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== Linear system == | == Linear system == | ||
| + | <pre> | ||
SYSTEM: y = 3x(t) - 10 | SYSTEM: y = 3x(t) - 10 | ||
a.) 1X1(t) --> SYSTEM --> 3Y1(t) - 10 | a.) 1X1(t) --> SYSTEM --> 3Y1(t) - 10 | ||
b.) 4X2(t) --> SYSTEM --> 12Y2(t) - 10 | b.) 4X2(t) --> SYSTEM --> 12Y2(t) - 10 | ||
| − | We can do the following proof to show that the above system is linear. Take two random constant numbers such as 9 and 6. Now multiply the output from "a" by 9. Then multiply the output from "b" by 6. Now take their sum. (27Y(t) - 90) + (72Y(t)-60)) = 99Y(t)-150 | + | We can do the following proof to show that the above system is linear. Take two random constant numbers |
| + | such as 9 and 6. Now multiply the output from "a" by 9. Then multiply the output from "b" by 6. Now take | ||
| + | their sum. (27Y(t) - 90) + (72Y(t)-60)) = 99Y(t)-150 | ||
| + | |||
| + | Now we will multiply the original signals by the constants, take their sum, and then send them through the system. | ||
| + | If we end up with 99Y(t)-150, then the system must be linear. So, (9*1X) + (6*4X) = 33x. This gives 99Y(t) - 150. | ||
| + | Therefore the system is linear. | ||
| + | </pre> | ||
| + | |||
| + | ==Non-Linear== | ||
| + | <pre> | ||
| + | SYSTEM: y = (x(t))^5 | ||
| + | a.) 1X1(t) --> SYSTEM --> (x(t))^5 | ||
| + | b.) 4X2(t) --> SYSTEM --> (4x(t))^5 | ||
| − | + | To prove that this system is non-linear, we multiply output "a" by 9, then multiply output "b" by 6. Then, we take their sum. The calculations are as follows: 9x(t)^5 + 24x(t)^5 = 33x(t)^5. | |
| − | + | Now we try multiplying the input in part "a" by 9 and the input of part "b" by 6. This gives us 9x(t) + 6x(t). | |
| − | + | Now we take their sum. This gives us 15x(t). Now, we run 15x(t) through the system to obtain: 759375(X(t))^5. | |
| + | Therefore, the system is not linear. | ||
</pre> | </pre> | ||
Latest revision as of 17:32, 10 September 2008
Linear system
SYSTEM: y = 3x(t) - 10 a.) 1X1(t) --> SYSTEM --> 3Y1(t) - 10 b.) 4X2(t) --> SYSTEM --> 12Y2(t) - 10 We can do the following proof to show that the above system is linear. Take two random constant numbers such as 9 and 6. Now multiply the output from "a" by 9. Then multiply the output from "b" by 6. Now take their sum. (27Y(t) - 90) + (72Y(t)-60)) = 99Y(t)-150 Now we will multiply the original signals by the constants, take their sum, and then send them through the system. If we end up with 99Y(t)-150, then the system must be linear. So, (9*1X) + (6*4X) = 33x. This gives 99Y(t) - 150. Therefore the system is linear.
Non-Linear
SYSTEM: y = (x(t))^5 a.) 1X1(t) --> SYSTEM --> (x(t))^5 b.) 4X2(t) --> SYSTEM --> (4x(t))^5 To prove that this system is non-linear, we multiply output "a" by 9, then multiply output "b" by 6. Then, we take their sum. The calculations are as follows: 9x(t)^5 + 24x(t)^5 = 33x(t)^5. Now we try multiplying the input in part "a" by 9 and the input of part "b" by 6. This gives us 9x(t) + 6x(t). Now we take their sum. This gives us 15x(t). Now, we run 15x(t) through the system to obtain: 759375(X(t))^5. Therefore, the system is not linear.
