A time-invariant (TIV) system is a system whose output does not depend explicitly on time. Such systems are regarded as a class of systems in the field of system analysis. Lack of time dependence is captured in the following mathematical property of such a system:
If the input signal produces an output then any time shifted input, , results in a time-shifted output
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.
In the context of a system schematic, this property can also be stated as follows:
If a system is time-invariant then the system block commutes with an arbitrary delay.
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.