Johnny Deng's Column: Dirac delta function

The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. The integral of the Dirac delta from any negative limit to any positive limit is 1. The discrete analog of the Dirac delta "function" is the Kronecker delta which is sometimes called a delta function even though it is a discrete sequence. It is also often referred to as the discrete unit impulse function. Note that the Dirac delta is not strictly a function, but a distribution that is also a measure.

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,

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

and which is also constrained to satisfy the identity

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

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 which differ from the above conceptualization. For example, $sinc(x / a) / a$ (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.

The defining characteristic

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

where f is a suitable test function, 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 $δ(x)$ implies that $δ(x)$ has dimensions reciprocal to those of dx.

Fourier transform

Using Fourier transforms, one has

$\int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f)$

and therefore:

$\int_{-\infty}^\infty e^{i 2\pi f_1 t} \left[e^{i 2\pi f_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (f_2 - f_1) t} \,dt = \delta(f_2 - f_1)$

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

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 $δ a (x)$ is sometimes called a nascent delta function. 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 $f$ .

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
$\delta_a(x) = \frac{1}{\pi} \frac{a}{a^2 + x^2} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} k x-\|ak\|}\;dk$	Limit of a Cauchy distribution
$\delta_a(x)=\frac{e^{-\|x/a\|}}{2a} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ikx}}{1+a^2k^2}\,dk$	Cauchy $\varphi$ (see note below)
$\delta_a(x)= \frac{\textrm{rect}(x/a)}{a} =\frac{1}{2\pi}\int_{-\infty}^\infty \textrm{sinc} \left( \frac{a k}{2 \pi} \right) e^{ikx}\,dk$	Limit of a rectangular function
$\delta_a(x)=\frac{1}{\pi x}\sin\left(\frac{x}{a}\right) =\frac{1}{2\pi}\int_{-1/a}^{1/a} \cos (k x)\;dk$	rectangular function $\varphi$ (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

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.

Dirac delta function
Probability density function 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	$x_0\,$ location (real)
Support	$x \in [x_0; x_0]$
Probability density function (pdf)	$\delta(x-x_0)\,$
Cumulative distribution function (cdf)	$H(x-x_0)\,$ (Heaviside)
Mean	$x_0\,$
Median	$x_0\,$
Mode	$x_0\,$
Variance	$0\,$
Skewness	$0\,$
Excess kurtosis	(undefined)
Entropy	$-\infty$
Moment-generating function (mgf)	$e^{tx_0}$
Characteristic function	$e^{itx_0}$

Johnny Deng's Column

Thursday 2 August 2007

Dirac delta function

Definitions

Fourier transform

Representations of the delta function

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