(New page: ==Linear system== A linear system is a system that will produce the same output for both of the following actions: 1. One puts signals through the system, multiplies the outcomes by a c...) |
(→Non-linear system) |
||
| (5 intermediate revisions by the same user not shown) | |||
| Line 11: | Line 11: | ||
| − | <math>y(t) = 15x(t)\!< | + | <math>y(t) = 15x(t)\!</math> |
| − | The proof for this is rather simple. Suppose you put <math>x(t) = t + 12\!< | + | The proof for this is rather simple. Suppose you put <math>x(t) = t + 12\!</math> and end up with <math>15t + 180\!</math>. You also send <math>z(t) = t - 2\pi\!</math> through the system and get <math>15t - 30\pi\!</math>. You multiply the first outcome by <math>2\!</math> and get <math>30t + 360\!</math>. You multiply the second result by <math>3\!</math> and you get <math>45t - 90\pi\!</math>. After summing the two, you get <math>75t + 360 - 90\pi\!</math>. |
| + | |||
| + | |||
| + | Now, we do it the second way. First, we multiply the signals by the given constants. <math>x(t) = t + 12\!</math> becomes <math>2t + 24\!</math>, and <math>z(t) = t - 2\pi\!</math> becomes <math>3t - 6\pi\!</math>. Then, we add them together, and we get <math>5t + 24 - 6\pi\!</math>. Finally, we run it through the system, which gives us <math>75t + 360 - 90\pi\!</math>. As you can see, it checks. | ||
| + | |||
| + | ==Non-linear system== | ||
| + | |||
| + | An example of a non-linear system is as follows: | ||
| + | |||
| + | |||
| + | <math>y(t) = [x(t)]^2\!</math> | ||
| + | |||
| + | |||
| + | Suppose you put <math>x(t) = 12t\!</math> and end up with <math>144t^2\!</math>. You also send <math>z(t) = 5t\!</math> through the system and get <math>25t^2\!</math>. You multiply the first outcome by <math>2\!</math> and get <math>288t^2\!</math>. You multiply the second result by <math>3\!</math> and you get <math>75t^2\!</math>. After summing the two, you get <math>363t^2\!</math>. | ||
| + | |||
| + | |||
| + | Now, we do it the second way. First, we multiply the signals by the given constants. <math>x(t) = 12t\!</math> becomes <math>24t\!</math>, and <math>z(t) = 5t\!</math> becomes <math>15t\!</math>. Then, we add them together, and we get <math>39t\!</math>. Finally, we run it through the system, which gives us <math>1521t^2\!</math>. As you can see, it does not checks at all. Thus, this system is non-linear. | ||
Latest revision as of 17:59, 10 September 2008
Linear system
A linear system is a system that will produce the same output for both of the following actions:
1. One puts signals through the system, multiplies the outcomes by a constant, and add the results together.
2. One multiplies the same signals by the same constants, adds the results together, and sends that outcome through the system.
An example of a linear system is as follows:
$ y(t) = 15x(t)\! $
The proof for this is rather simple. Suppose you put $ x(t) = t + 12\! $ and end up with $ 15t + 180\! $. You also send $ z(t) = t - 2\pi\! $ through the system and get $ 15t - 30\pi\! $. You multiply the first outcome by $ 2\! $ and get $ 30t + 360\! $. You multiply the second result by $ 3\! $ and you get $ 45t - 90\pi\! $. After summing the two, you get $ 75t + 360 - 90\pi\! $.
Now, we do it the second way. First, we multiply the signals by the given constants. $ x(t) = t + 12\! $ becomes $ 2t + 24\! $, and $ z(t) = t - 2\pi\! $ becomes $ 3t - 6\pi\! $. Then, we add them together, and we get $ 5t + 24 - 6\pi\! $. Finally, we run it through the system, which gives us $ 75t + 360 - 90\pi\! $. As you can see, it checks.
Non-linear system
An example of a non-linear system is as follows:
$ y(t) = [x(t)]^2\! $
Suppose you put $ x(t) = 12t\! $ and end up with $ 144t^2\! $. You also send $ z(t) = 5t\! $ through the system and get $ 25t^2\! $. You multiply the first outcome by $ 2\! $ and get $ 288t^2\! $. You multiply the second result by $ 3\! $ and you get $ 75t^2\! $. After summing the two, you get $ 363t^2\! $.
Now, we do it the second way. First, we multiply the signals by the given constants. $ x(t) = 12t\! $ becomes $ 24t\! $, and $ z(t) = 5t\! $ becomes $ 15t\! $. Then, we add them together, and we get $ 39t\! $. Finally, we run it through the system, which gives us $ 1521t^2\! $. As you can see, it does not checks at all. Thus, this system is non-linear.
