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Discrete time is non-continuous time. Sampling at non-continuous times results in discrete-time samples. For example, a newspaper may report the price of crude oil once every 24 hours. 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 an index 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.
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System clock
One of the fundamental concepts behind discrete time is an implied (actual or hypothetical) system clock1. 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 half the sampling frequency, Fs/2, in order to prevent aliasing. Likewise, all non-linear operations performed on discrete-time signals must be bandlimited to Fs/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.
Notes
- ^ "... digital systems [...] usually are discretized in time (there is a system clock)", Gershenfeld 1999, p.18
References
- Gershenfeld, Neil A. (1999). The Nature of mathematical Modeling. Cambridge University Press. ISBN 0 521 57095 6.
See also
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