kb/data/en.wikipedia.org/wiki/Discrete_time_and_continuous_time-0.md

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Discrete time and continuous time	1/2	https://en.wikipedia.org/wiki/Discrete_time_and_continuous_time	reference	science, encyclopedia	2026-05-05T06:33:55.043011+00:00	kb-cron

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 values of the variable "time". A discrete signal or discrete-time signal is a time series consisting of a sequence 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. 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. 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 (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. The function itself need not to be continuous. To contrast, a discrete-time signal has a countable domain, like the natural numbers. 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
    )
    ,
    
    t
    ∈
    
      R
    
  

{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }

A finite duration counterpart of the above signal could be:

    f
    (
    t
    )
    =
    sin
    ⁡
    (
    t
    )
    ,
    
    t
    ∈
    [
    −
    π
    ,
    π
    ]
  

{\displaystyle f(t)=\sin(t),\quad t\in [-\pi ,\pi ]}

and

    f
    (
    t
    )
    =
    0
  

{\displaystyle f(t)=0}

otherwise. The value of a finite (or infinite) duration signal may or may not be finite. For example,

    f
    (
    t
    )
    =
    
      
        1
        t
      
    
    ,
    
    t
    ∈
    [
    0
    ,
    1
    ]
  

{\displaystyle f(t)={\frac {1}{t}},\quad t\in [0,1]}

and

    f
    (
    t
    )
    =
    0
  

{\displaystyle f(t)=0}

otherwise, is a finite duration signal but it takes an infinite value for

    t
    =
    0
    
  

{\displaystyle 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
      
        −
        1
      
    
  

{\displaystyle t^{-1}}

signal is not integrable at infinity, but

      t
      
        −
        2
      
    
  

{\displaystyle t^{-2}}

is). Any analog signal is continuous by nature. Discrete-time signals, used in digital signal processing, 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.