Time-invariant system

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

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

${\displaystyle y(t)=f(x(t),t)=f(x(t)).}$

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, 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.

Simple example

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

• System A: ${\displaystyle y(t)=tx(t)}$
• System B: ${\displaystyle y(t)=10x(t)}$

Since the System Function ${\displaystyle y(t)}$ for system A explicitly depends on t outside of ${\displaystyle x(t)}$, 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 ${\displaystyle x(t)}$. 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.