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
In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can take on two particular real values such that it can also take on al ...
. 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 language ...
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. E ...
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.
Discrete-time signals may have several origins, but can usually be classified into one of two groups:
* By acquiring values of an
analog signal
An analog signal or analogue signal (see spelling differences) is any continuous signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the instantaneous signal voltage vari ...
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
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group)
Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...
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
In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can take on two particular real values such that it can also take on al ...
.
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 uni ...
(a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
)
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 num ...
. The function itself need not to be
continuous. To contrast, a
discrete-time
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 "po ...
signal has a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
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
An analog signal or analogue signal (see spelling differences) is any continuous signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the instantaneous signal voltage vari ...
. This (a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
) 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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, means that the signal value can be found at any arbitrary point in time.
A typical example of an infinite duration signal is:
:
A finite duration counterpart of the above signal could be:
: