In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... This page deals with mathematical distributions. ... The Dirac delta function, sometimes 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(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral...

for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ш). From the orthogonality of the Fourier series, an alternate definition may also be written: Ronald Newbold Bracewell (1921 – ) is the Lewis M. Terman Professor of Electrical Engineering, Emeritus of the Space, Telecommunications and Radioscience Laboratory at Stanford University. ... The Cyrillic alphabet (or azbuka, from the old name of the first letters) is an alphabet used to write six natural Slavic languages (Belarusian, Bulgarian, Macedonian, Russian, Serbian, and Ukrainian) and many other languages of the former Soviet Union, Asia and Eastern Europe. ... Sha (Ш, ш) is a letter of the Cyrillic alphabet, representing the consonant sound /ʃ/ or /ʃʲ/. This is equivalent to sh in English, ch in French, sch in German, ÅŸ in Turkish, or sz in Polish. ... The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ...

Properties

Many of the properties of the Dirac comb follow from the properties of the Dirac delta function.

• (scaling)
• The continuous Fourier transform of the Dirac comb is another Dirac comb:

In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

Sampling and aliasing

Multiplication of a continuous signal by a Dirac comb is sometimes called an ideal sampler with sampling rate T. When used as an ideal sampler, it can be used to understand the effects of aliasing and as a proof of the Shannon-Nyquist sampling theorem. A continuous signal or a continuous time signal is a varying quantity (a signal) that can be, or is expressed, as a continuous function of an independent variable, usually time. ... The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... In statistics, signal processing, and related disciplines, aliasing is an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when sampled. ... The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ...

See Shannon-Nyquist sampling theorem for a proof using the Dirac comb. The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ...

Reference

• Bracewell, R.N., The Fourier Transform and Its Applications (McGraw-Hill, 1965, 2nd ed. 1978, revised 1986)

See also

• Poisson summation formula