Dirac comb

# Dirac comb

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Dirac comb (also known as an impulse train and sampling function in electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

) is a periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

Schwartz distribution constructed from Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

s
Discussion

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Dirac comb (also known as an impulse train and sampling function in electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

) is a periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

Schwartz distribution constructed from Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

s

for some given period T. Some authors, notably Bracewell
Ronald N. Bracewell
Ronald Newbold Bracewell AO was the Lewis M. Terman Professor of Electrical Engineering, Emeritus of the at Stanford University.- Education :...

as well as some textbook authors in electrical engineering and circuit theory, refer to it as the Shah function (possibly because its graph resembles the shape of the Cyrillic
Cyrillic alphabet
The Cyrillic script or azbuka is an alphabetic writing system developed in the First Bulgarian Empire during the 10th century AD at the Preslav Literary School...

letter sha
Sha
For other uses, see Sha .Sha is a letter of the Cyrillic alphabet. It commonly represents the voiceless postalveolar fricative , like the pronunciation of ⟨sh⟩ in "sheep", or the somewhat similar voiceless retroflex fricative . It is used in every variation of the Cyrillic alphabet, for Slavic and...

Ш). Because the Dirac comb function is periodic, it can be represented as a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

:

## Scaling property

The scaling property follows directly from the properties of the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

## Fourier series

It is clear that ΔT(t) is periodic with period T. That is

for all t. The complex Fourier series for such a periodic function is

where the Fourier coefficients, cn are

All Fourier coefficients are 1/T resulting in

## Fourier transform

The Fourier transform
Continuous Fourier transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...

of a Dirac comb is also a Dirac comb.

Unitary transform to ordinary frequency domain (Hz):

Unitary transform to angular frequency domain (radian/s):

## Sampling and aliasing

Reconstruction of a continuous signal
Continuous signal
A continuous signal or a continuous-time signal is a varying quantity whose domain, which is often time, is a continuum . That is, the function's domain is an uncountable set. The function itself need not be continuous...

from samples taken at sampling interval
Sampling rate
The sampling rate, sample rate, or sampling frequency defines the number of samples per unit of time taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is hertz , sometimes noted as Sa/s...

T is done by some sort of interpolation, such as the Whittaker–Shannon interpolation formula
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...

. Mathematically, that process is often modelled as the output of a lowpass filter whose input is a Dirac comb whose teeth have been weighted by the sample values. Such a comb is equivalent to the product of a comb and the original continuous signal. That mathematical abstraction is often described as "sampling" for purposes of introducing the subjects of aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

and the Nyquist-Shannon sampling theorem.

## Use in directional statistics

In directional statistics, the Dirac comb of period 2π is equivalent to a wrapped
Wrapped distribution
In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere...

Dirac delta function, and is the analog of the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

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 real numbers, and whose integral from to is unity. In directional statistics, the random variable (θ) is distributed over the unit circle and the probability density of θ is a function whose domain is some interval of the real numbers of length 2π 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 2π with an arbitrary function of period 2π over the unit circle yields the value of that function at zero.