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]
