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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 \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. 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 ...
\delta 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 \deltas 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 \operatorname\,\,(t), where the period is omitted, represents a Dirac comb of unit period. This implies \operatorname_(t) \ = \frac\operatorname\ \!\!\!\left(\frac\right). 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: \operatorname_(t) = \frac\sum_^ e^. 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 1, or, alternatively, by using the ''rep'' operator (performing periodization) applied to the Dirac delta \delta. Formally, this yields (; ) \operatorname_T \ = \operatorname_T = \operatorname_T \, where \operatorname_T \ \triangleq \sum_^\infty \, f(kT) \, \delta(t - kT) and \operatorname_T \ \triangleq \sum_^\infty \, g(t - kT). 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 f(t) by ''multiplication'' with \operatorname_, and on the other hand it also allows the periodization of f(t) by ''convolution'' with \operatorname_T (). 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 \delta(t) = \frac\ \delta\!\left(\frac\right). for positive real numbers a, it follows that: \operatorname_\left(t\right) = \frac \operatorname\,\!\left( \frac \right), \operatorname_\left(t\right) = \frac \operatorname\,\!\left(\frac\right) = \frac \operatorname_\!\!\left(\frac\right). Note that requiring positive scaling numbers a instead of negative ones is not a restriction because the negative sign would only reverse the order of the summation within \operatorname_, which does not affect the result.


Fourier series

It is clear that \operatorname_(t) is periodic with period T. That is, \operatorname_(t + T) = \operatorname_(t) for all ''t''. The complex Fourier series for such a periodic function is \operatorname_(t) = \sum_^ c_n e^, where the Fourier coefficients are (symbolically) \begin c_n &= \frac \int_^ \operatorname_(t) e^\, dt \quad ( -\infty < t_0 < +\infty ) \\ &= \frac \int_^ \operatorname_(t) e^\, dt \\ &= \frac \int_^ \delta(t) e^\, dt \\ &= \frac e^ \\ &= \frac. \end All Fourier coefficients are 1/''T'' resulting in \operatorname_(t) = \frac\sum_^ \!\!e^. When the period is one unit, this simplifies to \operatorname\ \!(x) = \sum_^ \!\!e^. 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 \mathcal expressed in frequency domain (Hz) the Dirac comb \operatorname_ of period T transforms into a rescaled Dirac comb of period 1/T, i.e. for :\mathcal\left f \right\xi)= \int_^ dt f(t) e^, :\mathcal\left \operatorname_ \right\xi) = \frac \sum_^ \delta(\xi-k \frac) = \frac \operatorname_(\xi) ~ is proportional to another Dirac comb, but with period 1/T in frequency domain (radian/s). The Dirac comb \operatorname of unit period T=1 is thus an eigenfunction of \mathcal 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 ...
1. This result can be established () by considering the respective Fourier transforms S_(\xi)=\mathcal _\xi) of the family of functions s_(x) defined by :s_(x) = \tau^ e^ \sum_^ e^. Since s_(x) is a convergent series of Gaussian functions, and Gaussians transform into Gaussians, each of their respective Fourier transforms S_\tau(\xi) also results in a series of Gaussians, and explicit calculation establishes that :S_(\xi) = \tau^ \sum_^ e^ e^. The functions s_(x) and S_\tau(\xi) are thus each resembling a periodic function consisting of a series of equidistant Gaussian spikes \tau^ e^ and \tau^ e^ whose respective "heights" (pre-factors) are determined by slowly decreasing Gaussian envelope functions which drop to zero at infinity. Note that in the limit \tau \rightarrow 0 each Gaussian spike becomes an infinitely sharp Dirac impulse centered respectively at x=n and \xi=m for each respective n and m, and hence also all pre-factors e^ in S_(\xi) eventually become indistinguishable from e^. Therefore the functions s_(x) and their respective Fourier transforms S_(\xi) 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: :\lim_ s_(x) = \operatorname(), and \lim_ S_(\xi) = \operatorname(). Since S_=\mathcal _/math>, we obtain in this limit the result to be demonstrated: :\mathcal operatorname \operatorname. The corresponding result for period T can be found by exploiting the scaling property of 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 ...
, :\mathcal operatorname_T \frac \operatorname_. Another manner to establish that the Dirac comb transforms into another Dirac comb starts by examining continuous Fourier transforms of periodic functions in general, and then specialises to the case of the Dirac comb. In order to also show that the specific rule depends on the convention for the Fourier transform, this will be shown using angular frequency with \omega=2\pi \xi : for any periodic function f(t)=f(t+T) its Fourier transform :\mathcal\left f \right\omega)=F(\omega) = \int_^ dt f(t) e^ obeys: :F(\omega) (1 - e^) = 0 because Fourier transforming f(t) and f(t+T) leads to F(\omega) and F(\omega) e^. This equation implies that F(\omega)=0 nearly everywhere with the only possible exceptions lying at \omega= k \omega_0, with \omega_0=2\pi / T and k \in \mathbb. When evaluating the Fourier transform at F(k \omega_0) the corresponding Fourier series expression times a corresponding delta function results. For the special case of the Fourier transform of the Dirac comb, the Fourier series integral over a single period covers only the Dirac function at the origin and thus gives 1/T for each k. This can be summarised by interpreting the Dirac comb as a limit of the Dirichlet kernel such that, at the positions \omega= k \omega_0, all exponentials in the sum \sum\nolimits_^ e^ point into the same direction and add constructively. In other words, the continuous Fourier transform of periodic functions leads to :F(\omega)= 2 \pi \sum_^ c_k \delta(\omega-k\omega_0) with \omega_0=2 \pi/T, and :c_k = \frac \int_^ dt f(t) e^. The
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 '' ...
coefficients c_k=1/T for all k when f \rightarrow \operatorname_, i.e. :\mathcal\left \operatorname_ \right\omega) = \frac \sum_^ \delta(\omega-k \frac) is another Dirac comb, but with period 2 \pi/T in angular frequency domain (radian/s). As mentioned, the specific rule depends on the convention for the used Fourier transform. Indeed, when using the scaling property of the Dirac delta function, the above may be re-expressed in ordinary frequency domain (Hz) and one obtains again: \operatorname_(t) \stackrel \frac \operatorname_(\xi) = \sum_^\!\! e^, such that the unit period Dirac comb transforms to itself: \operatorname\ \!(t) \stackrel \operatorname\ \!(\xi). Finally, the Dirac comb is also an eigenfunction of the unitary continuous Fourier transform in
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
space to the eigenvalue 1 when T=\sqrt because for the unitary Fourier transform :\mathcal\left f \right\omega)=F(\omega) = \frac\int_^ dt f(t) e^, the above may be re-expressed as \operatorname_(t) \stackrel \frac \operatorname_(\omega) = \frac\sum_^ \!\!e^.


Sampling and aliasing

Multiplying any function by a Dirac comb transforms it into a train of impulses with integrals equal to the value of the function at the nodes of the comb. This operation is frequently used to represent sampling. (\operatorname_ x)(t) = \sum_^ \!\! x(t)\delta(t - kT) = \sum_^\!\! x(kT)\delta(t - kT). Due to the self-transforming property of the Dirac comb and the convolution theorem, this corresponds to convolution with the Dirac comb in the frequency domain. \operatorname_ x \ \stackrel\ \frac\operatorname_\frac * X Since convolution with a delta function \delta(t-kT) is equivalent to shifting the function by kT, convolution with the Dirac comb corresponds to replication or periodic summation: : (\operatorname_\! * X)(f) =\! \sum_^ \!\!X\!\left(f - \frac\right) This leads to a natural formulation of the Nyquist–Shannon sampling theorem. If the spectrum of the function x contains no frequencies higher than B (i.e., its spectrum is nonzero only in the interval (-B, B)) then samples of the original function at intervals \tfrac are sufficient to reconstruct the original signal. It suffices to multiply the spectrum of the sampled function by a suitable
rectangle function The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname(t) = \Pi(t) = \left\{\begin{array}{rl ...
, which is equivalent to applying a brick-wall lowpass filter. : \operatorname_ x\ \ \stackrel\ \ 2B\, \operatorname_ * X : \frac\Pi\left(\frac\right) (2B \,\operatorname_ * X) = X In time domain, this "multiplication with the rect function" is equivalent to "convolution with the sinc function" (, p.33-34). Hence, it restores the original function from its samples. This is known as the Whittaker–Shannon interpolation formula. Remark: Most rigorously, multiplication of the rect function with a generalized function, such as the Dirac comb, fails. This is due to undetermined outcomes of the multiplication product at the interval boundaries. As a workaround, one uses a Lighthill unitary function instead of the rect function. It is smooth at the interval boundaries, hence it yields determined multiplication products everywhere, see , p.62, Theorem 22 for details.


Use in directional statistics

In
directional statistics Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R''n''), axes ( lines through the origin in R''n'') or rotations in R''n''. ...
, the Dirac comb of period 2\pi is equivalent to a wrapped Dirac delta function and is the analog 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 ...
in linear statistics. In linear statistics, the random variable (x) is usually distributed over the real-number line, or some subset thereof, and the probability density of x is a function whose domain is the set of real numbers, and whose integral from -\infty to +\infty is unity. In directional statistics, the random variable (\theta) is distributed over the unit circle, and the probability density of \theta is a function whose domain is some interval of the real numbers of length 2\pi and whose integral over that interval is unity. Just as the integral of the product of a Dirac delta function with an arbitrary function over the real-number line yields the value of that function at zero, so the integral of the product of a Dirac comb of period 2\pi with an arbitrary function of period 2\pi over the unit circle yields the value of that function at zero.


See also

*
Comb filter In signal processing, a comb filter is a filter implemented by adding a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly space ...
* Frequency comb * Poisson summation formula


References


Further reading

* . * * . * . {{ProbDistributions, continuous-infinite Special functions Generalized functions Signal processing Directional statistics