# Discrete time and continuous time

In mathematics and, in particular, mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.

## Discrete time

Discrete sampled signal

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:[1]

• By acquiring values of an analog signal at constant or variable rate. This process is called sampling.[2]
• 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 for potentially only 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 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:

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

A finite duration counterpart of the above signal could be:

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

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

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

## Relevant contexts

Discrete time is often employed when empirical measurements are involved, because normally it is only possible to measure variables sequentially. For example, while economic 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 of quarterly values.

When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses time series or regression methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, yt might refer to the value of income observed in unspecified time period t, 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 mathematically tractable to construct theoretical models in continuous time, and often in areas such as physics an exact description requires the use of continuous time. In a continuous time context, the value of a variable y at an unspecified point in time is denoted as y(t) or, when the meaning is clear, simply as y.

## Types of equations

### Discrete time

Discrete time makes use of difference equations, also known as recurrence relations. An example, known as the logistic map or logistic equation, is

${\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}$

in which r is a parameter in the range from 2 to 4 inclusive, and x is a variable in the range from 0 to 1 inclusive whose value in period t nonlinearly affects its value in the next period, t+1. For example, if ${\displaystyle r=4}$ and ${\displaystyle x_{1}=1/3}$, then for t=1 we have ${\displaystyle x_{2}=4(1/3)(2/3)=8/9}$, and for t=2 we have ${\displaystyle x_{3}=4(8/9)(1/9)=32/81}$.

Another example models the adjustment of a price P in response to non-zero excess 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 the excess demand function.

### Continuous time

Continuous time makes use of differential equations. For example, the adjustment of a price P 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 the first 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.

## Graphical depiction

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

## References

1. ^ "Digital Signal Processing" Prentice Hall - Pages 11-12
2. ^ "Digital Signal Processing: Instant access." Butterworth-Heinemann - Page 8
• 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.
Continuous or discrete variable

In mathematics, a variable may be continuous or discrete. If it can take on two particular real values such that it can also take on all real values between them (even values that are arbitrarily close together), the variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts a variable can be discrete in some ranges of the number line and continuous in others.

Discretization

In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).

Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.

Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand.

The terms discretization and quantization often have the same denotation but not always identical connotations. (Specifically, the two terms share a semantic field.) The same is true of discretization error and quantization error.

Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold.

Hourglass

An hourglass (or sandglass, sand timer, sand clock or egg timer) is a device used to measure the passage of time. It comprises two glass bulbs connected vertically by a narrow neck that allows a regulated trickle of material (historically sand) from the upper bulb to the lower one. Factors affecting the time it measured include sand quantity, sand coarseness, bulb size, and neck width. Hourglasses may be reused indefinitely by inverting the bulbs once the upper bulb is empty. Depictions of hourglasses in art survive in large numbers from antiquity to the present day, as a symbol for the passage of time. These were especially common sculpted as epitaphs on tombstones or other monuments, also in the form of the winged hourglass, a literal depiction of the well-known Latin epitaph tempus fugit ("time flies").

MLDesigner

MLDesigner is an integrated modeling and simulation tool for the design and analysis of complex embedded and networked systems. MLDesigner speeds up modeling, simulation and analysis of discrete event, discrete time and continuous time systems concerning architecture, function and performance. The tools is based on ideas of the "Ptolemy Project", done at the University if California Berkeley (UC Berkeley). MLDesigner is developed by MLDesign Technologies Inc. Palo Alto, CA, USA in collaboration with Mission Level Design GmbH, Ilmenau, Germany.

Markov switching multifractal

In financial econometrics, the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates stochastic volatility components of heterogeneous durations. MSM captures the outliers, log-memory-like volatility persistence and power variation of financial returns. In currency and equity series, MSM compares favorably with standard volatility models such as GARCH(1,1) and FIGARCH both in- and out-of-sample. MSM is used by practitioners in the financial industry to forecast volatility, compute value-at-risk, and price derivatives.

Process calculus

In computer science, the process calculi (or process algebras) are a diverse family of related approaches for formally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions, communications, and synchronizations between a collection of independent agents or processes. They also provide algebraic laws that allow process descriptions to be manipulated and analyzed, and permit formal reasoning about equivalences between processes (e.g., using bisimulation). Leading examples of process calculi include CSP, CCS, ACP, and LOTOS. More recent additions to the family include the π-calculus, the ambient calculus, PEPA, the fusion calculus and the join-calculus.

Quantum walk

Quantum walks are quantum analogues of classical random walks. In contrast to the classical random walk, where the walker occupies definite states and the randomness arises due to stochastic transitions between states, in quantum walks randomness arises through: (1) quantum superposition of states, (2) non-random, reversible unitary evolution and (3) collapse of the wave function due to state measurements.

As with classical random walks, quantum walks admit formulations in both discrete time and continuous time.

Stochastic process

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space. The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

System on a chip

A system on a chip or system on chip (SoC es-oh-SEE or sock) is an integrated circuit (also known as a "chip") that integrates all components of a computer or other electronic system. These components typically (but not always) include a central processing unit (CPU), memory, input/output ports and secondary storage – all on a single substrate or microchip, the size of a coin. It may contain digital, analog, mixed-signal, and often radio frequency signal processing functions, depending on the application. As they are integrated on a single substrate, SoCs consume much less power and take up much less area than multi-chip designs with equivalent functionality. Because of this, SoCs are very common in the mobile computing (such as in Smartphones) and edge computing markets. Systems on chip are commonly used in embedded systems and the Internet of Things.

Systems on Chip are in contrast to the common traditional motherboard-based PC architecture, which separates components based on function and connects them through a central interfacing circuit board. Whereas a motherboard houses and connects detachable or replaceable components, SoCs integrate all of these components into a single integrated circuit, as if all these functions were built into the motherboard. An SoC will typically integrate a CPU, graphics and memory interfaces, hard-disk and USB connectivity, random-access and read-only memories and secondary storage on a single circuit die, whereas a motherboard would connect these modules as discrete components or expansion cards.

More tightly integrated computer system designs improve performance and reduce power consumption as well as semiconductor die area needed for an equivalent design composed of discrete modules, at the cost of reduced replaceability of components. By definition, SoC designs are fully or nearly fully integrated across different component modules. For these reasons, there has been a general trend towards tighter integration of components in the computer hardware industry, in part due to the influence of SoCs and lessons learned from the mobile and embedded computing markets. Systems-on-Chip can be viewed as part of a larger trend towards embedded computing and hardware acceleration.

A SoC integrates a microcontroller or microprocessor with advanced peripherals like graphics processing unit (GPU), Wi-Fi module, or one or more coprocessors. Similar to how a microcontroller integrates a microprocessor with peripheral circuits and memory, an SoC can be seen as integrating a microcontroller with even more advanced peripherals. For an overview of integrating system components, see system integration.

Term (time)

A term is a period of duration, time or occurrence, in relation to an event. To differentiate an interval or duration, common phrases are used to distinguish the observance of length are near-term or short-term, medium-term or mid-term and long-term.

It is also used as part of a calendar year, especially one of the three parts of an academic term and working year in the United Kingdom: Michaelmas term, Hilary term / Lent term or Trinity term / Easter term, the equivalent to the American semester. In America there is a midterm election held in the middle of the four-year presidential term, there are also academic midterm exams.

In economics, it is the period required for economic agents to reallocate resources, and generally reestablish equilibrium. The actual length of this period, usually numbered in years or decades, varies widely depending on circumstantial context. During the long term, all factors are variable.

In finance or financial operations of borrowing and investing, what is considered long-term is usually above 3 years, with medium-term usually between 1 and 3 years and short-term usually under 1 year. It is also used in some countries to indicate a fixed term investment such as a term deposit.

In law, the term of a contract is the duration for which it is to remain in effect (not to be confused with the meaning of "term" that denotes any provision of a contract). A fixed-term contract is one concluded for a pre-defined time, although it may also include provision for it to be extended. A contractor required to deliver against a term contract is often referred to as a "term contractor".

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