Dirac delta function

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Template:Probability distribution

The Dirac delta function, sometimes 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 degenerate distribution which is sometimes known as a delta function.

Contents

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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 can be 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 as defined in the strictest mathematical sense. 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 ostensibly 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. Precise 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 is defined as:

<math>\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases} ; \int_{-\infty}^\infty \delta(x) \, dx

= 1</math>

This function has great utility in sifting out a particular value of a function from an integral, as can be seen in this brief example where <math>f(0)</math> is sifted out of the integrand.

<math>\int_{-\infty}^\infty f(x) \, \delta(x) \, dx

= f(0)</math>

is valid for any continuous function <math>f</math>.

However, there is no actual function <math>\delta(x)</math> with this property. The Dirac delta is not a function; but it can be usefully treated as a distribution, as well as a measure.


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The Delta Function as a Measure

As a measure, <math>\delta (A)=1</math> if <math>0\in A</math>, and <math>\delta (A)=0</math> otherwise. Then,

<math>\int_{-\infty}^\infty f(x) \, d\delta(x)

= f(0)</math>

for all continuous <math>f</math>.

As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution.

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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

<math>\delta[\phi] = \phi(0)\,</math>

for every test function <math>\phi \ </math>. 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 <math>\delta : \mathbb{R} \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb{R})</math> as a distribution <math>\delta ( \xi )</math> whose indefinite integral is the function

<math>h : \mathbb{R} \ni \xi \longrightarrow \frac{1+{\rm sgn} \, \xi }{2} \in \mathbb{R}, </math>

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

<math>

\int^{x}_{-\infin} \delta (t) dt = h(x) \equiv \frac{1+{\rm sgn}(x) }{2} </math>

for all real numbers x.

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Delta function of more complicated arguments

A helpful identity is the scaling property:

<math>\int_{-\infty}^\infty \delta(\alpha x)\,dx

=\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}</math>

and so

<math>\delta(\alpha x) = \frac{\delta(x)}{|\alpha|}</math>

This concept may be generalized to:

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

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

<math>

\int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|} </math>

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

<math>

\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 </math>

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


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

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

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

<math>f(t) * \delta(t-T) = \int\limits_{-\infty}^\infty f(\tau) \cdot \delta(t-T-\tau) d\tau = f(t-T)</math>

means that the effect of convolving with the time-shifted Dirac delta is to time-shift <math>f(t)\,</math> by the same amount.


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Fourier transform

From the Fourier transform table, row 13, we have:


<math>\int_{-\infty}^\infty 1 \cdot e^{-j 2\pi f t}\,dt = \delta(f)</math>

and therefore:

<math>\int_{-\infty}^\infty e^{j 2\pi f_1 t} \cdot \left[e^{j 2\pi f_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-j 2\pi (f_2 - f_1) t} \,dt = \delta(f_2 - f_1)</math>

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


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Derivatives of the delta function

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

<math>\delta'[\phi] = -\phi'(0)\,</math>

for every test function <math>\phi \ </math>. From this it follows that

<math>x\delta'(x)=-\delta(x)\,</math>

The n-th derivative δ(n) is given by

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

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

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Representations of the delta function

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

<math>

\delta (x) = \lim_{a\to 0} \delta_a(x), </math>

where <math>\delta_a(x)</math> is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. 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:

<math>\delta_a(x) = \frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}</math> Limit of a Normal distribution
<math>\delta_a(x) = \frac{1}{\pi} {a \over a^2 + x^2}

=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} k x-|ak|}\;dk </math>

Limit of a Cauchy distribution
x/a|}}{2a}

=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ikx}}{1+a^2k^2}\,dk</math>

Cauchy <math>\varphi</math>(see note below)
<math>\delta_a(x)= \frac{\textrm{rect}(x/a)}{a}

=\frac{1}{2\pi}\int_{-\infty}^\infty \textrm{sinc}(ak/2)e^{ikx}\,dk </math>

Limit of a rectangular function
<math>

\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

</math>

rectangular function <math>\varphi</math>(see note below)
<math>

\delta_a(x)=\partial_x \frac{1}{1+\mathrm{e}^{-x/a}} 

            =-\partial_x \frac{1}{1+\mathrm{e}^{x/a}}

</math>

<math>

\delta_a(x)=\frac{a}{\pi x^2}\sin^2\left(\frac{x}{a}\right) </math>

<math>

\delta_a(x) = \frac{1}{a}A_i\left(\frac{x}{a}\right) </math>

Limit of the Airy function
<math> 
\delta_a(x) =

\frac{1}{a}J_{1/a} \left(\frac{x+1}{a}\right) </math>

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 δφ(a, x) can be built from its characteristic function as follows:

<math>\delta_\varphi(a,x)=\frac{1}{2\pi}~\frac{\varphi(1/a,x)}{\delta(1/a,0)}</math>

where

<math>\varphi(a,k)=\int_{-\infty}^\infty \delta(a,x)e^{-ikx}\,dx</math>

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.

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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.

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See also

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External links

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