Discrete time: Difference between revisions
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#redirect [[Discrete time and continuous time]] |
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{{about|discrete time in [[signal processing]]|discrete time in [[quantum physics]]|quantum time}} |
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'''Discrete time''' is the [[Classification of discontinuities|discontinuity]] of a [[Function (mathematics)|function]]'s [[time domain]] that results from [[Sampling (signal processing)|sampling]] a [[Variable (mathematics)|variable]] at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a [[frequency]] of once per 24 hours, and each number that's published is called a sample. The price is not [[Well-definition|defined]] by the newspaper in between the times that the numbers were published. Suppose it is necessary to know the price of the oil at 12:00PM on one particular day in the past; one must base the estimate on any number of samples that were obtained on the days before and after the event. Such a process is known as [[interpolation]]. In general, the sampling [[Frequency|period]] in discrete-time systems is constant, but in some cases nonuniform sampling is also used. |
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Discrete-time signals are typically written as a function of an index ''n'' (for example, ''x''(''n'') or ''x''<sub>''n''</sub> may represent a discretisation of ''x''(''t'') sampled every ''T'' seconds). In contrast to [[continuous time|continuous-time]] systems, where the behaviour of a system is often described by a set of linear [[differential equation]]s, discrete-time systems are described in terms of [[difference equation]]s. Most [[Monte Carlo Method|Monte Carlo]] simulations utilize a discrete-timing method, either because the system cannot be efficiently represented by a set of equations, or because no such set of equations exists. Transform-domain analysis of discrete-time systems often makes use of the [[Z transform]]. |
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==System clock== |
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One of the fundamental concepts behind discrete time is an implied (actual or hypothetical) system clock.<ref name="Gershenfeld-1999">"... digital systems [...] usually are discretized in time (there is a system clock)", Gershenfeld 1999, p.18</ref> If one wishes, one might imagine some [[atomic clock]] to be the de facto system clock. |
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==Time signals== |
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Uniformly sampled discrete-time signals can be expressed as the [[time domain|time-domain]] multiplication between a [[pulse train]] and a continuous time signal. This time-domain multiplication is equivalent to a [[convolution]] in the [[frequency domain]]. Practically, this means that a signal must be [[bandlimited]] to less than half the sampling frequency, i.e. ''F<sub>s</sub>/2 - epsilon'', in order to prevent [[aliasing]]. Likewise, all non-linear operations performed on discrete-time signals must be bandlimited to ''F<sub>s</sub>/2 - epsilon''. Wagner's book Analytical Transients proves why equality is not permissible.<ref name="Wagner-1959">Wagner 1959</ref> |
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'''Usage:''' when the phrase "discrete time" is used as a noun it should not be hyphenated; when it is a compound adjective, as when one writes of a "discrete-time [[stochastic process]]", then, at least according to traditional punctuation rules, it should be hyphenated. See [[hyphen]] for more. |
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==See also== |
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*[[Bernoulli process]] |
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*[[Digital]] |
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*[[Discrete signal]] |
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*[[Discrete system]] |
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*[[Nyquist frequency]] |
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*[[System dynamics]] |
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== Notes == |
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{{reflist|2}} |
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== References == |
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*{{cite book |
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| last = Gershenfeld |
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| first = Neil A. |
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| authorlink = |
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| title = The Nature of mathematical Modeling |
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| publisher = Cambridge University Press |
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| year = 1999 |
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| isbn = 0-521-57095-6}} |
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*{{cite book |
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| last = Wagner |
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| first = Thomas Charles Gordon |
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| authorlink = |
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| coauthors = |
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| title = Analytical transients |
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| publisher = Wiley |
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| year = 1959 |
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| pages = |
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| doi = |
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{{Time measurement and standards}} |
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{{DEFAULTSORT:Discrete Time}} |
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[[Category:Time]] |
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[[Category:Signal processing]] |
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[[Category:Time series analysis]] |
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[[pt:Tempo discreto]] |
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[[sv:Tidsdiskret]] |
Latest revision as of 15:19, 14 August 2013
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