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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. 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 languag ...
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. Ex ...
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 called ...
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 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 signals may have several origins, but can usually be classified into one of two groups: * By acquiring values of an analog signal 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 infinitesimally short amount of time. Between any two points in time there are an infinite number of other points in time. The variable "time" ranges over the entire real number line, 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 multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
(a signal) whose domain, which is often time, is a continuum (e.g., a connected 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 nu ...
. 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 g ...
. To contrast, a discrete-time signal has a countable 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 ...
s. A signal of continuous amplitude and time is known as a continuous-time signal or an analog signal. This (a signal) 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, 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, where two space dimensions are used.


Relevant contexts

Discrete time is often employed when empirical measurements are involved, because normally it is only possible to measure variables sequentially. For example, while economic activity 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 monetary 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 subjective nature this measure is of ...
will show a sequence of quarterly 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. Ex ...
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 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 ...
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 equations, also known as recurrence relations. An example, known as the logistic map 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 In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
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 ...
''P'' in response to non-zero excess demand 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, ...
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 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 which each time period is given a region on the horizontal axis 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 val ...
, since the domain of time is considered to be the entire real axis or at least some connected portion of it.


See also

* Aliasing * Bernoulli process *
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 exampl ...
* Discrete calculus *
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 ...
* Discretization * Normalized frequency * Nyquist–Shannon sampling theorem *
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