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Homework 2 Ben Horst: [[HW2-A Ben Horst _ECE301Fall2008mboutin| A]] :: [[HW2-B Ben Horst _ECE301Fall2008mboutin| B]] :: [[HW2-C Ben Horst _ECE301Fall2008mboutin| C]] :: [[HW2-D Ben Horst _ECE301Fall2008mboutin| D]] :: [[HW2-E Ben Horst _ECE301Fall2008mboutin| E]] | Homework 2 Ben Horst: [[HW2-A Ben Horst _ECE301Fall2008mboutin| A]] :: [[HW2-B Ben Horst _ECE301Fall2008mboutin| B]] :: [[HW2-C Ben Horst _ECE301Fall2008mboutin| C]] :: [[HW2-D Ben Horst _ECE301Fall2008mboutin| D]] :: [[HW2-E Ben Horst _ECE301Fall2008mboutin| E]] | ||
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| + | ==Linear Systems== | ||
| + | A linear system is one whose output based on input can also be shown as a sum of each input before being run through the system. | ||
| + | Stated another way, in1 + in2 -> out3 where in1 -> out1, in2 -> out2, and out1 + out2 = out3. | ||
| + | |||
| + | ==Example of a Linear System== | ||
| + | Given the system y(t) = 3x(t): | ||
| + | |||
| + | |||
| + | x1(t) = t -> y1(t) = 3t | ||
| + | |||
| + | x2(t) = 4t -> y2(t) = 12t | ||
| + | |||
| + | |||
| + | x3(t) = t + 4t -> y3(t) = 3(t + 4t) = 3t + 12t = 15t | ||
| + | |||
| + | y1(t) + y2(t) = 15t | ||
| + | |||
| + | |||
| + | Since y3 is equal to y1 + y2, the system is linear. | ||
| + | |||
| + | ==Example of a Non-Linear System== | ||
| + | Given the system y(t) = 12x(t) + 5: | ||
| + | |||
| + | x1(t) = t -> y1(t) = 12t + 5 | ||
| + | |||
| + | x2(t) = 4t -> y2(t) = 48t + 5 | ||
| + | |||
| + | |||
| + | x3(t) = t + 4t -> y3(t) = 12(t + 4t) + 5 = 12t + 48t + 5 = 60t + 5 | ||
| + | |||
| + | y1(t) + y2(t) = 60t + 10 | ||
| − | + | Since y3 does not equal y1 + y2, the system is non-linear. | |
Latest revision as of 08:43, 11 September 2008
Homework 2 Ben Horst: A :: B :: C :: D :: E
Linear Systems
A linear system is one whose output based on input can also be shown as a sum of each input before being run through the system. Stated another way, in1 + in2 -> out3 where in1 -> out1, in2 -> out2, and out1 + out2 = out3.
Example of a Linear System
Given the system y(t) = 3x(t):
x1(t) = t -> y1(t) = 3t
x2(t) = 4t -> y2(t) = 12t
x3(t) = t + 4t -> y3(t) = 3(t + 4t) = 3t + 12t = 15t
y1(t) + y2(t) = 15t
Since y3 is equal to y1 + y2, the system is linear.
Example of a Non-Linear System
Given the system y(t) = 12x(t) + 5:
x1(t) = t -> y1(t) = 12t + 5
x2(t) = 4t -> y2(t) = 48t + 5
x3(t) = t + 4t -> y3(t) = 12(t + 4t) + 5 = 12t + 48t + 5 = 60t + 5
y1(t) + y2(t) = 60t + 10
Since y3 does not equal y1 + y2, the system is non-linear.
