Dirac delta function

Schematic representation of the Dirac delta function by a line surmounted by an arrow. 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.

The Dirac delta function as the limit (in the sense of distributions) of the sequence of Gaussians $\delta _{a}(x)={\frac {1}{a{\sqrt {\pi }}}}\mathrm {e} ^{-x^{2}/a^{2}}$ as $a\to 0.$

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.

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, at least not a usual one with domain in reals. For example, the objects 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 , 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,

$\delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}$

and which is also constrained to satisfy the identity

$\int _{-\infty }^{\infty }\delta (x)\,dx=1.$

This is merely a heuristic definition. The Dirac delta is not a real function, as no real function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, ${\textrm {sinc}}(x/a)/a$ (where sinc is the sinc function) behaves as a delta function in the limit of $a\rightarrow 0$, 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 Dirac delta function can be rigorously defined either as a distribution or as a measure.

In terms of dimensional analysis, this definition of $\delta (x)$ implies that $\delta (x)$ has dimensions reciprocal to those of dx.

The delta function as a measure

As a measure, $\delta (A)=1$ if $0\in A$, and $\delta (A)=0$ otherwise. Then,

$\int _{-\infty }^{\infty }f(x)\,\delta (x)dx=f(0)$

for all functions $f$.

The delta function as a distribution

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

$\delta [\phi ]=\phi (0)\,$

for every test function $\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 garden-variety (Riemann or Lebesgue) integral.

Thus, the Dirac delta function may be interpreted as a probability distribution. 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 $\delta :\mathbb {R} \ni \xi \mapsto \delta (\xi )\in \delta (\mathbb {R} )$ as a distribution $\delta (\xi )$ whose indefinite integral is the function

$h:\mathbb {R} \ni \xi \longrightarrow {\frac {1+{\rm {sgn}}\,\xi }{2}}\in \mathbb {R} ,$

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

$\int _{-\infty }^{x}\delta (t)dt=h(x)\equiv {\frac {1+{\rm {sgn}}(x)}{2}}$

for all real numbers x. It is important to realize this "density" interpretation is a notational convenience; if dt is Lebesgue measure, then no such density $\delta$ exists. However, by choosing to interpret $\delta$ as a singular measure giving point mass to 0, one can move beyond mere notational convenience and state something both logically coherent and actually true, namely,

$\int _{-\infty }^{x}d\delta ={\begin{cases}0&{\text{if }}x<0,\\1&{\text{if }}x\geq 0\end{cases}}$

Delta function of more complicated arguments

A helpful identity is the scaling property ($\alpha$ is non-zero),

$\int _{-\infty }^{\infty }\delta (\alpha x)\,dx=\int _{-\infty }^{\infty }\delta (u)\,{\frac {du}{|\alpha |}}={\frac {1}{|\alpha |}}$

and so

$\delta (\alpha x)={\frac {\delta (x)}{|\alpha |}}$ (Eq.1)

The scaling property may be generalized to:

$\delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}$

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

$\delta (x^{2}-\alpha ^{2})={\frac {1}{2|\alpha |}}[\delta (x+\alpha )+\delta (x-\alpha )]$

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

$\int _{-\infty }^{\infty }f(x)\,\delta (g(x))\,dx=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}$

In an n-dimensional space with position vector $\mathbf {r}$, this is generalized to:

$\int _{V}f(\mathbf {r} )\,\delta (g(\mathbf {r} ))\,d^{n}r=\int _{\partial V}{\frac {f(\mathbf {r} )}{|\mathbf {\nabla } g|}}\,d^{n-1}r$

where the integral on the right is over $\partial V$, the n-1  dimensional surface defined by $g(\mathbf {r} )=0$.

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

$\int \limits _{-\infty }^{\infty }f(t)\delta (t-T)\,dt=f(T)$

(the sifting property). The delta function is said to "sift out" the value at $t=T\,$.

It follows that the convolution:

$f(t)*\delta (t-T)\,$	$\ {\stackrel {\mathrm {def} }{=}}\ \int \limits _{-\infty }^{\infty }f(\tau )\cdot \delta (t-T-\tau )\ d\tau$
	$=\int \limits _{-\infty }^{\infty }f(\tau )\cdot \delta (\tau -(t-T))\ d\tau$       (using  Eq.1 with $\alpha =-1$)
	$=f(t-T)\,$

means that the effect of convolving with the time-delayed Dirac delta is to time-delay $f(t)\,$ by the same amount.

Fourier transform

Using Fourier transforms, one finds that

$\int _{-\infty }^{\infty }1\cdot e^{-i2\pi ft}\,dt=\delta (f)$

and therefore:

$\int _{-\infty }^{\infty }e^{i2\pi f_{1}t}\left[e^{i2\pi f_{2}t}\right]^{*}\,dt=\int _{-\infty }^{\infty }e^{-i2\pi (f_{2}-f_{1})t}\,dt=\delta (f_{2}-f_{1})$

which is a statement of the orthogonality property for the Fourier kernel. Equating these non-converging improper integrals to $\delta (x)$ is not mathematically rigorous. However, they behave in the same way under a definite integral. That is,

${\begin{aligned}\int _{-\infty }^{\infty }F(f)\left(\int _{-\infty }^{\infty }e^{-i2\pi ft}dt\right)df&=F(0)\end{aligned}}$

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:

$\int _{0}^{\infty }\delta (t-a)e^{-st}\,dt=e^{-as}$

a curious identity using Euler's formula $2\cos(as)=e^{-ias}+e^{ias}$ allows us to find the Laplace inverse transform for the cosine

$2{\frac {1}{2\pi {i}}}\int _{c-i\,\infty }^{c+i\,\infty }\cos(as)e^{st}\,ds=\delta (t+ia)+\delta (t-ia)$ and a similar identity holds for $\sin(as)$.

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

$\left\langle \partial ^{\alpha }\delta _{a},\varphi \right\rangle =(-1)^{|\alpha |}\left\langle \delta _{a},\partial ^{\alpha }\varphi \right\rangle =\left.(-1)^{|\alpha |}\partial ^{\alpha }\varphi (x)\right|_{x=a}{\mbox{ for all }}\varphi \in S(U).$

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

$\int _{-\infty }^{\infty }\delta '(x-a)f(x)dx=-f'(a)$.

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

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

$\delta (x)=\lim _{a\to 0}\delta _{a}(x),$

where $\delta _{a}(x)$ is sometimes called a nascent delta function (and should not be confused with the Dirac's delta centered at a, denoted by the same symbol in the previous section). This limit is in the sense that

$\lim _{a\to 0}\int _{-\infty }^{\infty }\delta _{a}(x)f(x)dx=f(0)\$

for all continuous bounded $f$.

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:

$\delta _{a}(x)={\frac {1}{a{\sqrt {\pi }}}}\mathrm {e} ^{-x^{2}/a^{2}}$	Limit of a normal distribution
$\delta _{a}(x)={\frac {1}{\pi }}{\frac {a}{a^{2}+x^{2}}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\mathrm {e} ^{\mathrm {i} kx-|ak|}\;dk$	Limit of a Cauchy distribution
$\delta _{a}(x)={\frac {e^{-|x/a|}}{2a}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\frac {e^{ikx}}{1+a^{2}k^{2}}}\,dk$	Cauchy φ (see note below)
$\delta _{a}(x)={\frac {{\textrm {rect}}(x/a)}{a}}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\textrm {sinc}}\left({\frac {ak}{2\pi }}\right)e^{ikx}\,dk$	Limit of a rectangular function
$\delta _{a}(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{a}}\right)={\frac {1}{2\pi }}\int _{-1/a}^{1/a}\cos(kx)\;dk$	rectangular function φ (see note below)
$\delta _{a}(x)=\partial _{x}{\frac {1}{1+\mathrm {e} ^{-x/a}}}=-\partial _{x}{\frac {1}{1+\mathrm {e} ^{x/a}}}$	Derivative of the sigmoid (or Fermi-Dirac) function
$\delta _{a}(x)={\frac {a}{\pi x^{2}}}\sin ^{2}\left({\frac {x}{a}}\right)$
$\delta _{a}(x)={\frac {1}{a}}A_{i}\left({\frac {x}{a}}\right)$	Limit of the Airy function
$\delta _{a}(x)={\frac {1}{a}}J_{1/a}\left({\frac {x+1}{a}}\right)$	Limit of a Bessel function
$\delta _{a}(x)={\begin{cases}{\frac {1}{a}},&-{\frac {a}{2}}<x<{\frac {a}{2}}\\0,&{\mbox{otherwise}}\end{cases}}$

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:

$\delta _{\varphi }(a,x)={\frac {1}{2\pi }}~{\frac {\varphi (1/a,x)}{\delta (1/a,0)}}$

where

$\varphi (a,k)=\int _{-\infty }^{\infty }\delta (a,x)e^{-ikx}\,dx$

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.

There are also series and integral representations of the Dirac delta function in terms of special functions, such as integrals of products of Airy functions, of Bessel functions, of Coulomb wave functions and of parabolic cylinder functions, and also series of products of orthogonal polynomials.

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.

How the Dirac delta is a generalization of the Kronecker delta

Let ξ be a Hermitian operator on a finite-dimensional Hilbert space, i.e. ξ is a complex-valued, n-by-n Hermitian matrix. Then it can be shown that ξ is unitarily diagonalizable, i.e. that Cⁿ has an orthonormal basis consisting entirely of the eigenvectors of ξ.

In the bra-ket notation, use $|\xi '\rangle$ to denote an eigenvector of ξ, and ξ' to denote its corresponding eigenvalue. For simplicity's sake, assume furthermore that each eigenvalue ξ' is distinct. Also note that, since the operator ξ is Hermitian, each ξ' is real-valued.

Then, because the collection of eigenvectors $|\xi '\rangle$ is an orthonormal basis, we may expand any arbitrary vector $|P\rangle$ uniquely as a linear combination of the vectors $|\xi '\rangle$. Because, by assumption, each eigenvalue of ξ corresponds to a single eigenvector, this expansion can be thought of as a function ψ that associates each eigenvalue ξ' with a complex number ψ(ξ'). (This is in fact the definition of a wave function.) Symbolically,

$|P\rangle =\sum _{\xi '}\psi (\xi ')|\xi '\rangle$

i.e.,

$\psi (\xi ')=\langle \xi '|P\rangle .$

Suppose now that $|P\rangle =|\xi ''\rangle$ is some specific eigenvector of ξ. If we consider its associated mapping ψ, it can be easily seen that ψ(x) = 0 whenever x ≠ ξ" , and ψ(ξ") = 1. In other words, ψ(x) = δ_xξ" where δ is the Kronecker delta.

If, on the other hand, ξ is a Hermitian operator on an infinite-dimensional Hilbert space, the analysis becomes more complicated. The eigenvalues of ξ may lie in a continuous range. For example it may be the case that every number ξ' in [0,1] is an eigenvalue corresponding to some eigenvector $|\xi '\rangle$.

In this case, we can no longer expand any arbitrary vector $|P\rangle$ as a sum of eigenvectors. Instead, the expansion will take the form of an integral such as

$|P\rangle =\int \psi (\xi ')|\xi '\rangle d\xi '$

If we again consider $|P\rangle =|\xi ''\rangle$ to be a specific eigenvector, this time we see it is not enough for ψ(x) = δ_xξ" as in the case when the eigenvalues were discrete, since the integral of a function that is zero almost everywhere is simply zero. No matter by what number we scale δ_xξ", the integral above will still vanish. Thus we need ψ to be something stronger than a delta function, and stronger than a function at all.

The answer comes in the form of the Dirac delta. If we set ψ(x) = δ(x - ξ"), where this time δ denotes the Dirac delta, the integral above evaluates to $|\xi ''\rangle$ as we wanted.

It is in the above sense that the Dirac delta generalizes the concept of the Kronecker delta. In summary: in the continuous and discrete cases respectively, they succinctly describe the wave function of an eigenvector of a Hermitian operator.

See also

• Dirac comb
• Logarithmically-spaced Dirac comb
• Green's function
• Dirac measure

External links

• Delta Function on MathWorld
• Dirac Delta Function on PlanetMath
• Integral and Series Representations of the Dirac Delta Function
• The Dirac delta measure is a hyperfunction
• We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure
• Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure.