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*Example of a '''time variant''' system: | *Example of a '''time variant''' system: | ||
− | <math>y(t) = \sin t x(t) \mapsto y(t - t_0) \neq \ | + | <math>y(t) = \sin(t) x(t) \mapsto y(t - t_0) \neq \sin(t) x(t - t_0)</math> |
==[[Linearity_OldKiwi]]== | ==[[Linearity_OldKiwi]]== |
Revision as of 23:07, 17 June 2008
Contents
The six basic properties of Systems_OldKiwi
Memory_OldKiwi
A system with memory has outputs that depend on previous (or future) inputs.
- Example of a system with memory:
$ y(t) = x(t - \pi) $
- Example of a system without memory:
$ y(t) = x(t) $
Invertibility_OldKiwi
An invertible system is one in which there is a one-to-one correlation between inputs and outputs.
- Example of an invertible system:
$ y(t) = x(t) $
- Example of a non-invertible system:
$ y(t) = |x(t)| $
In the second example, both x(t) = -3 and x(t) = 3 yield the same result.
Causality_OldKiwi
A causal system has outputs that only depend on current and/or previous inputs.
- Example of a causal system:
$ y(t) = x(t) + x(t - 1) $
- Example of a non-causal system:
$ y(t) = x(t) + x(t + 1) $
Stability_OldKiwi
There are many types of stability, for this course, we first consider BIBO_OldKiwi (Bounded Input Bounded Output) stability.
A system is BIBO stable if, for all bounded inputs ($ \exist B \epsilon \Re, x(t) < B $), the output is also bounded ($ y(t) < \infty $)
Time Invariance_OldKiwi
A system is time invariant if a shift in the time domain corresponds to the same shift in the output.
- Example of a time invariant system:
$ y(t) = x(t) \mapsto y(t - t_0) = x(t - t_0) $
- Example of a time variant system:
$ y(t) = \sin(t) x(t) \mapsto y(t - t_0) \neq \sin(t) x(t - t_0) $