 x Discrete time Encyclopedia
Discrete time is the discontinuity
Classification of discontinuities
Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there...

of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

's time domain
Time domain
Time domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...

that results from sampling
Sampling (signal processing)
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....

a variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the 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
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

. 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 equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, discrete-time systems are described in terms of difference equations. Most Monte Carlo
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

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
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...

to be the de facto system clock.

## Time signals

Uniformly sampled discrete-time signals can be expressed as the time-domain
Time domain
Time domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...

multiplication between a pulse train and a continuous time signal. This time-domain multiplication is equivalent to a convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

in the frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

. Practically, this means that a signal must be bandlimited
Bandlimited
Bandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...

to less than half the sampling frequency, i.e. Fs/2 - epsilon, in order to prevent aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

. 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.

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
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

", then, at least according to traditional punctuation rules, it should be hyphenated. See hyphen
Hyphen
The hyphen is a punctuation mark used to join words and to separate syllables of a single word. The use of hyphens is called hyphenation. The hyphen should not be confused with dashes , which are longer and have different uses, or with the minus sign which is also longer...

for more.

• Bernoulli process
Bernoulli process
In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identical and independent...

• Digital
Digital
A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...

• Discrete signal
Discrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...

• Discrete system
Discrete system
A discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems, which may also be called analog systems. A final discrete system is often modeled with a directed graph and is analyzed for correctness and complexity according to...

• Nyquist frequency
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...

• System dynamics
System dynamics
System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use... 