In
theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumsc ...
, a discrete system is a system with 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 numbers; ...
number of
state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
s. Discrete systems may be contrasted with continuous systems, which may also be called
analog
Analog or analogue may refer to:
Computing and electronics
* Analog signal, in which information is encoded in a continuous variable
** Analog device, an apparatus that operates on analog signals
*** Analog electronics, circuits which use analog ...
systems. A final discrete system is often modeled with a directed
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
and is analyzed for correctness and
complexity
Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generall ...
according to
computational theory
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
. Because discrete systems have a countable number of states, they may be described in precise
mathematical models
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
.
A
computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
is a
finite state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...
that may be viewed as a discrete system. Because computers are often used to model not only other discrete systems but continuous systems as well, methods have been developed to represent real-world continuous systems as discrete systems. One such method involves
sampling a continuous signal at
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 ...
intervals.
See also
*
Digital control
Digital control is a branch of control theory that uses digital computers to act as system controllers.
Depending on the requirements, a digital control system can take the form of a microcontroller to an ASIC to a standard desktop computer.
Sin ...
*
Finite state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...
*
Frequency spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
*
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
*
Sample and hold
In electronics, a sample and hold (also known as sample and follow) circuit is an analog device that samples (captures, takes) the voltage of a continuously varying analog signal and holds (locks, freezes) its value at a constant level for a ...
*
Sample rate
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a value of the signal at a point in time and/or sp ...
*
Sample time
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a value of the signal at a point in time and/or s ...
*
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
References
*
*
Automata (computation)
Models of computation
Signal processing
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