Editing Dirac comb

{{Short description|Periodic distribution ("function") of "point-mass" Dirac delta sampling}}
{{use dmy dates|date=August 2024}}
[[Image:Dirac comb.svg|thumb|300px|The graph of the Dirac comb function is an infinite series of [[Dirac delta function]]s spaced at intervals of ''T'']]

In [[mathematics]], a '''Dirac comb''' (also known as '''sha function''', '''impulse train''' or '''sampling function''') is a [[periodic function|periodic]] [[Function (mathematics)|function]] with the formula
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) \ :=  \sum_{k=-\infty}^{\infty} \delta(t - k T) </math>
for some given period <math>T</math>.<ref name=":0">{{Cite web |title=The Dirac Comb and its Fourier Transform |url=https://dspillustrations.com/pages/posts/misc/the-dirac-comb-and-its-fourier-transform.html#:~:text=The%20Dirac%20Comb%20function&text=CT(t)=T,(t%E2%88%92nT).&text=As%20shown,%20the%20Dirac%20comb,distinct%20impulses%20are%20T%20apart.&text=or%20equivalently%20as%20a%20sum,%CF%80nt/T). |access-date=2022-06-28 |website=dspillustrations.com}}</ref> Here ''t'' is a real variable and the sum extends over all [[integer]]s ''k.'' The [[Dirac delta function]] <math>\delta</math> and the Dirac comb are [[Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform|tempered distributions]].<ref name="Schwartz 1951">{{cite book|last=Schwartz |first=L. |title=Théorie des distributions |volume=I-II |year=1951 |publisher=Hermann |location=Paris |authorlink=Laurent Schwartz}}</ref><ref name="Strichartz 1994">{{cite book |last=Strichartz |first=R. |title=A Guide to Distribution Theory and Fourier Transforms |year=1994 |publisher=CRC Press |isbn=0-8493-8273-4}}</ref> The graph of the function resembles a [[comb]] (with the <math>\delta</math>s as the comb's ''teeth''), hence its name and the use of the comb-like [[Cyrillic script|Cyrillic]] letter [[Sha (Cyrillic)|sha]] (Ш) to denote the function.

The symbol <math>\operatorname{\text{Ш}}\,\,(t)</math>, where the period is omitted, represents a Dirac comb of unit period. This implies<ref name=":0" />
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) \ = \frac{1}{T}\operatorname{\text{Ш}}\ \!\!\!\left(\frac{t}{T}\right).</math>

Because the Dirac comb function is periodic, it can be represented as a [[Fourier series]] based on the [[Dirichlet kernel]]:<ref name=":0" />
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} e^{i  2 \pi n \frac{t}{T}}.</math>

The Dirac comb function allows one to represent both [[Continuous function|continuous]] and [[Discrete mathematics|discrete]] phenomena, such as [[sampling (signal processing)|sampling]] and [[aliasing]], in a single framework of [[Fourier transform|continuous Fourier analysis]] on tempered distributions, without any reference to Fourier series.  The [[Fourier transform]] of a Dirac comb is another Dirac comb. Owing to the [[Convolution_theorem#Convolution theorem for tempered distributions|Convolution Theorem]] on tempered distributions which turns out to be the [[Poisson summation formula]], in [[signal processing]], the Dirac comb allows modelling sampling by ''[[multiplication]]'' with it, but it also allows modelling periodization by ''[[convolution]]'' with it.<ref name="Bracewell 1986">{{cite book |last1=Bracewell|first1=R. N.|title=The Fourier Transform and Its Applications|publisher=McGraw-Hill|edition=revised| year=1986 |orig-year=1st ed. 1965, 2nd ed. 1978}}</ref>

== Dirac-comb identity ==

The Dirac comb can be constructed in two ways, either by using the ''comb'' [[Operator (mathematics)|operator]] (performing [[sampling (signal processing)|sampling]]) applied to the function that is constantly <math>1</math>, or, alternatively, by using the ''rep'' operator (performing [[periodic summation|periodization]]) applied to the [[Dirac delta]] <math>\delta</math>. Formally, this yields the following:{{sfn|Woodward|1953}}{{sfn|Brandwood|2003}}
<math display="block">\operatorname{comb}_T \{1\} = \operatorname{\text{Ш}}_T = \operatorname{rep}_T \{\delta \}, </math>
where
<math display="block">
 \operatorname{comb}_T \{f(t)\} \triangleq \sum_{k=-\infty}^\infty \, f(kT) \, \delta(t - kT)
</math> and <math display="block">
 \operatorname{rep}_T \{g(t)\} \triangleq \sum_{k=-\infty}^\infty \, g(t - kT).
</math>

In [[signal processing]], this property on one hand allows [[sampling (signal processing)|sampling]] a function <math>f(t)</math> by ''multiplication'' with <math>\operatorname{\text{Ш}}_{\ T}</math>, and on the other hand it also allows the [[periodic summation|periodization]] of <math>f(t)</math> by ''convolution'' with <math>\operatorname{\text{Ш}}_T</math>.{{sfn|Bracewell|1986}}
The Dirac comb identity is a particular case of the [[Convolution_theorem#Convolution theorem for tempered distributions| Convolution Theorem]] for tempered distributions.

== Scaling ==

The scaling property of the Dirac comb follows from the properties of the [[Dirac delta function]]. 
Since <math>\delta(t) = \frac{1}{a}\  \delta\!\left(\frac{t}{a}\right)</math><ref name="Rahman 2011">{{cite book | first=M.| last=Rahman | year=2011|title=Applications of Fourier Transforms to Generalized Functions|publisher=WIT Press |location=Southampton |isbn=978-1-84564-564-9}}</ref> for positive real numbers <math>a</math>, it follows that:
<math display="block"> \operatorname{\text{Ш}}_{\ T}\left(t\right) = \frac{1}{T} \operatorname{\text{Ш}}\,\!\left( \frac{t}{T} \right), </math>
<math display="block">\operatorname{\text{Ш}}_{\ aT}\left(t\right) = \frac{1}{aT} \operatorname{\text{Ш}}\,\!\left(\frac{t}{aT}\right) = \frac{1}{a} \operatorname{\text{Ш}}_{\ T}\!\!\left(\frac{t}{a}\right).</math>
Note that requiring positive scaling numbers <math>a</math> instead of negative ones is not a restriction because the negative sign would only reverse the order of the summation within <math>\operatorname{\text{Ш}}_{\ T}</math>, which does not affect the result.

== Fourier series ==
{{see also|Dirichlet kernel}}
It is clear that <math>\operatorname{\text{Ш}}_{\ T}(t)</math> is periodic with period <math>T</math>. That is,
<math display="block">\operatorname{\text{Ш}}_{\ T}(t + T) = \operatorname{\text{Ш}}_{\ T}(t)</math>
for all ''t''. The complex Fourier series for such a periodic function is
<math display="block"> \operatorname{\text{Ш}}_{\ T}(t) = \sum_{n=-\infty}^{+\infty} c_n e^{i 2 \pi n \frac{t}{T}}, </math>
where (using [[Distribution_(mathematics)|distribution theory]]) the Fourier coefficients are
<math display="block">\begin{align}
c_n &= \frac{1}{T} \int_{t_0}^{t_0 + T} \operatorname{\text{Ш}}_{\ T}(t) e^{-i 2 \pi n \frac{t}{T}}\, dt \quad ( -\infty < t_0 < +\infty ) \\
    &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \operatorname{\text{Ш}}_{\ T}(t) e^{-i 2 \pi n \frac{t}{T}}\, dt \\
    &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \delta(t) e^{-i 2 \pi n \frac{t}{T}}\, dt \\
    &= \frac{1}{T} e^{-i 2 \pi n \frac{0}{T}} \\
    &= \frac{1}{T}.
\end{align}</math>

All Fourier coefficients are 1/''T'' resulting in
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} \!\!e^{i 2 \pi n \frac{t}{T}}.</math>

When the period is one unit, this simplifies to 
<math display="block">\operatorname{\text{Ш}}\ \!(x) = \sum_{n=-\infty}^{\infty} \!\!e^{i 2 \pi n x}.</math>
This is a [[divergent series]], when understood as a series of ordinary complex numbers, but becomes convergent in the sense of [[distribution (mathematics)|distributions]].

A "square root" of the Dirac comb is employed in some applications to physics, specifically:<ref>{{Cite book |last=Schleich |first=Wolfgang |title=Quantum optics in phase space |date=2001 |publisher=Wiley-VCH |isbn=978-3-527-29435-0 |edition=1st |pages=683–684}}</ref><math display="block">\delta_N^{(1 / 2)}(\xi) = \frac{1}{\sqrt{NT}} \sum_{\nu=0}^{N-1} e^{-i \frac{2\pi}{T}\xi \nu}, \quad 
\lim_{N \rightarrow \infty}\left|\delta_N^{(1 / 2)}(\xi)\right|^2= \sum_{k=-\infty}^{\infty} \delta(\xi - kT).</math>
However this is not a distribution in the ordinary sense.

== Fourier transform ==
The [[continuous Fourier transform|Fourier transform]] of a Dirac comb is also a Dirac comb. For the Fourier transform <math>\mathcal{F}</math> expressed in [[Fourier transform#Other conventions|frequency domain]] (Hz) the Dirac comb <math>\operatorname{\text{Ш}}_{T}</math> of period <math>T</math> transforms into a rescaled Dirac comb of period <math>1/T,</math> i.e. for
:<math>\mathcal{F}\left[ f \right](\xi)= \int_{-\infty}^{\infty} dt f(t) e^{- 2 \pi i\xi t}, </math> 
:<math>\mathcal{F}\left[ \operatorname{\text{Ш}}_{T} \right](\xi) = \frac{1}{T} \sum_{k=-\infty}^{\infty} \delta(\xi-k \frac{1}{T}) = \frac{1}{T} \operatorname{\text{Ш}}_{\ \frac{1}{T}}(\xi) ~</math>
is proportional to another Dirac comb, but with period <math>1/T</math> in frequency domain (radian/s). The Dirac comb <math>\operatorname{\text{Ш}}</math> of unit period <math>T=1</math> is thus an [[continuous Fourier transform#Eigenfunctions|eigenfunction]] of <math>\mathcal{F}</math> to the [[eigenvalue]] <math>1.</math>

This result can be established{{sfn|Bracewell|1986}} by considering the respective Fourier transforms <math>S_{\tau}(\xi)=\mathcal{F}[s_{\tau}](\xi)</math> of the family of functions <math>s_{\tau}(x)</math> defined by

:<math>s_{\tau}(x) = \tau^{-1} e^{-\pi \tau^2 x^2} \sum_{n=-\infty}^{\infty} e^{-\pi \tau^{-2} ( x-n)^{2} }.</math>

Since <math>s_{\tau}(x)</math> is a convergent series of [[Gaussian function|Gaussian functions]], and Gaussians [[Fourier transform#Square-integrable functions, one-dimensional|transform]] into [[Normal distribution#Fourier transform and characteristic function|Gaussians]], each of their respective Fourier transforms <math>S_\tau(\xi)</math> also results in a series of Gaussians, and explicit calculation establishes that

:<math>S_{\tau}(\xi) = \tau^{-1}  \sum_{m=-\infty}^{\infty}  e^{-\pi \tau^2 m^2} e^{-\pi \tau^{-2} ( \xi-m)^{2} }.</math>

The functions <math>s_{\tau}(x)</math> and <math>S_\tau(\xi)</math> are thus each resembling a periodic function consisting of a series of equidistant Gaussian spikes <math>\tau^{-1} e^{-\pi \tau^{-2} ( x-n)^{2} }</math> and <math>\tau^{-1} e^{-\pi \tau^{-2} ( \xi-m)^{2} }</math> whose respective  "heights" (pre-factors)  are determined by slowly decreasing Gaussian envelope functions which drop to zero at infinity.  Note that in the limit <math>\tau \rightarrow 0</math> each Gaussian spike becomes an infinitely sharp [[Dirac delta function|Dirac impulse]] centered respectively at <math>x=n</math> and <math>\xi=m</math> for each respective <math>n</math> and <math>m</math>, and hence also all pre-factors <math> e^{-\pi \tau^2 m^2}</math> in <math>S_{\tau}(\xi)</math> eventually become indistinguishable from <math> e^{-\pi \tau^2 \xi^2}</math>. Therefore  the functions <math>s_{\tau}(x)</math> and their respective Fourier transforms <math>S_{\tau}(\xi)</math> 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:

:<math>\lim_{\tau \rightarrow 0} s_{\tau}(x) = \operatorname{\text{Ш}}({x}),</math>   and   <math>\lim_{\tau \rightarrow 0} S_{\tau}(\xi) = \operatorname{\text{Ш}}({\xi}).</math>

Since <math>S_{\tau}=\mathcal{F}[s_{\tau}]</math>, we obtain in this limit the result to be demonstrated:
:<math>\mathcal{F}[\operatorname{\text{Ш}}]= \operatorname{\text{Ш}}.</math>
The corresponding result for period <math>T</math> can be found by exploiting the [[Fourier transform#Time scaling|scaling property]] of the [[Fourier transform]],
:<math>\mathcal{F}[\operatorname{\text{Ш}}_T]= \frac{1}{T} \operatorname{\text{Ш}}_{\frac{1}{T}}.</math>

Another manner to establish that the Dirac comb transforms into another Dirac comb starts by examining [[Fourier transform#Fourier transform for periodic functions|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 [[Fourier transform#Other conventions|convention]] for the Fourier transform, this will be shown using angular frequency with <math>\omega=2\pi \xi :</math> for any periodic function <math>f(t)=f(t+T)</math> its Fourier transform
:<math>\mathcal{F}\left[ f \right](\omega)=F(\omega) = \int_{-\infty}^{\infty} dt f(t) e^{-i\omega t} </math>  obeys:
:<math>F(\omega) (1 - e^{i \omega T}) = 0</math>
because Fourier transforming <math>f(t)</math> and <math>f(t+T)</math> leads to <math>F(\omega)</math> and <math>F(\omega) e^{i \omega T}.</math> This equation implies that <math>F(\omega)=0</math> nearly everywhere with the only possible exceptions lying at <math>\omega= k \omega_0,</math> with <math>\omega_0=2\pi / T</math> and <math>k \in \mathbb{Z}.</math> When evaluating the Fourier transform at <math>F(k \omega_0)</math> 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 <math>1/T</math> for each <math>k.</math> This can be summarised by interpreting the Dirac comb as a limit of the [[Dirichlet kernel#Relation to the periodic delta function|Dirichlet kernel]] such that, at the positions <math>\omega= k \omega_0,</math> all exponentials in the sum <math> \sum\nolimits_{m=-\infty}^{\infty} e^{\pm i \omega m T} </math> point into the same direction and add constructively. In other words, the [[Fourier transform#Fourier transform for periodic functions|continuous Fourier transform of periodic functions]] leads to
:<math>F(\omega)= 2 \pi \sum_{k=-\infty}^{\infty} c_k \delta(\omega-k\omega_0) </math> with <math>\omega_0=2 \pi/T,</math>
and
:<math>c_k = \frac{1}{T} \int_{-T/2 }^{+T/2} dt f(t) e^{-i 2 \pi k t/T}.</math>
The [[#Fourier series|Fourier series]] coefficients <math>c_k=1/T</math> for all <math>k</math> when <math>f \rightarrow \operatorname{\text{Ш}}_{T}</math>, i.e. 
:<math>\mathcal{F}\left[ \operatorname{\text{Ш}}_{T} \right](\omega) = \frac{2 \pi}{T} \sum_{k=-\infty}^{\infty} \delta(\omega-k \frac{2 \pi}{T})</math>
is another Dirac comb, but with period <math>2 \pi/T</math> in angular frequency domain (radian/s).

As mentioned, the specific rule depends on the [[Fourier transform#Other conventions|convention]] for the used Fourier transform. Indeed, when using the [[Dirac delta function#Scaling and symmetry|scaling property]] of the Dirac delta function, the above may be re-expressed in ordinary frequency domain (Hz) and one obtains again:
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{T} \operatorname{\text{Ш}}_{\ \frac{1}{T}}(\xi) = \sum_{n=-\infty}^{\infty}\!\! e^{-i 2\pi \xi n T},</math>

such that the unit period Dirac comb transforms to itself:
<math display="block">\operatorname{\text{Ш}}\ \!(t) \stackrel{\mathcal{F}}{\longleftrightarrow} \operatorname{\text{Ш}}\ \!(\xi).</math>

Finally, the Dirac comb is also an [[Fourier transform#Eigenfunctions|eigenfunction]] of the unitary continuous Fourier transform in [[Fourier transform#Other conventions|angular frequency]] space to the eigenvalue 1 when <math>T=\sqrt{2 \pi}</math> because for the unitary Fourier transform 
:<math>\mathcal{F}\left[ f \right](\omega)=F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dt f(t) e^{-i\omega t}, </math>
the above may be re-expressed as 
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{\sqrt{2\pi}}{T} \operatorname{\text{Ш}}_{\ \frac{2\pi}{T}}(\omega) = \frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty} \!\!e^{-i\omega nT}.</math>

==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.
<math display="block"> (\operatorname{\text{Ш}}_{\ T} x)(t) = \sum_{k=-\infty}^{\infty} \!\! x(t)\delta(t - kT) = \sum_{k=-\infty}^{\infty}\!\! x(kT)\delta(t - kT).</math>

Due to the [[Dirac comb#Fourier transform|self-transforming]] property of the Dirac comb and the [[convolution theorem]], this corresponds to convolution with the Dirac comb in the frequency domain.
<math display="block"> \operatorname{\text{Ш}}_{\ T} x \ \stackrel{\mathcal{F}}{\longleftrightarrow}\  \frac{1}{T}\operatorname{\text{Ш}}_\frac{1}{T} * X</math>

Since convolution with a delta function <math>\delta(t-kT)</math> is equivalent to shifting the function by <math>kT</math>, convolution with the Dirac comb corresponds to replication or [[periodic summation]]:
:<math> (\operatorname{\text{Ш}}_{\ \frac{1}{T}}\! * X)(f) =\! \sum_{k=-\infty}^{\infty} \!\!X\!\left(f - \frac{k}{T}\right) </math>

This leads to a natural formulation of the [[Nyquist–Shannon sampling theorem]]. If the spectrum of the function <math>x</math> contains no frequencies higher than B  (i.e., its spectrum is nonzero only in the interval <math>(-B, B)</math>) then samples of the original function at intervals <math>\tfrac{1}{2B}</math> are sufficient to reconstruct the original signal. It suffices to multiply the spectrum of the sampled function by a suitable [[rectangle function]], which is equivalent to applying a brick-wall [[lowpass filter]].
:<math> \operatorname{\text{Ш}}_{\ \!\frac{1}{2B}} x\ \ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \ 2B\, \operatorname{\text{Ш}}_{\ 2B} * X</math>

:<math> \frac{1}{2B}\Pi\left(\frac{f}{2B}\right) (2B \,\operatorname{\text{Ш}}_{\ 2B} * X) = X</math>

In time domain, this "multiplication with the rect function" is equivalent to "convolution with the sinc function."{{sfn|Woodward|1953|pp=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 {{harvnb|Lighthill|1958|p=62}}, Theorem 22 for details.

== Use in directional statistics ==
{{unreferenced section|date=October 2017}}

In [[directional statistics]], the Dirac comb of period <math>2\pi</math> is equivalent to a [[wrapped distribution|wrapped]] Dirac delta function and is the analog of the [[Dirac delta function]] in linear statistics.

In linear statistics, the random variable <math>(x)</math> is usually distributed over the real-number line, or some subset thereof, and the probability density of <math>x</math> is a function whose domain is the set of real numbers, and whose integral from <math>-\infty</math> to <math>+\infty</math> is unity. In directional statistics, the random variable <math>(\theta)</math> is distributed over the unit circle, and the probability density of <math>\theta</math> is a function whose domain is some interval of the real numbers of length <math>2\pi</math> 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 <math>2\pi</math> with an arbitrary function of period <math>2\pi</math> over the unit circle yields the value of that function at zero.

==See also==

*[[Comb filter]]
*[[Frequency comb]]
*[[Poisson summation formula]]
*[[Theta function]]

== Notes ==
{{Reflist}}

== References ==
* {{Cite book |last=Brandwood |first=D. |title=Fourier Transforms in Radar and Signal Processing |publisher=Artech House |year=2003 |isbn=1580531741 |location=Boston |lccn=2002044073}}
* {{Cite book |last=Lighthill |first=M.J. |title=An Introduction to Fourier Analysis and Generalized Functions |publisher=Cambridge University Press |year=1958 |location=Cambridge |doi=10.1017/CBO9781139171427|isbn=978-0-521-05556-7 }}
* {{Cite book |last=Woodward |first=P. M. |title=Probability and Information Theory, with Applications to Radar |publisher=Pergamon Press |year=1953 |oclc=6570386}}

== Further reading ==
* {{Cite journal |last=Córdoba |first=A |year=1989 |title=Dirac combs |journal=Letters in Mathematical Physics |volume=17 |issue=3 |pages=191–196 |bibcode=1989LMaPh..17..191C |doi=10.1007/BF00401584 |s2cid=189883287}}


{{ProbDistributions|continuous-infinite}}

[[Category:Special functions]]
[[Category:Generalized functions]]
[[Category:Signal processing]]
[[Category:Directional statistics]]