \to \to System \to e^{ax1(2t)+bx2(2t)}=e^{ax1(2t)}e^{bx2(2t)} $
Those two yielded different outputs, thus it is not linear.
