(New page: == Homework 2 part C == <br> ==What is a linear system?== A system is linear if:<br> 1. The sum of any two inputs sent through a system is equal to any two inputs sent through a system tha...) |
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==What is a linear system?== | ==What is a linear system?== | ||
A system is linear if:<br> | A system is linear if:<br> | ||
| − | 1. The | + | 1. The output of summing any two inputs togerther then sending the result through a system is equal to any two inputs sent through a system than added together.<br> |
| − | 2. | + | 2. Multiplying an input then sending it through the system equals the input sent through the system then multiplied by the constant.<br> |
| + | |||
| + | == proving this...== | ||
| + | '''Example of a Linear System''' | ||
| + | <pre> | ||
| + | First I will sum the inputs before the system | ||
| + | the system is y(t) = 5x(t). | ||
| + | |||
| + | a(t) = 3t | ||
| + | b(t) = 4t | ||
| + | |||
| + | c(t) = a(t) + b(t) | ||
| + | c(t) = 3t + 4t | ||
| + | y(t) = 5(3t + 4t) = 35t | ||
| + | If it works backwards it is linear: | ||
| + | y1(t) = 5(3t) = 15t | ||
| + | y2(t) = 5(4t) = 20t | ||
| + | y3(t) = 15t+20t = 35t | ||
| + | Both approaches yield the same result. Therefore it is linear. | ||
| + | |||
| + | </pre> | ||
| + | '''Example of a non-linear System''' | ||
| + | Given the system | ||
| + | y(t) = 10x(t) + 3: | ||
| + | x1(t) = t | ||
| + | y1(t) = 10t + 3 | ||
| + | x2(t) = 4t | ||
| + | y2(t) = 40t + 3 | ||
| + | |||
| + | y1(t) + y2(t) = 50t + 6 | ||
| + | Do this backwards: | ||
| + | z(t) = x1(t)+y1(t) | ||
| + | z(t) = 5t | ||
| + | y[z(t)] = 50t + 6 | ||
| + | |||
| + | 60t+10 is not equal to 60t + 5 | ||
Latest revision as of 15:06, 11 September 2008
Homework 2 part C
What is a linear system?
A system is linear if:
1. The output of summing any two inputs togerther then sending the result through a system is equal to any two inputs sent through a system than added together.
2. Multiplying an input then sending it through the system equals the input sent through the system then multiplied by the constant.
proving this...
Example of a Linear System
First I will sum the inputs before the system
the system is y(t) = 5x(t).
a(t) = 3t
b(t) = 4t
c(t) = a(t) + b(t)
c(t) = 3t + 4t
y(t) = 5(3t + 4t) = 35t
If it works backwards it is linear:
y1(t) = 5(3t) = 15t
y2(t) = 5(4t) = 20t
y3(t) = 15t+20t = 35t
Both approaches yield the same result. Therefore it is linear.
Example of a non-linear System Given the system
y(t) = 10x(t) + 3:
x1(t) = t
y1(t) = 10t + 3
x2(t) = 4t
y2(t) = 40t + 3
y1(t) + y2(t) = 50t + 6
Do this backwards:
z(t) = x1(t)+y1(t)
z(t) = 5t
y[z(t)] = 50t + 6
60t+10 is not equal to 60t + 5
