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Output signal y(t) can be <math>10e^t</math> by system<br> | Output signal y(t) can be <math>10e^t</math> by system<br> | ||
Prove.<br> | Prove.<br> | ||
| − | |||
| − | |||
| − | + | <math>e^t</math> is changed to <math>e^{(t-t0)}</math> by time delay.<br> | |
| − | + | <math>e^{(t-t0)} -> 10e^{(t-t0)}</math> by system.<br> | |
| + | |||
| + | <math>e^t -> 10e^t </math> by system.<br> | ||
| + | <math>10e^t -> 10e^{(t-t0)}</math> by time delay.<br> | ||
The output signals are same. Then we can say that the system is time-invariant.<br> | The output signals are same. Then we can say that the system is time-invariant.<br> | ||
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| + | == Example of a time-variant system == | ||
| + | |||
| + | Input signal is <math> x(t). </math><br> | ||
| + | Output signal y(t) can be <math>x(2t)</math> by system.<br> | ||
| + | Prove.<br> | ||
| + | |||
| + | <math>x(t)</math> is changed to <math>x{(t-t0)}</math> by time delay.<br> | ||
| + | <math>x{(t-t0)} -> x{(2t-t0)}</math> by system.<br> | ||
| + | |||
| + | <math>x(t)</math> is changed to <math> x(2t) </math> by system.<br> | ||
| + | <math>x(2t) is changed to x{(2(t-t0))}</math> by time delay.<br> | ||
| + | |||
| + | The output signals are same. Then we can say that the system is time-variant.<br> | ||
Latest revision as of 14:29, 9 September 2008
A time-invariant system
For any input signal x(t), a system yelids y(t). Now, suppose input signal shifted t0, x(t-t0). Then output signal also shifted t0, y(t-t0). Then we can say a system is time-invariant.
Example of a tume-invariant system
x(t) = $ e^t $
Output signal y(t) can be $ 10e^t $ by system
Prove.
$ e^t $ is changed to $ e^{(t-t0)} $ by time delay.
$ e^{(t-t0)} -> 10e^{(t-t0)} $ by system.
$ e^t -> 10e^t $ by system.
$ 10e^t -> 10e^{(t-t0)} $ by time delay.
The output signals are same. Then we can say that the system is time-invariant.
Example of a time-variant system
Input signal is $ x(t). $
Output signal y(t) can be $ x(2t) $ by system.
Prove.
$ x(t) $ is changed to $ x{(t-t0)} $ by time delay.
$ x{(t-t0)} -> x{(2t-t0)} $ by system.
$ x(t) $ is changed to $ x(2t) $ by system.
$ x(2t) is changed to x{(2(t-t0))} $ by time delay.
The output signals are same. Then we can say that the system is time-variant.
