A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".
Mathematically speaking, "time-invariance" of a system is the following property::p. 50
- Given a system with a time-dependent output function and a time-dependent input function the system will be considered time-invariant if a time-delay on the input directly equates to a time-delay of the output function. For example, if time is "elapsed time", then "time-invariance" implies that the relationship between the input function and the output function is constant with respect to time :
In the language of signal processing, this property can be satisfied if the transfer function of the system is not a direct function of time except as 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.
If a time-invariant system is also linear, it is the subject of linear time-invariant theory (linear time-invariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear time-invariant systems lack a comprehensive, governing theory. Discrete time-invariant systems are known as shift-invariant systems. Systems which lack the time-invariant property are studied as time-variant systems.
To demonstrate how to determine if a system is time-invariant, consider the two systems:
- System A:
- System B:
Since the System Function for system A explicitly depends on t outside of , it is not time-invariant because the time-dependence is not explicitly a function of the input function.
In contrast, system B's time-dependence is only a function of the time-varying input . This makes system B time-invariant.
The Formal Example below shows in more detail that while System B is a Shift-Invariant System as a function of time, t, System A is not.
A more formal proof of why systems A and B above differ is now presented. To perform 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.
More generally, the relationship between the input and output is
and its variation with time is
For time-invariant systems, the system properties remain constant with time,
Applied to Systems A and B above:
- in general, so it is not time-invariant,
- so it is time-invariant.
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
- ^ Oppenheim, Alan; Willsky, Alan (1997). Signals and Systems (second ed.). Prentice Hall.