# Time-invariant system

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". Block diagram illustrating the time invariance for a deterministic continuous-time single-input single-output system. The system is time-invariant if and only if y 2 ( t ) = y 1 ( t − t 0 ) {\displaystyle y_{2}(t)=y_{1}(t-t_{0})} for all time t {\displaystyle t} , for all real constant t 0 {\displaystyle t_{0}} and for all input x 1 ( t ) {\displaystyle x_{1}(t)} . Click image to expand it.

Mathematically speaking, "time-invariance" of a system is the following property::p. 50

Given a system with a time-dependent output function $y(t),$ and a time-dependent input function $x(t);$ the system will be considered time-invariant if a time-delay on the input $x(t+\delta )$ directly equates to a time-delay of the output $y(t+\delta )$ function. For example, if time $t$ is "elapsed time", then "time-invariance" implies that the relationship between the input function $x(t)$ and the output function $y(t)$ is constant with respect to time $t$ :
$y(t)=f(x(t),t)=f(x(t)).$ 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, as shown in the figure to the right:

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.