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Input: <math>X_k[n] = d[n-k] \,</math> | Input: <math>X_k[n] = d[n-k] \,</math> | ||
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| + | Prob: Which variable represent time ? | ||
| + | |||
| + | ===1st Assumption: n represents time=== | ||
| + | |||
| + | <math> d[n-k] --> [system] --> (k+1)^2 d[n - (k+1)] --> [timedelay -1] --> (k+1)^2 d[(n-1) -( k+1)] ,\</math> | ||
| + | |||
| + | yields the same result as: | ||
| + | |||
| + | <math> d[n-k] --> [timedelay -1] --> d[(n-1) - k] --> [system] --> (k+1)^2 d[(n-1) - (k+1)] ,\</math> | ||
| + | |||
| + | |||
| + | ===2st Assumption: k represents time=== | ||
| + | |||
| + | <math> d[n-k] --> [system] --> (k+1)^2 d[n - (k+1)] --> [timedelay -1] --> (k+1)^2 d[n-k] ,\</math> | ||
| + | |||
| + | yields not the same result as: | ||
| + | |||
| + | <math> d[n-k] --> [timedelay -1] --> d[n - (k-1)] --> [system] --> k^2 d[n-k] ,\</math> | ||
| + | |||
| + | |||
| + | |||
| + | Remember: Time delay only occurs on function, not variable on equations. | ||
| + | |||
| + | Concluded, it is time invariant if we say n represents time, | ||
| + | and it is time variant if we say k represents time. | ||
Revision as of 18:44, 10 September 2008
Time Invariance? or Time Variance?
System: $ Y_k[n] = (k+1)^2 d[n - (k+1)] \, $
Input: $ X_k[n] = d[n-k] \, $
Prob: Which variable represent time ?
1st Assumption: n represents time
$ d[n-k] --> [system] --> (k+1)^2 d[n - (k+1)] --> [timedelay -1] --> (k+1)^2 d[(n-1) -( k+1)] ,\ $
yields the same result as:
$ d[n-k] --> [timedelay -1] --> d[(n-1) - k] --> [system] --> (k+1)^2 d[(n-1) - (k+1)] ,\ $
2st Assumption: k represents time
$ d[n-k] --> [system] --> (k+1)^2 d[n - (k+1)] --> [timedelay -1] --> (k+1)^2 d[n-k] ,\ $
yields not the same result as:
$ d[n-k] --> [timedelay -1] --> d[n - (k-1)] --> [system] --> k^2 d[n-k] ,\ $
Remember: Time delay only occurs on function, not variable on equations.
Concluded, it is time invariant if we say n represents time,
and it is time variant if we say k represents time.
