(→Time-Invariance) |
(→Time-variant System) |
||
| (7 intermediate revisions by the same user not shown) | |||
| Line 9: | Line 9: | ||
==Time-invariant System== | ==Time-invariant System== | ||
| + | |||
| + | An example of a time-invariant system would be the system I used for my linearity problem. Therefore the system is a linear, time-invariant system. | ||
| + | |||
| + | |||
| + | <math>x(t) \rightarrow system \rightarrow y(t) = 2x(t) </math> | ||
| + | |||
| + | |||
| + | Proof: | ||
| + | |||
| + | <math>x(t) \rightarrow system \rightarrow 2x(t) \rightarrow time-delay \rightarrow 2x(t-t_0)</math> | ||
| + | |||
| + | |||
| + | <math>x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2x(t-t_0)</math> | ||
| + | |||
| + | |||
| + | Since the output is the same for both configurations the system is time-invariant. | ||
| + | |||
==Time-variant System== | ==Time-variant System== | ||
| + | |||
| + | An example for a time-variant system would be <math>x(t) \rightarrow system \rightarrow y(t) = 2tx(t)</math> | ||
| + | |||
| + | Proof: | ||
| + | |||
| + | <math>x(t) \rightarrow system \rightarrow 2tx(t) \rightarrow time-delay \rightarrow 2(t-t_0)x(t-t_0)</math> | ||
| + | |||
| + | |||
| + | <math>x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2tx(t-t_0)</math> | ||
| + | |||
| + | |||
| + | Since the output is not similar the system is time-variant. | ||
Latest revision as of 15:20, 11 September 2008
Time Invariance
A time-invariant system is a system in which the output gets time-shifted when the input is time-shifted.
$ x(t - t_0) \rightarrow system \rightarrow y(t - t_0) $
Time-invariant System
An example of a time-invariant system would be the system I used for my linearity problem. Therefore the system is a linear, time-invariant system.
$ x(t) \rightarrow system \rightarrow y(t) = 2x(t) $
Proof:
$ x(t) \rightarrow system \rightarrow 2x(t) \rightarrow time-delay \rightarrow 2x(t-t_0) $
$ x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2x(t-t_0) $
Since the output is the same for both configurations the system is time-invariant.
Time-variant System
An example for a time-variant system would be $ x(t) \rightarrow system \rightarrow y(t) = 2tx(t) $
Proof:
$ x(t) \rightarrow system \rightarrow 2tx(t) \rightarrow time-delay \rightarrow 2(t-t_0)x(t-t_0) $
$ x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2tx(t-t_0) $
Since the output is not similar the system is time-variant.
