Dirac delta function

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Dirac delta function
Probability density function Image:Dirac distribution PDF.png Schematic representation of the Dirac delta function for x₀ = 0. A line surmounted by an arrow is usually used to schematically represent the Dirac delta function. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.
Cumulative distribution function Using the half-maximum convention, with x₀ = 0
Parameters	Failed to parse (Missing texvc executable; please see math/README to configure.): x_0\, location (real)
Support	Failed to parse (Missing texvc executable; please see math/README to configure.): x \in [x_0; x_0]
Probability density function (pdf)	Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x-x_0)\,
Cumulative distribution function (cdf)	Failed to parse (Missing texvc executable; please see math/README to configure.): H(x-x_0)\, (Heaviside)
Mean	Failed to parse (Missing texvc executable; please see math/README to configure.): x_0\,
Median	Failed to parse (Missing texvc executable; please see math/README to configure.): x_0\,
Mode	Failed to parse (Missing texvc executable; please see math/README to configure.): x_0\,
Variance	Failed to parse (Missing texvc executable; please see math/README to configure.): 0\,
Skewness	(undefined)
Excess kurtosis	(undefined)
Entropy	Failed to parse (Missing texvc executable; please see math/README to configure.): -\infty
Moment-generating function (mgf)	Failed to parse (Missing texvc executable; please see math/README to configure.): e^{tx_0}
Characteristic function	Failed to parse (Missing texvc executable; please see math/README to configure.): e^{itx_0}

The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function δ(x) that has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1. It is a continuous analogue of the discrete Kronecker delta. In the context of signal processing it is often referred to as the unit impulse function. Note that the Dirac delta is not strictly a function. While for many purposes it can be manipulated as such, formally it can be defined as a distribution that is also a measure.

1 Overview
2 Definitions
3 The delta function as a measure
4 The delta function as a probability density function
5 Delta function of more complicated arguments
6 Fourier transform
7 Laplace transform
8 Distributional derivatives
9 Representations of the delta function
10 The Dirac comb
11 See also
12 External links

Overview

A Dirac function can be of any size in which case its 'strength' A is defined by duration multiplied by amplitude. The graph of the delta function is usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function.)

Despite its name, the delta function is not truly a function. One reason for this is because the functions f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g are identical. 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 (an impulse). It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball 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^{[citation needed]}, since it can be used as a continuous analogue of the discrete Kronecker delta.

Definitions

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases}

and which is also constrained to satisfy the identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty \delta(x) \, dx = 1.

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 of the delta function which differ from the above conceptualization. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): \textrm{sinc}(x/a)/a

(where sinc is the sinc function) behaves as a delta function in the limit of Failed to parse (Missing texvc executable; please see math/README to configure.): a\rightarrow 0

, yet this function does not approach zero for values of x outside the origin, rather it oscillates between 1/x and -1/x more and more rapidly as a approaches infinity.

The defining characteristic

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty f(x) \, \delta(x) \, dx = f(0)

where f is a suitable test function, cannot be achieved by any function^{[citation needed]}, 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x)

implies that Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x)
has dimensions reciprocal to those of dx.

The delta function as a measure

As a measure, Failed to parse (Missing texvc executable; please see math/README to configure.): \delta (A)=1

if Failed to parse (Missing texvc executable; please see math/README to configure.): 0\in A

, and Failed to parse (Missing texvc executable; please see math/README to configure.): \delta (A)=0

otherwise.  Then,

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty f(x) \, \delta(x) \, dx = f(0)

for all continuous Failed to parse (Missing texvc executable; please see math/README to configure.): f .

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta[\phi] = \phi(0)\,

for every test function Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \ . It is a distribution with compact support (the support being {0}). 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.

Equivalently, one may define Failed to parse (Missing texvc executable; please see math/README to configure.): \delta : \mathbb{R} \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb{R})

 as a distribution   Failed to parse (Missing texvc executable; please see math/README to configure.): \delta ( \xi )
whose indefinite integral is the function

Failed to parse (Missing texvc executable; please see math/README to configure.): h : \mathbb{R} \ni \xi \longrightarrow \frac{1+{\rm sgn} \, \xi }{2} \in \mathbb{R},

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int^{x}_{-\infin} \delta (t) dt = h(x) \equiv \frac{1+{\rm sgn}(x)}{2}

for all real numbers x.

Delta function of more complicated arguments

A helpful identity is the scaling property (Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

is non-zero),

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty \delta(\alpha x)\,dx =\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}

and so

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(\alpha x) = \frac{\delta(x)}{|\alpha|}

The scaling property may be generalized to:

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(g(x)) = \sum_{i}\frac{\delta(x-x_i)}{|g'(x_i)|}

and, Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(\alpha g(x)) = \frac{1}{|\alpha|}\delta(g(x))

where x_i are the real roots of g(x) (assumed simple roots). Thus, for example

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x^2-\alpha^2) = \frac{1}{2|\alpha|}[\delta(x+\alpha)+\delta(x-\alpha)]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}

In an n-dimensional space with position vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r} , this is generalized to:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_V f(\mathbf{r}) \, \delta(g(\mathbf{r})) \, d^nr = \int_{\partial V}\frac{f(\mathbf{r})}{|\mathbf{\nabla}g|}\,d^{n-1}r

where the integral on the right is over Failed to parse (Missing texvc executable; please see math/README to configure.): \partial V , the n-1 dimensional surface defined by Failed to parse (Missing texvc executable; please see math/README to configure.): g(\mathbf{r})=0 .

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int\limits_{-\infty}^\infty f(t) \delta(t-T)\,dt = f(T)

(the sifting property). The delta function is said to "sift out" the value at Failed to parse (Missing texvc executable; please see math/README to configure.): t=T\, . Similarly, the convolution:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(t) * \delta(t-T) = \int\limits_{-\infty}^\infty f(\tau) \cdot \delta(t-T-\tau) d\tau = f(t-T)

means that the effect of convolving with the time-delayed Dirac delta is to time-delay Failed to parse (Missing texvc executable; please see math/README to configure.): f(t)\,

by the same amount.

Fourier transform

Using Fourier transforms, one finds that

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f)

and therefore:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^\infty e^{i 2\pi f_1 t} \left[e^{i 2\pi f_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (f_2 - f_1) t} \,dt = \delta(f_2 - f_1)

which is a statement of the orthogonality property for the Fourier kernel. Equating these non-converging improper integrals to Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x)

is not mathematically rigorous.  However, they behave in the same way under a definite integral.  That is,

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \int_{-\infty}^\infty F(f) \left(\int_{-\infty}^\infty e^{-i 2\pi f t} dt\right) df &= F(0) \end{align}

according to the definition of the Fourier transform. Therefore, the bracketed term is considered equivalent to the Dirac delta function.

Laplace transform

The direct Laplace transform of the delta function is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{0}^{\infty}\delta (t-a)e^{-st} \, dt=e^{-as}

a curious identity using Euler's formula Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \cos(as)=e^{-ias}+e^{ias}

allows us to find the Laplace inverse transform for the cosine

Failed to parse (Missing texvc executable; please see math/README to configure.): 2\frac{1}{2\pi {i}}\int_{c-i\infty}^{c+i\infty} \cos(as)e^{st} \, ds=2[\delta (t+ia) +\delta (t-ia)]

and a similar identity holds for Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(as)

Distributional derivatives

As a tempered distribution, the Dirac delta distribution is infinitely differentiable. Let U be an open subset of Euclidean space Rⁿ and let S(U) denote the Schwartz space of smooth, rapidly decaying real-valued functions on U. Let a be a point of U and let δ_a be the Dirac delta distribution centred at a. If α = (α₁, ..., α_n) is any multi-index and ∂^α denotes the associated mixed partial derivative operator, then the α^th derivative ∂^αδ_a of δ_a is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \left\langle \partial^{\alpha} \delta_{a}, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = \left. (-1)^{| \alpha |} \partial^{\alpha} \varphi (x) \right|_{x = a} \mbox{ for all } \varphi \in S(U).

That is, the α^th derivative of δ_a is the distribution whose value on any test function φ is the α^th derivative of φ at a (with the appropriate positive or negative sign). This is rather convenient, since the Dirac delta distribution δ_a applied to φ is just φ(a). For the α=1 case this means

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^{\infty} \delta'(x-a)f(x)dx = -f'(a)

The first derivative of the delta function is referred to as a doublet (or the doublet function). [1] Its schematic representation looks like that of δ_a(t) and -δ_a(t) superposed.

Representations of the delta function

Image:Dirac function approximation.gif

A sequence of normal distributions that in the limit approximates the Dirac delta function (see the formula in the text).

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta (x) = \lim_{a\to 0} \delta_a(x),

where Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x)

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{a\to 0} \int_{-\infty}^{\infty}\delta_a(x)f(x)dx = f(0) \

for all continuous Failed to parse (Missing texvc executable; please see math/README to configure.): f .

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 (also on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions.

Some nascent delta functions are:

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x) = \frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}	Limit of a normal distribution
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x) = \frac{1}{\pi} \frac{a}{a^2 + x^2} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} k x-\|ak\|}\;dk	Limit of a Cauchy distribution
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x)=\frac{e^{-\|x/a\|}}{2a} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ikx}}{1+a^2k^2}\,dk	Cauchy Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (see note below)
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x)= \frac{\textrm{rect}(x/a)}{a} =\frac{1}{2\pi}\int_{-\infty}^\infty \textrm{sinc} \left( \frac{a k}{2 \pi} \right) e^{ikx}\,dk	Limit of a rectangular function
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x)=\frac{1}{\pi x}\sin\left(\frac{x}{a}\right) =\frac{1}{2\pi}\int_{-1/a}^{1/a} \cos (k x)\;dk	rectangular function Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (see note below)
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x)=\partial_x \frac{1}{1+\mathrm{e}^{-x/a}} =-\partial_x \frac{1}{1+\mathrm{e}^{x/a}}	Derivative of the sigmoid (or Fermi-Dirac) function
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x)=\frac{a}{\pi x^2}\sin^2\left(\frac{x}{a}\right)
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x) = \frac{1}{a}A_i\left(\frac{x}{a}\right)	Limit of the Airy function
Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_a(x) = \frac{1}{a}J_{1/a} \left(\frac{x+1}{a}\right)	Limit of a Bessel function

Note: If δ(a, x) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always non-negative between -∞ and +∞) then another nascent delta function δ_φ(a, x) can be built from its characteristic function as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_\varphi(a,x)=\frac{1}{2\pi}~\frac{\varphi(1/a,x)}{\delta(1/a,0)}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi(a,k)=\int_{-\infty}^\infty \delta(a,x)e^{-ikx}\,dx

is the characteristic function of the nascent delta function δ(a, x). This result is related to the localization property of the continuous Fourier transform.

The Dirac comb

Main article: Dirac comb

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the shah distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis.