Abstract Example
We can denote the shift operator by where is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system
can be represented in this abstract notation by
where is a function given by
with the system yielding the shifted output
So is an operator that advances the input vector by 1.
Suppose we represent a system by an operator . This system is time-invariant if it commutes with the shift operator, i.e.,
If our system equation is given by
then it is time-invariant if we can apply the system operator on followed by the shift operator, or we can apply the shift operator followed by the system operator, with the two computations yielding equivalent results.
Applying the system operator first gives
Applying the shift operator first gives
If the system is time-invariant, then
Read more about this topic: Time-invariant System
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