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Dirac delta function PDF Print E-mail
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Dirac delta function

Dirac delta function
Probability density function
Plot of the Dirac delta function
Schematic representation of the Dirac delta function for x0 = 0. A line with 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
Plot of the Heaviside step 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\,
Kurtosis (undefined)
Entropy -\infty
mgf e^{tx_0}
Char. func. e^{itx_0}

The Dirac delta function, 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, the value zero elsewhere. The integral from minus infinity to plus infinity is 1. The discrete analog of the delta "function" is the Kronecker delta 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 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 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, 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 (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.

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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.


The delta function as a measure

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

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

for all continuous f.

As distributions, the Heaviside step function is an antiderivative 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

\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 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 \delta : \mathbb{R} \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb{R}) as a distribution δ(ξ) 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^{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:

\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|}

This concept may be generalized to:

\delta(g(x)) = \sum_{i}\frac{\delta(x-x_i)}{|g'(x_i)|}

where xi are the roots of g(x). In the integral form it is equivalent to

\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^nr = \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-shifted Dirac delta is given by:

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

Thus, the delta function is said to "sift out" the function f(t)\, at the value t=T\,, when integrated over all time.
Similarly, the convolution:

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-shifted Dirac delta is to time-shift f(t)\, by the same amount.


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.


Laplace transform

The direct Laplace transform of the delta function is:

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

as a curious identity using Euler's formula 2cos(as) = e ias + eias we got 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=2[\delta (t+ia) +\delta (t-ia)] and a similar identity holds for sin(as).

Derivatives of the delta function

The derivative of the Dirac delta function (also called a doublet) is the distribution δ' defined by

\delta'[\phi] = -\phi'(0)\,

for every test function \phi \. From this it follows that

x\delta'(x)=-\delta(x)\,

The n-th derivative δ(n) is given by

\delta^{(n)}[\phi] = (-1)^n \phi^{(n)}(0)\,

The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials.


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 δa(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 (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(ax) 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 δφ(ax) 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(ax). 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, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis.

From Wikipedia, the free encyclopedia

 





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