In mathematical dynamics,discrete time andcontinuous time are two alternative frameworks within whichvariables that evolve over time are modeled.

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 adiscrete 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 sequentialinteger values of the variable "time".

Adiscrete signal ordiscrete-time signal is atime series consisting of asequence 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 bysampling from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associatedsampling rate.

Discrete-time signals may have several origins, but can usually be classified into one of two groups:^[1]

• By acquiring values of ananalog signal at constant or variable rate. This process is calledsampling.^[2]
• By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.

In contrast,continuous time views variables as having a particular value only for aninfinitesimally short amount of time. Between any two points in time there are aninfinite number of other points in time. The variable "time" ranges over the entirereal number line, or depending on the context, over some subset of it such as the non-negative reals. Thus time is viewed as acontinuous variable.

Acontinuous signal or acontinuous-time signal is a varyingquantity (asignal) whose domain, which is often time, is acontinuum (e.g., aconnected interval of thereals). That is, the function's domain is anuncountable set. The function itself need not to becontinuous. To contrast, adiscrete-time signal has acountable domain, like thenatural numbers.

A signal of continuous amplitude and time is known as a continuous-time signal or ananalog signal. This (asignal) 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 ofreal numbers, means that the signal value can be found at any arbitrary point in time.

A typical example of an infinite duration signal is:

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

A finite duration counterpart of the above signal could be:

${\displaystyle f(t)=\sin (t),t\in [-\pi ,\pi ]}$ and${\displaystyle f(t)=0}$ otherwise.

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

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

is a finite duration signal but it takes an infinite value for${\displaystyle t=0}$.

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${\displaystyle t^{-1}}$ signal is not integrable at infinity, but${\displaystyle t^{-2}}$ is).

Any analog signal is continuous by nature.Discrete-time signals, used indigital signal processing, can be obtained bysampling andquantization 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 inimage processing, where two space dimensions are used.

Discrete time is often employed whenempiricalmeasurements are involved, because normally it is only possible to measure variables sequentially. For example, whileeconomic 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 ofquarterly values.

When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one usestime series orregression 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 ofincome observed in unspecified time periodt,y₃ 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 mathematicallytractable to constructtheoretical models in continuous time, and often in areas such asphysics an exact description requires the use of continuous time. In a continuous time context, the value of a variabley at an unspecified point in time is denoted asy(t) or, when the meaning is clear, simply asy.

Discrete time makes use ofdifference equations, also known as recurrence relations. An example, known as thelogistic map or logistic equation, is

${\displaystyle x_{t+1}=r{x}_t(1-x_t),}$

in whichr is aparameter in the range from 2 to 4 inclusive, andx is a variable in the range from 0 to 1 inclusive whose value in periodtnonlinearly affects its value in the next period,t+1. For example, if${\displaystyle r=4}$ and${\displaystyle x_1=1/3}$, then fort=1 we have${\displaystyle x_2=4(1/3)(2/3)=8/9}$, and fort=2 we have${\displaystyle x_3=4(8/9)(1/9)=32/81}$.

Another example models the adjustment of apriceP in response to non-zeroexcess demand for a product as

${\displaystyle P_{t+1}=P_t+\delta \cdot f(P_t,...)}$

where${\displaystyle \delta }$ is the positive speed-of-adjustment parameter which is less than or equal to 1, and where${\displaystyle f}$ is theexcess demand function.

Continuous time makes use ofdifferential equations. For example, the adjustment of a priceP in response to non-zero excess demand for a product can be modeled in continuous time as

${\displaystyle \frac{dP}{dt}=\lambda \cdot f(P,...)}$

where the left side is thefirst derivative of the price with respect to time (that is, the rate of change of the price),${\displaystyle \lambda }$ is the speed-of-adjustment parameter which can be any positive finite number, and${\displaystyle f}$ is again the excess demand function.

A variable measured in discrete time can be plotted as astep function, in which each time period is given a region on thehorizontal 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 acontinuous function, since the domain of time is considered to be the entire real axis or at least some connected portion of it.

• Gershenfeld, Neil A. (1999).The Nature of mathematical Modeling. Cambridge University Press.ISBN 0-521-57095-6.

• Wagner, Thomas Charles Gordon (1959).Analytical transients. Wiley.

• Time in science
• Dynamical systems

• Articles with short description
• Short description matches Wikidata