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<math> Y_1(t) = A*x(t) = Z_1(t)</math> | <math> Y_1(t) = A*x(t) = Z_1(t)</math> | ||
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<math> | <math> | ||
Y_2(t) = X(At) = Z_2(t)</math> | Y_2(t) = X(At) = Z_2(t)</math> | ||
| + | |||
<math> | <math> | ||
Z_1(t) = Z_2(t) </math> | Z_1(t) = Z_2(t) </math> | ||
for any number A | for any number A | ||
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| + | ===Non-linear=== | ||
| + | |||
| + | <math>x(t) = t^2</math> | ||
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| + | Is an example of a non linear system. The order of linear operations before and after the system affect the result of the cascade | ||
| + | |||
| + | for t = 2 and A = 2 | ||
| + | <math> Y_1(2) = 2*x(2) = 8</math> | ||
| + | |||
| + | <math>Y_2(2) = x(2*2) = 16</math> | ||
| + | |||
| + | <math> Z_1(t) \neq Z_2(t)</math> | ||
Latest revision as of 08:45, 12 September 2008
Part C: Linearity
My definition of linearity in terms of systems is:
A system whose output combined with a linear shift is equivalent to the output if the linear shift is on the input of the system.
An example of a linear system is:
$ x(t) = t + 3 $
To prove this:
$ Y_1(t) = A*x(t) = Z_1(t) $
$ Y_2(t) = X(At) = Z_2(t) $
$ Z_1(t) = Z_2(t) $
for any number A
Non-linear
$ x(t) = t^2 $
Is an example of a non linear system. The order of linear operations before and after the system affect the result of the cascade
for t = 2 and A = 2 $ Y_1(2) = 2*x(2) = 8 $
$ Y_2(2) = x(2*2) = 16 $
$ Z_1(t) \neq Z_2(t) $
