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[[Homework 3_ECE301Fall2008mboutin|<< Back to Homework 3]] | [[Homework 3_ECE301Fall2008mboutin|<< Back to Homework 3]] | ||
| − | Homework 3 Ben Horst: [[ | + | Homework 3 Ben Horst: [[HW3.A Ben Horst _ECE301Fall2008mboutin| A]] :: [[HW3.B Ben Horst _ECE301Fall2008mboutin| B]] :: [[HW3.C Ben Horst _ECE301Fall2008mboutin| C]] |
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==Formal Definition of Linearity== | ==Formal Definition of Linearity== | ||
| Line 32: | Line 32: | ||
For any x(t): | For any x(t): | ||
| − | x(t) -> |system| -> y(t) | + | x(t) -> |system| -> y(t) |
| − | then | + | then |
| − | x(t - t_0) -> |system| -> y(t - t_0) | + | x(t - t_0) -> |system| -> y(t - t_0) |
If this is not true, the system is not time invariant. | If this is not true, the system is not time invariant. | ||
| Line 42: | Line 42: | ||
In dumbed-down math jibberish, that is: | In dumbed-down math jibberish, that is: | ||
| − | There exists some number (sigma) such that sigma's magnitude is greater than the input for all values of [t] | + | |
| + | There exists some number (sigma) such that sigma's magnitude is greater than the input for all values of [t] | ||
| + | |||
ALSO | ALSO | ||
| − | There exists some other number (M) such that M's magnitude is greater than the output for all values of [t] | + | |
| + | There exists some other number (M) such that M's magnitude is greater than the output for all values of [t] | ||
Latest revision as of 14:57, 25 September 2008
Homework 3 Ben Horst: A :: B :: C
Contents
Formal Definition of Linearity
A system is linear if the following conditions are met:
An input x1 yields output y1.
An input x2 yields output y2.
An input that is the sum of a*x1 and b*x2 yields output that is the sum of a*y1 and b*y2.
a and b are any complex constants.
If these conditions are not met, the system is non-linear.
Formal Definition of Memoryless Systems
A system is memoryless only if it does not depend on the output of the signal at any other point in time. Example:
y(t) = x(t) + x(t-1) --> has memory
y(t) = x(t) + t - 1 --> memoryless (see that "t - 1" is a constant, not an input to a function)
Formal Definition of Causal Systems
A system is causal if the output at any given time only depends on the input in present and past. The system's output may NOT depend on the future. A memoryless system is by definition, also causal.
Formal Definition of A Time Invariant System
A system is time invariant if the following is true:
For any x(t): x(t) -> |system| -> y(t) then x(t - t_0) -> |system| -> y(t - t_0)
If this is not true, the system is not time invariant.
Formal Definition of A Stable System
A system is stable if and only if bounded inputs yield bounded outputs.
In dumbed-down math jibberish, that is:
There exists some number (sigma) such that sigma's magnitude is greater than the input for all values of [t]
ALSO
There exists some other number (M) such that M's magnitude is greater than the output for all values of [t]
