Discrete time

This article is about discrete time in signal processing. For discrete time in quantum physics, see quantum time.

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 an index n (for example, x(n) or x_n 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. F_s/2 - epsilon, in order to prevent aliasing. Likewise, all non-linear operations performed on discrete-time signals must be bandlimited to F_s/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

• Bernoulli process
• Digital
• Discrete signal
• Discrete system
• Nyquist frequency
• System dynamics

Notes

1. ^ "... digital systems [...] usually are discretized in time (there is a system clock)", Gershenfeld 1999, p.18
2. ^ Wagner 1959

References

• Gershenfeld, Neil A. (1999). The Nature of mathematical Modeling. Cambridge University Press. ISBN 0 521 57095 6. 

• Wagner, Thomas Charles Gordon (1959). Analytical transients. Wiley.