Content deleted Content added
oops |
merge from Continuous signal |
||
Line 1:
{{unreferenced|date=March 2016}}
{{merge from
In [[mathematics]] and in particular [[Dynamical system|mathematical dynamics]], '''discrete time''' and '''continuous time''' are two alternative frameworks within which to model [[Variable (mathematics)|variables]] that evolve over time.
Line 10:
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 be [[continuous function|continuous]]. To contrast, a [[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.
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:
<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>.
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 <math>t^{-1}</math> signal is not integrable at infinity, but <math>t^{-2}</math> is).
Any analog signal is continuous by nature. [[Discrete-time signal]]s, used in [[digital signal processing]], can be obtained by [[Sampling (signal processing)|sampling]] and [[Quantization (signal processing)|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==
| |||