In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Dirac comb (also known as shah function, impulse train or sampling function) is a
periodic function with the formula
for some given period
.
Here ''t'' is a real variable and the sum extends over all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''k.'' The
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
and the Dirac comb are
tempered distributions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
.
The graph of the function resembles a
comb
A comb is a tool consisting of a shaft that holds a row of teeth for pulling through the hair to clean, untangle, or style it. Combs have been used since prehistoric times, having been discovered in very refined forms from settlements dating ba ...
(with the
s as the comb's ''teeth''), hence its name and the use of the comb-like
Cyrillic
The Cyrillic script ( ), Slavonic script or the Slavic script, is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking co ...
letter
sha (Ш) to denote the function.
The symbol
, where the period is omitted, represents a Dirac comb of unit period. This implies
Because the Dirac comb function is periodic, it can be represented as a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
based on the
Dirichlet kernel:
The Dirac comb function allows one to represent both
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
and
discrete phenomena, such as
sampling and
aliasing, in a single framework of
continuous Fourier analysis on tempered distributions, without any reference to Fourier series. The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a Dirac comb is another Dirac comb. Owing to the
Convolution Theorem on tempered distributions which turns out to be the
Poisson summation formula, in
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the Dirac comb allows modelling sampling by ''
multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
'' with it, but it also allows modelling periodization by ''
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
'' with it.
[; 1st ed. 1965, 2nd ed. 1978.]
Dirac-comb identity
The Dirac comb can be constructed in two ways, either by using the ''comb''
operator (performing
sampling) applied to the function that is constantly
, or, alternatively, by using the ''rep'' operator (performing
periodization) applied to the
Dirac delta . Formally, this yields (; )
where
and
In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, this property on one hand allows
sampling a function
by ''multiplication'' with
, and on the other hand it also allows the
periodization of
by ''convolution'' with
().
The Dirac comb identity is a particular case of the
Convolution Theorem for tempered distributions.
Scaling
The scaling property of the Dirac comb follows from the properties of the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
.
Since
[.] for positive real numbers
, it follows that:
Note that requiring positive scaling numbers
instead of negative ones is not a restriction because the negative sign would only reverse the order of the summation within
, which does not affect the result.
Fourier series
It is clear that
is periodic with period
. That is,
for all ''t''. The complex Fourier series for such a periodic function is
where the Fourier coefficients are (symbolically)
All Fourier coefficients are 1/''T'' resulting in
When the period is one unit, this simplifies to
Remark: Most rigorously, Riemann or Lebesgue integration over any products including a Dirac delta function yields zero. For this reason, the integration above (Fourier series coefficients determination) must be understood "in the generalized functions sense". It means that, instead of using the characteristic function of an interval applied to the Dirac comb, one uses a so-called Lighthill unitary function as cutout function, see , p.62, Theorem 22 for details.
Fourier transform
The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a Dirac comb is also a Dirac comb. For the Fourier transform
expressed in
frequency domain (Hz) the Dirac comb
of period
transforms into a rescaled Dirac comb of period
i.e. for
:
:
is proportional to another Dirac comb, but with period
in frequency domain (radian/s). The Dirac comb
of unit period
is thus an
eigenfunction of
to the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
This result can be established () by considering the respective Fourier transforms
of the family of functions
defined by
:
Since
is a convergent series of
Gaussian functions, and Gaussians
transform into
Gaussians, each of their respective Fourier transforms
also results in a series of Gaussians, and explicit calculation establishes that
:
The functions
and
are thus each resembling a periodic function consisting of a series of equidistant Gaussian spikes
and
whose respective "heights" (pre-factors) are determined by slowly decreasing Gaussian envelope functions which drop to zero at infinity. Note that in the limit
each Gaussian spike becomes an infinitely sharp
Dirac impulse centered respectively at
and
for each respective
and
, and hence also all pre-factors
in
eventually become indistinguishable from
. Therefore the functions
and their respective Fourier transforms
converge to the same function and this limit function is a series of infinite equidistant Gaussian spikes, each spike being multiplied by the same pre-factor of one, i.e. the Dirac comb for unit period:
:
and
Since