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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>. | 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>. | ||
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| + | 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. | ||
Revision as of 17:51, 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.
