# Time-invariant system

Last updated

In control theory, a time-invariant (TI) 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".

## Contents

Mathematically speaking, "time-invariance" of a system is the following property: [4] :p. 50

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.

## 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 ${\displaystyle x_{d}(t)=x(t+\delta )}$
${\displaystyle y(t)=tx(t)}$
${\displaystyle y_{1}(t)=tx_{d}(t)=tx(t+\delta )}$
Now delay the output by ${\displaystyle \delta }$
${\displaystyle y(t)=tx(t)}$
${\displaystyle y_{2}(t)=y(t+\delta )=(t+\delta )x(t+\delta )}$
Clearly ${\displaystyle y_{1}(t)\neq y_{2}(t)}$, therefore the system is not time-invariant.
System B: Start with a delay of the input ${\displaystyle x_{d}(t)=x(t+\delta )}$
${\displaystyle y(t)=10x(t)}$
${\displaystyle y_{1}(t)=10x_{d}(t)=10x(t+\delta )}$
Now delay the output by ${\displaystyle \delta }$
${\displaystyle y(t)=10x(t)}$
${\displaystyle y_{2}(t)=y(t+\delta )=10x(t+\delta )}$
Clearly ${\displaystyle y_{1}(t)=y_{2}(t)}$, therefore the system is time-invariant.

More generally, the relationship between the input and output is

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

and its variation with time is

${\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}.}$

For time-invariant systems, the system properties remain constant with time,

${\displaystyle {\frac {\partial f}{\partial t}}=0.}$

Applied to Systems A and B above:

${\displaystyle f_{A}=tx(t)\qquad \implies \qquad {\frac {\partial f_{A}}{\partial t}}=x(t)\neq 0}$ in general, so it is not time-invariant,
${\displaystyle f_{B}=10x(t)\qquad \implies \qquad {\frac {\partial f_{B}}{\partial t}}=0}$ so it is time-invariant.

## Abstract example

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

${\displaystyle x(t+1)=\delta (t+1)*x(t)}$

can be represented in this abstract notation by

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

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

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

with the system yielding the shifted output

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

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

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

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

If our system equation is given by

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

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

Applying the system operator first gives

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

Applying the shift operator first gives

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

If the system is time-invariant, then

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

## Related Research Articles

In mathematics, the Dirac delta distribution, also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's equation, that is,

A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency depends on the filter design. A high-pass filter is usually modeled as a linear time-invariant system. It is sometimes called a low-cut filter or bass-cut filter in the context of audio engineering. High-pass filters have many uses, such as blocking DC from circuitry sensitive to non-zero average voltages or radio frequency devices. They can also be used in conjunction with a low-pass filter to produce a band-pass filter.

In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).

In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.

In special relativity, electromagnetism and wave theory, the d'Alembert operator, also called the d'Alembertian, wave operator, box operator or sometimes quabla operator is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.

In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.

In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions.

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. For example,

In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.

In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928.

In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (xh)(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication, as is frequently employed by the symbol in computer languages). What's more, there are systematic methods for solving any such system (determining h(t)), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set.

In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).

In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities. Strictly speaking, "invariant" would mean that the estimates themselves are unchanged when both the measurements and the parameters are transformed in a compatible way, but the meaning has been extended to allow the estimates to change in appropriate ways with such transformations. The term equivariant estimator is used in formal mathematical contexts that include a precise description of the relation of the way the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the use of "equivariance" in more general mathematics.

In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.

A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.

## References

1. Bessai, Horst J. (2005). MIMO Signals and Systems. Springer. p. 28. ISBN   0-387-23488-8.
2. Sundararajan, D. (2008). A Practical Approach to Signals and Systems. Wiley. p. 81. ISBN   978-0-470-82353-8.
3. Roberts, Michael J. (2018). Signals and Systems: Analysis Using Transform Methods and MATLAB® (3 ed.). McGraw-Hill. p. 132. ISBN   978-0-07-802812-0.
4. Oppenheim, Alan; Willsky, Alan (1997). Signals and Systems (second ed.). Prentice Hall.