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<math>x(t-T) = 2e^{t-5} </math> -> [SYSTEM] -> <math>35*2*e^{t-5}</math> | <math>x(t-T) = 2e^{t-5} </math> -> [SYSTEM] -> <math>35*2*e^{t-5}</math> | ||
| − | <math>x(t) = 2e^{t} -> [SYSTEM] -> <math>y(t-T)=35*2*e^{t-5}</math> | + | <math>x(t) = 2e^{t}</math> -> [SYSTEM] -> <math>y(t-T)=35*2*e^{t-5}</math> |
As you can see these two outputs are the same, so the system is time invariant. | As you can see these two outputs are the same, so the system is time invariant. | ||
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<math>x(t-T) = 2e^{t-5} </math> -> [SYSTEM] -> <math>2*e^{35t-5}</math> | <math>x(t-T) = 2e^{t-5} </math> -> [SYSTEM] -> <math>2*e^{35t-5}</math> | ||
| − | <math>x(t) = 2e^{t} -> [SYSTEM] -> <math>y(t-T)=2*e^{35(t-5)} = 2*e^{35t - 175}</math> | + | <math>x(t) = 2e^{t}</math> -> [SYSTEM] -> <math>y(t-T)=2*e^{35(t-5)} = 2*e^{35t - 175}</math> |
Clearly these two results are not the same, so the system is time variant. | Clearly these two results are not the same, so the system is time variant. | ||
Latest revision as of 16:32, 11 September 2008
Time Invariance
A system is time invariant if for a certain x(t) that produces an output y(t) if you shift the input to x(t-T) it just yields the same output shifted by the same T. y(t-T).
Time Invariant System
I propose that a system where
$ x(t) $ -> [SYSTEM] -> $ y(t) = 35x(t) $ is time invariant. Let's check.
Let $ x(t)=2e^t $ and $ T=5 $
$ x(t-T) = 2e^{t-5} $ -> [SYSTEM] -> $ 35*2*e^{t-5} $
$ x(t) = 2e^{t} $ -> [SYSTEM] -> $ y(t-T)=35*2*e^{t-5} $
As you can see these two outputs are the same, so the system is time invariant.
Time Variant System
I propose that a system where
$ x(t) $ -> [SYSTEM] -> $ x(35t) $
Is time variant. Let's use the same T and x(t) for this example.
$ x(t-T) = 2e^{t-5} $ -> [SYSTEM] -> $ 2*e^{35t-5} $
$ x(t) = 2e^{t} $ -> [SYSTEM] -> $ y(t-T)=2*e^{35(t-5)} = 2*e^{35t - 175} $
Clearly these two results are not the same, so the system is time variant.
