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{{Short description|Frameworks for modeling variables that evolve over time}}
In [[Dynamical system|mathematical dynamics]], '''discrete time''' and '''continuous time''' are two alternative frameworks within which to model [[Variable (mathematics)|variables]] that evolve over time.▼
{{redirect-distinguish|Discrete signal|Discrete variable}}
▲In
==Discrete time==
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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 (signal processing)|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]].
Discrete-time signals may have several origins, but can usually be classified into one of two groups:<ref>"Digital Signal Processing", Prentice Hall -
* By acquiring values of an [[analog signal]] at constant or variable rate. This process is called [[Sampling (signal processing)|sampling]].<ref>"Digital Signal Processing: Instant access
* 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
▲In contrast, '''continuous time''' views variables as having a particular value for potentially only an [[infinitesimal]]ly short amount of time. Between any two points in time there are an [[infinity|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]] (a [[signal (information theory)|signal]])
whose ___domain, which is often time, is a [[Continuum (set theory)|continuum]] (e.g., a [[connected space|connected]] interval of the [[real number|reals]]). That is, the function's ___domain is an [[uncountable set]]. The function itself need not to be [[continuous function|continuous]]. To contrast, a [[discrete time|discrete-time]] signal has a [[countable set|countable]] ___domain, like the [[natural number]]s.
A signal of continuous amplitude and time is known as a continuous-time signal or an [[analog signal]]. This (a [[Signal (electrical engineering)|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.
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A typical example of an infinite duration signal is:
:<math>f(t) = \sin(t), \quad t \in \mathbb{R}</math>
A finite duration counterpart of the above signal could be:
:<math>f(t) = \sin(t), \quad t \in [-\pi,\pi]</math> and <math>f(t) = 0</math> otherwise.
The value of a finite (or infinite) duration signal may or may not be finite. For example,
:<math>f(t) = \frac{1}{t}, \quad t \in [0,1]</math> and <math>f(t) = 0</math> otherwise,
is a finite duration signal but it takes an infinite value for <math>t = 0\,</math>.
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==Relevant contexts==
Discrete time is often employed when [[empirical]] [[measurement]]s 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]] will show a sequence of [[Calendar year#Quarters|quarterly]] values.
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===Discrete time===
Discrete time makes use of [[difference equation]]s, also known as recurrence relations. An example, known as the [[logistic map]] or logistic equation, is
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===Continuous time===
Continuous time makes use of [[differential equation]]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
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==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.
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*[[Discretization]]
*[[Normalized frequency (digital signal processing)|Normalized frequency]]
*[[Nyquist–Shannon sampling theorem]]
*[[Time-scale calculus]]
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