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 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.
Encyclopedia
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 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.
Despite its name, the delta function is not a function. One reason for this is because the functions
f = δ and
g = 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 . 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 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,
-
and which is also constrained to satisfy the identity
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, behaves as a delta function in the limit of , yet this function does not approach zero for values of x outside the origin.
The defining characteristic
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 implies that has dimensions reciprocal to those of
dx.
The delta function as a measure
As a measure, if , and otherwise. Then,
-
for all continuous .
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
-
for every test function . It is a distribution with compact support . 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 as a distribution whose
indefinite integral is the function
usually called the
Heaviside step function or commonly the
unit step function. That is, it satisfies the integral equation
for all real numbers
x.
Delta function of more complicated arguments
A helpful identity is the scaling property:
and so
The scaling property may be generalized to:
-
where x
i are the roots of g. Thus, for example
In the integral form the generalized scaling property may be written as
-
In an n-dimensional space with position vector , this is generalized to:
-
where the integral on the right is over , the
n-1 dimensional surface defined by .
The integral of the time-shifted Dirac delta is given by
:Thus, the delta function is said to "sift out" the function at the value , when integrated over all time.
Similarly, the convolution
:means that the effect of convolving with the time-shifted Dirac delta is to time-shift by the same amount.
Fourier transform
Using Fourier transforms, one has
and therefore:
which is a statement of the orthogonality property for the Fourier kernel.
Laplace transform
The direct
Laplace transform of the delta function is:
-
a curious identity using Euler's formula allows us to find the Laplace inverse transform for the cosine
and a similar identity holds for .
Derivatives of the delta function
The derivative of the Dirac delta function is the distribution δ' defined by
-
for every test function . From this it follows that
-
The
n-th derivative δ
is given by
-
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
where is sometimes called a
nascent delta function. This limit is in the sense that
for all continuous .
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 . There the condition is made that the limiting sequence should be of positive functions.
Some nascent delta functions are:
Note: If δ is a nascent delta function which is a probability distribution over the whole real line
then another nascent delta function δφ can be built from its characteristic function as follows:
where
is the characteristic function of the nascent delta function δ. 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 and discrete time signal analysis.
See also
External links
