Discrete time: Difference between revisions
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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 decision 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 period in discrete-time systems is constant, but in some cases nonuniform sampling is also used. |
'''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 decision 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 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 |
Discrete-time signals are typically written as a function of a POOPOO ''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== |
==System clock== |
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Revision as of 16:49, 19 July 2011
Discrete time is the discontinuity of a function's time domain that results from sampling a 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 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 decision 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 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 a POOPOO n (for example, x(n) or xn may represent a discretisation of x(t) sampled every T seconds). In contrast to continuous-time systems, where the behaviour of a system is often described by a set of linear differential equations, discrete-time systems are described in terms of difference equations. Most 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.[1] 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 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. Fs/2 - epsilon, in order to prevent aliasing. Likewise, all non-linear operations performed on discrete-time signals must be bandlimited to Fs/2 - epsilon. Wagner's book Analytical Transients proves why equality is not permissible.[2]
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
Notes
References
- Gershenfeld, Neil A. (1999). The Nature of mathematical Modeling. Cambridge University Press. ISBN 0 521 57095 6.
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- Wagner, Thomas Charles Gordon (1959). Analytical transients. Wiley.
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