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There are many types of stability, for this course, we first consider [[BIBO_OldKiwi]] (Bounded Input Bounded Output) stability. | 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 (<math>\exist B \epsilon \Re, x(t) < B</math>), the output is also bounded (<math>y(t) < \infty</math>) | + | A system is BIBO stable if, for all bounded inputs (<math>\exist B \epsilon \Re, |x(t)| < B</math>), the output is also bounded (<math>|y(t)| < \infty</math>) |
==[[Time Invariance_OldKiwi]]== | ==[[Time Invariance_OldKiwi]]== | ||
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*Example of a '''time invariant''' system: | *Example of a '''time invariant''' system: | ||
| − | <math> | + | <math>y_1(t) = x_1(t) \mapsto y_2(t - t_0) = x_2(t - t_0)</math> |
*Example of a '''time variant''' system: | *Example of a '''time variant''' system: | ||
| − | <math> | + | <math>y_1(t) = \sin(t) x_1(t) \mapsto y_2(t - t_0) = \sin(t) x_2(t - t_0)</math> |
| + | |||
| + | In the first example, <math>y_2</math> is the shifted version of <math>y_1</math>. This is not true of the second example. | ||
==[[Linearity_OldKiwi]]== | ==[[Linearity_OldKiwi]]== | ||
| + | |||
| + | A system is linear if the [[superposition_OldKiwi]] property holds, that is, that linear combinations of inputs lead to the same linear combinations of the outputs. | ||
| + | |||
| + | A system with inputs <math>x_1</math> and <math>x_2</math> and corresponding outputs <math>y_1</math> and <math>y_2</math> is linear if: | ||
| + | <math>ax_1 + bx_2 = ay_1 + by_2</math> for any constants ''a'' and ''b''. | ||
| + | |||
| + | *Example of a '''linear''' system: | ||
| + | <math>y(t) = 10x(t)</math> | ||
| + | |||
| + | *Example of a '''nonlinear''' system: | ||
| + | <math>y(t) = x(t)^2</math> | ||
| + | |||
| + | [[Category:ECE 301 San Summer 2008]] | ||
Latest revision as of 20:52, 18 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_1(t) = x_1(t) \mapsto y_2(t - t_0) = x_2(t - t_0) $
- Example of a time variant system:
$ y_1(t) = \sin(t) x_1(t) \mapsto y_2(t - t_0) = \sin(t) x_2(t - t_0) $
In the first example, $ y_2 $ is the shifted version of $ y_1 $. This is not true of the second example.
Linearity_OldKiwi
A system is linear if the superposition_OldKiwi property holds, that is, that linear combinations of inputs lead to the same linear combinations of the outputs.
A system with inputs $ x_1 $ and $ x_2 $ and corresponding outputs $ y_1 $ and $ y_2 $ is linear if: $ ax_1 + bx_2 = ay_1 + by_2 $ for any constants a and b.
- Example of a linear system:
$ y(t) = 10x(t) $
- Example of a nonlinear system:
$ y(t) = x(t)^2 $
