Time-variant system

# Time-variant system

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A time-variant system is a system
System
System is a set of interacting or interdependent components forming an integrated whole....

that is not time invariant
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.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....

(TIV). Roughly speaking, characteristics of its output depend explicitly upon time.

## Overview

There are many well developed techniques
LTI system theory
Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...

for dealing with the response of linear time invariant systems, such as Laplace and Fourier
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

transforms. However, these techniques are not strictly valid for time-varying systems. A system undergoing slow time variation in comparison to its time constants can usually be considered to be time invariant: they are close to time invariant on a small scale. An example of this is the aging and wear of electronic components, which happens on a scale of years, and thus does not result in any behaviour qualitatively different from that observed in a time invariant system: day-to-day, they are effectively time invariant, though year to year, the parameters may change. Other linear time variant systems may behave more like nonlinear systems, if the system changes quickly – significantly differently between measurements.

The following things can be said about a time-variant system:
• It has explicit dependence on time.
• It does not have an impulse response
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

in the normal sense. The system can be characterized by an impulse response except the impulse response must be known at each and every time instant.
• It is not stationary

## Examples of time-variant systems

The following time varying systems cannot be modelled by assuming that they are time invariant:
• Aircraft – Time variant characteristics are caused by different configuration of control surfaces during take off, cruise and landing as well as constantly decreasing weight due to consumption of fuel.
• The Earth's thermodynamic response to incoming solar radiation varies with time due to changes in the Earth's albedo
Albedo
Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...

and the presence of greenhouse gasses in the atmosphere.
• The human vocal tract is a time variant system, with its transfer function at any given time dependent on the shape of the vocal organs. As with any fluid-filled tube, resonances (called formant
Formant
Formants are defined by Gunnar Fant as 'the spectral peaks of the sound spectrum |P|' of the voice. In speech science and phonetics, formant is also used to mean an acoustic resonance of the human vocal tract...

s) change as the vocal organs such as the tongue
Tongue
The tongue is a muscular hydrostat on the floors of the mouths of most vertebrates which manipulates food for mastication. It is the primary organ of taste , as much of the upper surface of the tongue is covered in papillae and taste buds. It is sensitive and kept moist by saliva, and is richly...

and velum
Velum
Velum may refer to:* Superior medullary velum, part of the nervous system that stretches between parts of the brain* Veil , the veil-like membrane of immature mushrooms extending from the margin of the cap to the stem and torn by growth...

move. Mathematical models of the vocal tract are therefore time-variant, with transfer functions often linearly interpolated
Linear interpolation
Linear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation, which can also be used as a verb .-Linear interpolation between two known points:...

between states over time.
• Linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

time varying processes such as amplitude modulation
Amplitude modulation
Amplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent...

occur on a time scale similar to or faster than that of the input signal. In practice amplitude modulation is often implemented using time invariant nonlinear elements such as diode
Diode
In electronics, a diode is a type of two-terminal electronic component with a nonlinear current–voltage characteristic. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material connected to two electrical terminals...

s.
• The Discrete Wavelet Transform
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...

, often used in modern signal processing, is time variant because it makes use of the decimation
Decimation (signal processing)
In digital signal processing, decimation is a technique for reducing the number of samples in a discrete-time signal. The element which implements this technique is referred to as a decimator.Decimation is a two-step process:...

operation.

## Time-variant system: Elaboration

By definition, the input–output characteristics vary with time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

.
Let:
• x(t) be an excitation signal.
• T(x(t), t) be the describe the input–output map of a system
System
System is a set of interacting or interdependent components forming an integrated whole....

in relaxed state.
• y(t) be the system's output response to the excitation signal.

If the excitation signal is delayed by time k (i.e., x(t-k)) and the output response is not equivalent to a delayed version of the original output , then the system is time variant.

• Control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...

• Control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

• System analysis
System analysis
System analysis in the field of electrical engineering characterizes electrical systems and their properties. System Analysis can be used to represent almost anything from population growth to audio speakers, electrical engineers often use it because of its direct relevance to many areas of their...

• Time-invariant system
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.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....

: examples how to prove if a system is time-variant or time-invariant.