Phase correlation

2014-01-22T05:50:29Z

82.14.27.98: /* Rationale */

{{Unreferenced|date=September 2007}}

A '''time-variant system''' is a [[system]] that is not [[time-invariant system|time invariant]] (TIV). Roughly speaking, characteristics its output depend explicitly upon time.

== Overview ==
There are many well developed [[LTI system theory|techniques]] for dealing with the response of linear time invariant systems, such as [[Laplace Transform|Laplace]] and [[Fourier Transform|Fourier]] 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]] 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]] 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]]s) change as the vocal organs such as the [[tongue]] and [[Soft palate|velum]] move. Mathematical models of the vocal tract are therefore time-variant, with transfer functions often [[linear interpolation|linearly interpolated]] between states over time.
* [[Linear]] time varying processes such as [[amplitude modulation]] 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]]s.
* The [[Discrete Wavelet Transform]], often used in modern signal processing, is time variant because it makes use of the [[decimation (signal processing)|decimation]] operation.

== Time-variant system: Elaboration ==
By definition, the input–output characteristics vary with [[time]].
Let:
*''x''(''t'') be an excitation [[Wiktionary:signal|signal]].
*''T''(''x''(''t''), ''t'') describe the input–output map of a [[system]] in relaxed state.
*''y''(''t'') be the system's output response <math>y(t) = T(x(t), t)</math> to the excitation signal.
If the excitation signal is delayed by time ''k'' (i.e., ''x(t-k)'') and the output response <math>T( x(t-k), t )</math> is ''not'' equivalent to a delayed version of the original output <math>y(t-k)=T( x(t-k), t-k )</math>, then the system is time variant.

== See also ==
*[[Control system]]
*[[Control theory]]
*[[System analysis]]
*[[Time-invariant system]]: examples how to prove if a system is time-variant or time-invariant.''

== References ==

[[Category:Control theory]]

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Phase correlation