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In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.


Discrete time

Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a
discrete variable In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can take on two particular real values such that it can also take on all re ...
. Thus a non-time variable jumps from one value to another as time moves from one time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
values of the variable "time". A discrete signal or discrete-time signal is a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
consisting of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of quantities. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated
sampling rate In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or spac ...
. Discrete-time signals may have several origins, but can usually be classified into one of two groups: * By acquiring values of an
analog signal An analog signal or analogue signal (see spelling differences) is any continuous signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the instantaneous signal voltage varies c ...
at constant or variable rate. This process is called sampling."Digital Signal Processing: Instant access", Butterworth-Heinemann - page 8 * By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.


Continuous time

In contrast, continuous time views variables as having a particular value only for an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
ly short amount of time. Between any two points in time there are an
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
number of other points in time. The variable "time" ranges over the entire
real number line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, or depending on the context, over some subset of it such as the non-negative reals. Thus time is viewed as a continuous variable. A continuous signal or a continuous-time signal is a varying
quantity Quantity or amount is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate discontinuity (mathematics), discontinuity and continuum (theory), continuity. Quantities can be compared in terms o ...
(a
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
) whose domain, which is often time, is a
continuum Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number ...
(e.g., a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
interval of the reals). That is, the function's domain is an
uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
. The function itself need not to be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
. To contrast, a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
signal has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
domain, like the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. A signal of continuous amplitude and time is known as a continuous-time signal or an
analog signal An analog signal or analogue signal (see spelling differences) is any continuous signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the instantaneous signal voltage varies c ...
. This (a
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
) will have some value at every instant of time. The electrical signals derived in proportion with the physical quantities such as temperature, pressure, sound etc. are generally continuous signals. Other examples of continuous signals are sine wave, cosine wave, triangular wave etc. The signal is defined over a domain, which may or may not be finite, and there is a functional mapping from the domain to the value of the signal. The continuity of the time variable, in connection with the law of density of
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, means that the signal value can be found at any arbitrary point in time. A typical example of an infinite duration signal is: :f(t) = \sin(t), \quad t \in \mathbb A finite duration counterpart of the above signal could be: :f(t) = \sin(t), \quad t \in \pi,\pi/math> and f(t) = 0 otherwise. The value of a finite (or infinite) duration signal may or may not be finite. For example, :f(t) = \frac, \quad t \in ,1/math> and f(t) = 0 otherwise, is a finite duration signal but it takes an infinite value for t = 0\,. In many disciplines, the convention is that a continuous signal must always have a finite value, which makes more sense in the case of physical signals. For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the t^ signal is not integrable at infinity, but t^ is). Any analog signal is continuous by nature. Discrete-time signals, used in
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
, can be obtained by sampling and quantization of continuous signals. Continuous signal may also be defined over an independent variable other than time. Another very common independent variable is space and is particularly useful in
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
, where two space dimensions are used.


Relevant contexts

Discrete time is often employed when
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
s are involved, because normally it is only possible to measure variables sequentially. For example, while
economic activity Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely. For this reason, published data on, for example,
gross domestic product Gross domestic product (GDP) is a money, monetary Measurement in economics, measure of the market value of all the final goods and services produced and sold (not resold) in a specific time period by countries. Due to its complex and subjec ...
will show a sequence of
quarterly A magazine is a periodical literature, periodical publication, generally published on a regular schedule (often weekly or monthly), containing a variety of content (media), content. They are generally financed by advertising, newsagent's shop, ...
values. When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
or regression methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, ''y''''t'' might refer to the value of
income Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For ...
observed in unspecified time period ''t'', ''y''''3'' to the value of income observed in the third time period, etc. Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often the theory itself is expressed in discrete time in order to facilitate the development of a time series or regression model. On the other hand, it is often more mathematically tractable to construct
theoretical model A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
s in continuous time, and often in areas such as
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
an exact description requires the use of continuous time. In a continuous time context, the value of a variable ''y'' at an unspecified point in time is denoted as ''y''(''t'') or, when the meaning is clear, simply as ''y''.


Types of equations


Discrete time

Discrete time makes use of
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s, also known as recurrence relations. An example, known as the
logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
or logistic equation, is : x_ = rx_t(1-x_t), in which ''r'' is a
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
in the range from 2 to 4 inclusive, and ''x'' is a variable in the range from 0 to 1 inclusive whose value in period ''t'' nonlinearly affects its value in the next period, ''t''+1. For example, if r=4 and x_1 = 1/3, then for ''t''=1 we have x_2=4(1/3)(2/3)=8/9, and for ''t''=2 we have x_3=4(8/9)(1/9)=32/81. Another example models the adjustment of a
price A price is the (usually not negative) quantity of payment or compensation given by one party to another in return for goods or services. In some situations, the price of production has a different name. If the product is a "good" in the c ...
''P'' in response to non-zero
excess demand In economics, a shortage or excess demand is a situation in which the demand for a product or service exceeds its supply in a market. It is the opposite of an excess supply (surplus). Definitions In a perfect market (one that matches a sim ...
for a product as :P_ = P_t + \delta \cdot f(P_t,...) where \delta is the positive speed-of-adjustment parameter which is less than or equal to 1, and where f is the excess demand function.


Continuous time

Continuous time makes use of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. For example, the adjustment of a price ''P'' in response to non-zero excess demand for a product can be modeled in continuous time as :\frac=\lambda \cdot f(P,...) where the left side is the
first derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the price with respect to time (that is, the rate of change of the price), \lambda is the speed-of-adjustment parameter which can be any positive finite number, and f is again the excess demand function.


Graphical depiction

A variable measured in discrete time can be plotted as a
step function In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only ...
, in which each time period is given a region on the
horizontal axis A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of the same length as every other time period, and the measured variable is plotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graph appears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-axis point. In this technique, the graph appears as a set of dots. The values of a variable measured in continuous time are plotted as a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
, since the domain of time is considered to be the entire real axis or at least some connected portion of it.


See also

*
Aliasing In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when a ...
*
Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) 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. Th ...
*
Digital data Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of values from some alphabet, such as letters or digits. An example i ...
*
Discrete calculus Discrete calculus or the calculus of discrete functions, is the mathematical study of ''incremental'' change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word ''ca ...
*
Discrete system In theoretical computer science, 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 ...
*
Discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
* Normalized frequency *
Nyquist–Shannon sampling theorem The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that pe ...
*
Time-scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...


References

* *{{cite book , author1 = Wagner, Thomas Charles Gordon , title = Analytical transients , publisher = Wiley , year = 1959 Time in science Dynamical systems