2620:0:1000:7200:D0FA:6FB:708C:5F8B at 23:52, 22 September 2014

2014-09-22T23:52:46Z

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2014-01-22T05:50:29Z

Rationale

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			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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2012-07-10T17:58:44Z

Method: rearranged a little

New page

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Phase correlation - Revision history

2620:0:1000:7200:D0FA:6FB:708C:5F8B at 23:52, 22 September 2014

82.14.27.98: /* Rationale */

en>Tarchon: /* Method */ rearranged a little