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Time invariant system

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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:

Contents

If the input signal x ( t ) produces an output y ( t ) then any time shifted input, x ( t + δ ) , results in a time-shifted output y ( t + δ )

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 ) = 10 x ( 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 + δ ) y ( t ) = t x ( t ) y 1 ( t ) = t x d ( t ) = t x ( t + δ ) Now delay the output by δ y ( t ) = t x ( t ) y 2 ( t ) = y ( t + δ ) = ( t + δ ) x ( t + δ ) Clearly y 1 ( t ) ≠ y 2 ( t ) , therefore the system is not time-invariant.

    System B:

    Start with a delay of the input x d ( t ) = x ( t + δ ) y ( t ) = 10 x ( t ) y 1 ( t ) = 10 x d ( t ) = 10 x ( t + δ ) Now delay the output by δ y ( t ) = 10 x ( t ) y 2 ( t ) = y ( t + δ ) = 10 x ( t + δ ) Clearly 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 ( t , x ( t ) ) , and its variation with time is

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

    Abstract example

    We can denote the shift operator by 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 ) = δ ( t + 1 ) ∗ x ( t )

    can be represented in this abstract notation by

    x ~ 1 = T 1 x ~

    where x ~ is a function given by

    x ~ = x ( t ) ∀ t ∈ R

    with the system yielding the shifted output

    x ~ 1 = x ( t + 1 ) ∀ t ∈ R

    So T 1 is an operator that advances the input vector by 1.

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

    T r H = H T r ∀ r

    If our system equation is given by

    y ~ = H x ~

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

    Applying the system operator first gives

    T r H x ~ = T r y ~ = y ~ r

    Applying the shift operator first gives

    H T r x ~ = H x ~ r

    If the system is time-invariant, then

    H x ~ r = y ~ r

    References

    Time-invariant system Wikipedia
    (Text) CC BY-SA