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There are many types of stability, for this course, we first consider [[BIBO_Old Kiwi]] (Bounded Input Bounded Output) stability.
 
There are many types of stability, for this course, we first consider [[BIBO_Old Kiwi]] (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>)
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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_Old Kiwi]]==
 
==[[Time Invariance_Old Kiwi]]==
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*Example of a '''time invariant''' system:
 
*Example of a '''time invariant''' system:
<math>y(t) = x(t) \mapsto y(t - t_0) = x(t - t_0)</math>
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<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>y(t) = \sin t x(t) \mapsto y(t - t_0) \neq \sint t x(t - t_0)</math>
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<math>y_1(t) = \sin(t) x_1(t) \mapsto y_2(t - t_0) = \sin(t) x_2(t - t_0)</math>
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 +
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_Old Kiwi]]==
 
==[[Linearity_Old Kiwi]]==
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 +
A system is linear if the [[superposition_Old Kiwi]] 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:
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<math>y(t) = 10x(t)</math>
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 +
*Example of a '''nonlinear''' system:
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<math>y(t) = x(t)^2</math>
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[[Category:ECE 301 San Summer 2008]]

Latest revision as of 20:52, 18 June 2008

The six basic properties of Systems_Old Kiwi

Memory_Old Kiwi

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_Old Kiwi

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_Old Kiwi

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_Old Kiwi

There are many types of stability, for this course, we first consider BIBO_Old Kiwi (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_Old Kiwi

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_Old Kiwi

A system is linear if the superposition_Old Kiwi 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 $

Alumni Liaison

Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett