# Time-invariant system

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.*

If a time-invariant system is also linear, it is the subject of LTI system 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.

## Simple example

To demonstrate how to determine if a system is time-invariant then consider the two systems:

Since system A explicitly depends on *t* outside of and , it is not time-invariant. System B, however, does not depend explicitly on *t* so it is time-invariant.

## Formal example

A more formal proof of why system A & B from 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. 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

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