- Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.:If the input signal produces an output then any time shifted input, , results in a time-shifted output
Formal: If is the shifting operator (),then the operator is called time-invariant, if
:
This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output.This property can also be stated in another way in terms of a schematic
:If a system is time-invariant then the system block is commutative with an arbitrary delay.
Simple example
To demonstrate how to determine if a system is time-invariant then consider the two systems:
* System A:
* System B:Since system A explicitly depends on "t" outside of and it is time-variant. System B, however, does not depend explicitly on "t", so it is time-invariant.
Formal example
A more formal proof of the previous example is now presented.For this proof, the second definition will be used.
System A::Start with a delay of the input :::Now delay the output by :::::Clearly , therefore the system is not time-invariant.
System B::Start with a delay of the input :::Now delay the output by :::::Clearly , therefore the system is time-invariant. Although there are many other proofs, this is the easiest.
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
:
See also
*
Finite impulse response
*LTI system theory
*Sheffer sequence
*State space (controls)
*System analysis
*Time-variant system
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