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#redirect [[Discrete time and continuous time]]
{{about|discrete time in [[signal processing]]|discrete time in [[quantum physics]]|quantum time}}

'''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.

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]].

==System clock==
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.

==Time signals==

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>

'''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.

==See also==

*[[Bernoulli process]]
*[[Digital]]
*[[Discrete signal]]
*[[Discrete system]]
*[[Nyquist frequency]]
*[[System dynamics]]

== Notes ==
{{reflist|2}}

== References ==
*{{cite book
| last = Gershenfeld
| first = Neil A.
| authorlink =
| coauthors =
| title = The Nature of mathematical Modeling
| publisher = Cambridge University Press
| year = 1999
| location =
| pages =
| url =
| doi =
| id =
| isbn = 0-521-57095-6}}

*{{cite book
| last = Wagner
| first = Thomas Charles Gordon
| authorlink =
| coauthors =
| title = Analytical transients
| publisher = Wiley
| year = 1959
| location =
| pages =
| url =
| doi =
| id =
| isbn = }}

{{Time measurement and standards}}

{{DEFAULTSORT:Discrete Time}}
[[Category:Time]]
[[Category:Signal processing]]
[[Category:Time series analysis]]

[[pt:Tempo discreto]]
[[sv:Tidsdiskret]]

Latest revision as of 15:19, 14 August 2013