* Analog-to-digital converter, a circuit which converts continuous analog signals to discrete digital values

Therefore, at the design stage either digital components are mapped into the continuous domain and the design is carried out in the continuous domain, or analog components are mapped in to discrete domain and design is carried out there.

Beads only have meaning in discrete up and down states, not in analog in-between states.

Even if this process is more complex than analog processing and has a discrete value range, the application of computational power to digital signal processing allows for many advantages over analog processing in many applications, such as error detection and correction in transmission as well as data compression.

Signals can be either analog, in which case the signal varies continuously according to the information, or digital, in which case the signal varies according to a series of discrete values representing the information.

Very different mathematical treatments apply to the design of filters termed infinite impulse response ( IIR ) filters, characteristic of mechanical and analog electronics systems, and finite impulse response ( FIR ) filters, which can be implemented by discrete time systems such as computers ( then termed digital signal processing ).

In digital modulation, an analog carrier signal is modulated by a discrete signal.

These are methods to transfer a digital bit stream over an analog baseband channel ( a. k. a. lowpass channel ) using a pulse train, i. e. a discrete number of signal levels, by directly modulating the voltage or current on a cable.

Note also that in most spoken languages, the sounds representing successive letters blend into each other in a process termed coarticulation, so the conversion of the analog signal to discrete characters can be a very difficult process.

Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

Digital electronics, or digital ( electronic ) circuits, represent signals by discrete bands of analog levels, rather than by a continuous range.

Relatively small changes to the analog signal levels due to manufacturing tolerance, signal attenuation or parasitic noise do not leave the discrete envelope, and as a result are ignored by signal state sensing circuitry.

Most useful digital systems must translate from continuous analog signals to discrete digital signals.

The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton ; in essence, it is the Newton interpolation formula, first published in his Principia Mathematica in 1687, namely the discrete analog of the continuum Taylor expansion,

The resolution of the converter indicates the number of discrete values it can produce over the range of analog values.

While analog transmission is the transfer of a continuously varying analog signal, digital communications is the transfer of discrete messages.

For instance where there are integral transforms in harmonic analysis for studying continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals.

A fuzzy control system is a control system based on fuzzy logic — a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, which operates on discrete values of either 1 or 0 ( true or false, respectively ).

The discrete time and level of the binary signal allow a decoder to recreate the analog signal upon replay.

Newton's law of cooling is a discrete analog of Fourier's law, while Ohm's law is the electrical analogue of Fourier's law.

This is the special case of Gauss – Bonnet, where the curvature is concentrated at discrete points ( the vertices ).

Thinking of curvature as a measure, rather than as a function, Descartes ' theorem is Gauss – Bonnet where the curvature is a discrete measure, and Gauss – Bonnet for measures generalizes both Gauss – Bonnet for smooth manifolds and Descartes ' theorem.

# The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.

It was spurred by the 1987 monograph of Gromov " Hyperbolic groups " that introduced the notion of a hyperbolic group ( also known as word-hyperbolic or Gromov-hyperbolic or negatively curved group ), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monograph Asymptotic Invariants of Inifinite Groups, that outlined Gromov's program of understanding discrete groups up to quasi-isometry.

Define, where is the discrete valuation corresponding to the ideal P < sub > i </ sub >.

In mass spectrometry, discrete peaks appeared corresponding to molecules with the exact mass of sixty or seventy or more carbon atoms.

It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform ( DHT ), but it was subsequently argued that a specialized real-input DFT algorithm ( FFT ) can typically be found that requires fewer operations than the corresponding DHT algorithm ( FHT ) for the same number of inputs.

Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with as above by setting the interval length short enough to exclude any of the discrete masses.

where k is the index of the discrete probability mass corresponding to observation x.

Note that the Bernstein-von Mises theorem asserts here the asymptotic convergence to the " true " distribution because the probability space corresponding to the discrete set of events is finite ( see above section on asymptotic behaviour of the posterior ).

In principle, there are actually four additional types of discrete sine transform ( Martucci, 1994 ), corresponding to real-odd DFTs of logically odd order, which have factors of N + 1 / 2 in the denominators of the sine arguments.

If boundary conditions are applied to this system ( e. g., at two physical locations in space ), then only certain values of satisfy the boundary conditions, generating corresponding discrete eigenvalues.

* If I is a small discrete category ( i. e. its only morphisms are the identity morphisms ), then a functor from I to C essentially consists of a family of objects of C, indexed by I ; the functor category C < sup > I </ sup > can be identified with the corresponding product category: its elements are families of objects in C and its morphisms are families of morphisms in C.

In particular, if S is an inverse limit of discrete R algebras, we can define F ( S ) to be the inverse limit of the corresponding groups.

Let be an observable, and suppose that it has discrete eigenstates ( in bra-ket notation ) for and corresponding eigenvalues, no two of which are equal.

Noting that Spec K is the generic point of Spec R and discrete valuation rings are precisely the regular local one-dimensional rings, one may rephrase the criterion: given a regular curve on Y ( corresponding to the morphism s: Spec R → Y ) and given a lift of the generic point of this curve to X, f is proper if and only if there is exactly one way to complete the curve.

One can think of conditioning on conjugate priors as defining a kind of ( discrete time ) dynamical system: from a given set of hyperparameters, incoming data updates these hyperparameters, so one can see the change in hyperparameters as a kind of " time evolution " of the system, corresponding to " learning ".

* Sinusoidal waves do not have location – they spread across the whole space – but do have phase – the second and third waves are translations of each other, corresponding to being 90 ° out of phase, like cosine and sine, of which these are discrete versions.

A function on G is called the time domain representation of the function, while the corresponding expression in terms of characters is called the frequency domain representation of the function: changing from the time domain description to the frequency domain description is called the discrete Fourier transform, and the opposite direction is called the inverse discrete Fourier transform.

In discrete time, the change in a stock variable from one point in time to another point in time one time unit later is equal to the corresponding flow variable per unit of time.

The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup Γ into an algebraic group over a field which is technically easier to work with.

A bit plane of a digital discrete signal ( such as image or sound ) is a set of bits corresponding to a given bit position in each of the binary numbers representing the signal.

Since a discrete signal is a sequence ( merely a series of symbols ; typically, numbers ) it contains no direct information as to determine the frequency of the corresponding continuous signal.

As the above suggest, if a group has a universal covering group ( if it is path-connected, locally path-connected, and semilocally simply connected ), with discrete center, then the set of all topologically groups that are covered by the universal covering group form a lattice, corresponding to the lattice of subgroups of the center of the universal covering group: inclusion of subgroups corresponds to covering of quotient groups.

Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue λ ( see spectral theorem for Hermitian operators for the mathematical background ):

where X < sub > i </ sub > is the feature vector for instance i, β < sub > k </ sub > is the vector of weights corresponding to category k, and score ( X < sub > i </ sub >, k ) is the score associated with assigning instance i to category k. In discrete choice theory, where instances represent people and categories represent choices, the score is considered the utility associated with person i choosing category k.