(→What is Time Invariance?) |
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==What is Time Invariance?== | ==What is Time Invariance?== | ||
| − | Time Invariance describes | + | Time Invariance describes the property of a system such that the input of signal shifted k units in time equals it's respective output shifted k units in time. |
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| + | ==An example of time invariance== | ||
| + | |||
| + | Suppose a system is modeled mathematically as: | ||
| + | <math>\ f(x)= 5x + 6 \ln(x) + 9</math> | ||
| + | |||
| + | Let <math>\ x(t)=Ct </math> be the input to this system. Then the output is <math> \ y(t) = f(x(t)) </math> | ||
| + | Therefore | ||
| + | |||
| + | <math>\ x(t) \longrightarrow f(t) \longrightarrow y(t)= 5Ct + 6 \ln(Ct) + 9 \,</math> | ||
| + | |||
| + | Now if we introduce a time delay into the input signal: | ||
| + | |||
| + | <math>x(t) \longrightarrow x(t-t_0) \longrightarrow f(x) \longrightarrow 5C(t-t_0)+6\ln(C(t-t_0)) + 9 =Z(t)</math> | ||
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| + | Now instead of introducing the time delay at the input, we move it to the output: | ||
| + | |||
| + | |||
| + | <math>x(t) \longrightarrow f(x) \longrightarrow 5C(t)+6\ln(C t) + 9 = y(t) \rightarrow y(t-t_0) = 5C(t-t_0) + 6\ln(C(t-t_0)) + 9</math> | ||
| + | |||
| + | Since<math>\ Z(t)= y(t) </math> system is time invariant | ||
Latest revision as of 15:36, 11 September 2008
What is Time Invariance?
Time Invariance describes the property of a system such that the input of signal shifted k units in time equals it's respective output shifted k units in time.
An example of time invariance
Suppose a system is modeled mathematically as: $ \ f(x)= 5x + 6 \ln(x) + 9 $
Let $ \ x(t)=Ct $ be the input to this system. Then the output is $ \ y(t) = f(x(t)) $ Therefore
$ \ x(t) \longrightarrow f(t) \longrightarrow y(t)= 5Ct + 6 \ln(Ct) + 9 \, $
Now if we introduce a time delay into the input signal:
$ x(t) \longrightarrow x(t-t_0) \longrightarrow f(x) \longrightarrow 5C(t-t_0)+6\ln(C(t-t_0)) + 9 =Z(t) $
Now instead of introducing the time delay at the input, we move it to the output:
$ x(t) \longrightarrow f(x) \longrightarrow 5C(t)+6\ln(C t) + 9 = y(t) \rightarrow y(t-t_0) = 5C(t-t_0) + 6\ln(C(t-t_0)) + 9 $
Since$ \ Z(t)= y(t) $ system is time invariant
