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 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 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 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 signal 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. 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 - ε, in order to prevent aliasing. Likewise, all non-linear operations performed on discrete-time signals must be bandlimited to F_s/2 - ε. Wagner's book Analytical Transients proves why equality is not permissible.

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

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.