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Dirac delta function

The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British United Kingdom

The United Kingdom of Great Britain and Northern Ireland is a country and sovereign state [i] tha ... 

 theoretical physicist Paul Dirac Paul Dirac

Paul Adrien Maurice Dirac, OM [i], FRS [i] was a British [i]... 

, can usually be informally thought of as a function δ that has the value of infinity Infinity

he word infinity comes from the Latin [i] infinitas or "unboundedness." It refers to several distinc ... 

 for x = 0, the value zero elsewhere. The integral Integral

In calculus [i], the integral of a function [i] is an extension of the concept of a sum. ... 

 from minus infinity to plus infinity is 1. The discrete Discrete signal

A discrete signal or discrete-time signal is a time series [i], perhaps a signal [i] that h ... 

 analog of the delta "function" is the Kronecker delta Kronecker delta

In mathematics [i], the Kronecker delta or Kronecker's delta, named after Leopold Kronecker [i], i ... 

 which is sometimes known as a delta function. Note that the Dirac delta is not a function, but a distribution that is also a measure.

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Encyclopedia

The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British United Kingdom

The United Kingdom of Great Britain and Northern Ireland is a country and sovereign state [i] tha ... 

 theoretical physicist Paul Dirac Paul Dirac

Paul Adrien Maurice Dirac, OM [i], FRS [i] was a British [i]... 

, can usually be informally thought of as a function δ that has the value of infinity Infinity

he word infinity comes from the Latin [i] infinitas or "unboundedness." It refers to several distinc ... 

 for x = 0, the value zero elsewhere. The integral Integral

In calculus [i], the integral of a function [i] is an extension of the concept of a sum. ... 

 from minus infinity to plus infinity is 1. The discrete Discrete signal

A discrete signal or discrete-time signal is a time series [i], perhaps a signal [i] that h ... 

 analog of the delta "function" is the Kronecker delta Kronecker delta

In mathematics [i], the Kronecker delta or Kronecker's delta, named after Leopold Kronecker [i], i ... 

 which is sometimes known as a delta function. Note that the Dirac delta is not a function, but a distribution that is also a measure.

Overview



Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph Graph of a function

In mathematics, the graph of a function [i] f is the collection of all ordered pair [i]s). ... 

 of the delta function is usually thought of as following the whole x-axis and the positive y-axis.

Despite its name, the delta function is not a function. One reason for this is because the functions f = δ and g = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory Lebesgue integration

In mathematics [i], the integral [i] of a nonnegative function can be regarded in the simplest case as the ... 

, if f, g are functions such that f = g almost everywhere, then f is integrable iff g is integrable and the integrals of f and g are the same. Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions.

The Dirac delta is very useful as an approximation for a tall narrow spike function . It is the same type of abstraction Abstraction

Abstraction is the process of reducing the information content [i] of a concept [i], typically in order... 

 as a point charge, point mass Mass

Mass is a property of a physical [i] object that quantifies the amount of matter [i] and energy [i] ... 

 or electron Electron

The electron is a fundamental [i] subatomic particle [i] that carries an electric charge [i]... 

 point. For example, in calculating the dynamics of a baseball Baseball

Baseball is a team sport [i] popular in North America [i], parts of Latin America [i], the Caribbean [i] ... 

 being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball.

The Dirac delta function was named after the Kronecker delta Kronecker delta

In mathematics [i], the Kronecker delta or Kronecker's delta, named after Leopold Kronecker [i], i ... 

, since it can be used as a continuous analogue of the discrete Kronecker delta.

Definitions


The Dirac delta function can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,




and which is also constrained to satisfy the identity

This heuristic definition should not be taken too seriously though. Firstly, the Dirac delta is not a function, as no function has the above properties. Moreover there exist descriptions which differ from the above conceptualization. For example, behaves as a delta function in the limit of , yet this function does not approach zero for values of x outside the origin.

The defining characteristic
where f is a suitable test function Test Function

Sorry, no overview for this topic 

, cannot be achieved by any function, but the Dirac delta function can be rigorously defined either as a distribution or as a measure.

In terms of dimensional analysis, this definition of implies that has dimensions reciprocal to those of dx.

The delta function as a measure


As a measure, if , and otherwise. Then,




for all continuous .

As distributions, the Heaviside step function Heaviside step function

The Heaviside step function, sometimes called the unit step function [i] and named in honor of Oliver Heaviside [i] ... 

 is an antiderivative Antiderivative

In calculus [i], an antiderivative, primitive or indefinite integral of a function [i] ... 

 of the Dirac delta distribution.

The delta function as a probability density function


As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by


for every test function . It is a distribution with compact support . Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience, and not a true integral.

Thus, the Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function,
so that all moments are zero. The cumulative distribution function is the Heaviside step function Heaviside step function

The Heaviside step function, sometimes called the unit step function [i] and named in honor of Oliver Heaviside [i] ... 

.

Equivalently, one may define as a distribution whose indefinite integral Antiderivative

In calculus [i], an antiderivative, primitive or indefinite integral of a function [i] ... 

 is the function

usually called the Heaviside step function Heaviside step function

The Heaviside step function, sometimes called the unit step function [i] and named in honor of Oliver Heaviside [i] ... 

or commonly the unit step function. That is, it satisfies the integral equation

for all real numbers x.

Delta function of more complicated arguments


A helpful identity is the scaling property:

and so

The scaling property may be generalized to:




where xi are the roots of g. Thus, for example

In the integral form the generalized scaling property may be written as




In an n-dimensional space with position vector , this is generalized to:




where the integral on the right is over , the n-1  dimensional surface defined by .

The integral of the time-shifted Dirac delta is given by:

Thus, the delta function is said to "sift out" the function at the value , when integrated over all time.

Similarly, the convolution:

means that the effect of convolving with the time-shifted Dirac delta is to time-shift by the same amount.

Fourier transform


Using Fourier transforms, one has

and therefore:

which is a statement of the orthogonality property for the Fourier kernel.

Laplace transform


The direct Laplace transform Laplace transform

In mathematics [i], the Laplace transform is a powerful technique for analyzing linear time-invariant [i] ... 

 of the delta function is:




a curious identity using Euler's formula allows us to find the Laplace inverse transform for the cosine

and a similar identity holds for .

Derivatives of the delta function


The derivative of the Dirac delta function is the distribution δ' defined by




for every test function . From this it follows that




The n-th derivative δ is given by




The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomial Polynomial

In mathematics [i], a polynomial is an expression [i] in which a finite number of constants ... 

s.

Representations of the delta function


The delta function can be viewed as the limit of a sequence of functions

where is sometimes called a nascent delta function. This limit is in the sense that

for all continuous .

The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation . There the condition is made that the limiting sequence should be of positive functions.

Some nascent delta functions are:
 Limit of a Normal distribution Normal distribution

The normal distribution, also called Gaussian distribution , is an extremely important probability distribution [i] ... 

 Limit of a Cauchy distribution Cauchy distribution

The Cauchy-Lorentz distribution, named after Augustin Cauchy [i] and Hendrik Lorentz [i], is a continuou ... 

 Cauchy
 Limit of a rectangular function Rectangular function

The rectangular function is defined as,

... 

 rectangular function
 Derivative of the sigmoid  function
  
 Limit of the Airy function Airy function

In mathematics [i], the Airy function Ai(x) is a special function [i], i.e., a function that appears ... 

 Limit of a Bessel function Bessel function

In mathematics [i], Bessel functions, first defined by the mathematician [i] Daniel Bernoulli [i] and ge ... 





Note: If δ is a nascent delta function which is a probability distribution Probability distribution

In mathematics [i] and statistics [i], a probability distribution, more properly called a probability... 

 over the whole real line
then another nascent delta function δφ can be built from its characteristic function as follows:

where

is the characteristic function of the nascent delta function δ. This result is related to the localization property of the continuous Fourier transform.

The Dirac comb

Main article: Dirac comb Dirac comb

In mathematics [i], a Dirac comb is a periodic [i] Schwartz distribution [i] construct ... 



A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb Dirac comb

In mathematics [i], a Dirac comb is a periodic [i] Schwartz distribution [i] construct ... 

,or as the shah distribution, creates a sampling function, often used in digital signal processing  and discrete time signal analysis.

See also


  • Kronecker delta Kronecker delta

    In mathematics [i], the Kronecker delta or Kronecker's delta, named after Leopold Kronecker [i], i ... 

  • Dirac comb Dirac comb

    In mathematics [i], a Dirac comb is a periodic [i] Schwartz distribution [i] construct ... 

  • Logarithmically-spaced Dirac comb
  • Green's function

External links

  • on MathWorld
  • on PlanetMath PlanetMath

    PlanetMath is a free, collaborative, online mathematics [i] encyclopedia [i]. ...