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 $x(t)$ produces an output $y(t)$ then any time shifted input, $x(t+\delta )$, results in a time-shifted output $y(t+\delta )$

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, consider the two systems:

• System A: $y(t)=t\,x(t)$
• System B: $y(t)=10x(t)$

Since system A explicitly depends on t outside of $x(t)$ and $y(t)$, 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 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 $x_{d}(t)=\,\!x(t+\delta )$

$y(t)=t\,x(t)$

$y_{1}(t)=t\,x_{d}(t)=t\,x(t+\delta )$

Now delay the output by $\delta$

$y(t)=t\,x(t)$

$y_{2}(t)=\,\!y(t+\delta )=(t+\delta )x(t+\delta )$

Clearly $y_{1}(t)\,\!\neq y_{2}(t)$, therefore the system is not time-invariant.

System B:

Start with a delay of the input $x_{d}(t)=\,\!x(t+\delta )$

$y(t)=10\,x(t)$

$y_{1}(t)=10\,x_{d}(t)=10\,x(t+\delta )$

Now delay the output by $\,\!\delta$

$y(t)=10\,x(t)$

$y_{2}(t)=y(t+\delta )=10\,x(t+\delta )$

Clearly $y_{1}(t)=\,\!y_{2}(t)$, therefore the system is time-invariant.

Abstract example

We can denote the shift operator by $\mathbb {T} _{r}$ where $r$ is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system

$x(t+1)=\,\!\delta (t+1)*x(t)$

can be represented in this abstract notation by

${\tilde {x}}_{1}=\mathbb {T} _{1}\,{\tilde {x}}$

where ${\tilde {x}}$ is a function given by

${\tilde {x}}=x(t)\,\forall \,t\in \mathbb {R}$

with the system yielding the shifted output

${\tilde {x}}_{1}=x(t+1)\,\forall \,t\in \mathbb {R}$

So $\mathbb {T} _{1}$ is an operator that advances the input vector by 1.

Suppose we represent a system by an operator $\mathbb {H}$. This system is time-invariant if it commutes with the shift operator, i.e.,

$\mathbb {T} _{r}\,\mathbb {H} =\mathbb {H} \,\mathbb {T} _{r}\,\,\forall \,r$

If our system equation is given by

${\tilde {y}}=\mathbb {H} \,{\tilde {x}}$

then it is time-invariant if we can apply the system operator $\mathbb {H}$ on ${\tilde {x}}$ followed by the shift operator $\mathbb {T} _{r}$, or we can apply the shift operator $\mathbb {T} _{r}$ followed by the system operator $\mathbb {H}$, with the two computations yielding equivalent results.

Applying the system operator first gives

$\mathbb {T} _{r}\,\mathbb {H} \,{\tilde {x}}=\mathbb {T} _{r}\,{\tilde {y}}={\tilde {y}}_{r}$

Applying the shift operator first gives

$\mathbb {H} \,\mathbb {T} _{r}\,{\tilde {x}}=\mathbb {H} \,{\tilde {x}}_{r}$

If the system is time-invariant, then

$\mathbb {H} \,{\tilde {x}}_{r}={\tilde {y}}_{r}$

See also

• Finite impulse response
• Sheffer sequence
• State space (controls)
• Signal-flow graph
• LTI system theory