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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:^{[1]:p. 50}

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.

Simple example

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

• System A: y(t)=tx(t){displaystyle y(t)=t,x(t)}
  


• System B: y(t)=10x(t){displaystyle y(t)=10x(t)}
  

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

y(t)=tx(t){displaystyle y(t)=t,x(t)}

y1(t)=txd(t)=tx(t+δ){displaystyle y_{1}(t)=t,x_{d}(t)=t,x(t+delta )}

Now delay the output by δ{displaystyle delta }

y(t)=tx(t){displaystyle y(t)=t,x(t)}

y2(t)=y(t+δ)=(t+δ)x(t+δ){displaystyle y_{2}(t)=,!y(t+delta )=(t+delta )x(t+delta )}

Clearly y1(t)≠y2(t){displaystyle y_{1}(t),!neq y_{2}(t)}, therefore the system is not time-invariant.

System B:

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

y(t)=10x(t){displaystyle y(t)=10,x(t)}

y1(t)=10xd(t)=10x(t+δ){displaystyle y_{1}(t)=10,x_{d}(t)=10,x(t+delta )}

Now delay the output by δ{displaystyle ,!delta }

y(t)=10x(t){displaystyle y(t)=10,x(t)}

y2(t)=y(t+δ)=10x(t+δ){displaystyle y_{2}(t)=y(t+delta )=10,x(t+delta )}

Clearly y1(t)=y2(t){displaystyle y_{1}(t)=,!y_{2}(t)}, therefore the system is time-invariant.

More generally, the relationship between the input and output is y(t)=f(x(t),t){displaystyle y(t)=f(x(t),t)}, and its variation with time is

dydt=∂f∂t+∂f∂xdxdt{displaystyle {frac {mathrm {d} y}{mathrm {d} t}}={frac {partial f}{partial t}}+{frac {partial f}{partial x}}{frac {mathrm {d} x}{mathrm {d} t}}}.

For time-invariant systems, the system properties remain constant with time, ∂f/∂t=0{displaystyle partial f/partial t=0}. Applied to Systems A and B above:

fA=tx(t)⟹∂fA∂t=x(t)≠0{displaystyle f_{A}=tx(t)qquad implies qquad {frac {partial f_{A}}{partial t}}=x(t)neq 0} in general, so not time-invariant

fB=10x(t)⟹∂fB∂t=0{displaystyle f_{B}=10x(t)qquad implies qquad {frac {partial f_{B}}{partial t}}=0} so time-invariant.

Abstract example

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

x(t+1)=δ(t+1)∗x(t){displaystyle x(t+1)=,!delta (t+1)*x(t)}

can be represented in this abstract notation by

x~1=T1x~{displaystyle {tilde {x}}_{1}=mathbb {T} _{1},{tilde {x}}}

where x~{displaystyle {tilde {x}}} is a function given by

x~=x(t)∀t∈R{displaystyle {tilde {x}}=x(t),forall ,tin mathbb {R} }

with the system yielding the shifted output

x~1=x(t+1)∀t∈R{displaystyle {tilde {x}}_{1}=x(t+1),forall ,tin mathbb {R} }

So T1{displaystyle mathbb {T} _{1}} is an operator that advances the input vector by 1.

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

TrH=HTr∀r{displaystyle mathbb {T} _{r},mathbb {H} =mathbb {H} ,mathbb {T} _{r},,forall ,r}

If our system equation is given by

y~=Hx~{displaystyle {tilde {y}}=mathbb {H} ,{tilde {x}}}

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

Applying the system operator first gives

TrHx~=Try~=y~r{displaystyle mathbb {T} _{r},mathbb {H} ,{tilde {x}}=mathbb {T} _{r},{tilde {y}}={tilde {y}}_{r}}

Applying the shift operator first gives

HTrx~=Hx~r{displaystyle mathbb {H} ,mathbb {T} _{r},{tilde {x}}=mathbb {H} ,{tilde {x}}_{r}}

If the system is time-invariant, then

Hx~r=y~r{displaystyle mathbb {H} ,{tilde {x}}_{r}={tilde {y}}_{r}}

See also

• Finite impulse response


• Sheffer sequence


• State space (controls)


• Signal-flow graph


• LTI system theory

References