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