periodic, summation, mathematics, integrable, function, displaystyle, made, into, periodic, function, displaystyle, with, period, summing, translations, function, displaystyle, integer, multiples, this, called, periodic, summation, fourier, transform, variatio. In mathematics any integrable function s t displaystyle s t can be made into a periodic function s P t displaystyle s P t with period P by summing the translations of the function s t displaystyle s t by integer multiples of P This is called periodic summation A Fourier transform and 3 variations caused by periodic sampling at interval T and or periodic summation at interval P of the underlying time domain function s P t n s t n P displaystyle s P t sum n infty infty s t nP When s P t displaystyle s P t is alternatively represented as a Fourier series the Fourier coefficients are equal to the values of the continuous Fourier transform S f F s t displaystyle S f triangleq mathcal F s t at intervals of 1 P displaystyle tfrac 1 P 1 2 That identity is a form of the Poisson summation formula Similarly a Fourier series whose coefficients are samples of s t displaystyle s t at constant intervals T is equivalent to a periodic summation of S f displaystyle S f which is known as a discrete time Fourier transform The periodic summation of a Dirac delta function is the Dirac comb Likewise the periodic summation of an integrable function is its convolution with the Dirac comb Quotient space as domain editIf a periodic function is instead represented using the quotient space domain R P Z displaystyle mathbb R P mathbb Z nbsp then one can write f P R P Z R displaystyle varphi P mathbb R P mathbb Z to mathbb R nbsp f P x t x s t displaystyle varphi P x sum tau in x s tau nbsp The arguments of f P displaystyle varphi P nbsp are equivalence classes of real numbers that share the same fractional part when divided by P displaystyle P nbsp Citations edit Pinsky Mark 2001 Introduction to Fourier Analysis and Wavelets Brooks Cole ISBN 978 0534376604 Zygmund Antoni 1988 Trigonometric Series 2nd ed Cambridge University Press ISBN 978 0521358859 See also editDirac comb Circular convolution Discrete time Fourier transform Retrieved from https en wikipedia org w index php title Periodic summation amp oldid 1139771547, wikipedia, wiki, book, books, library,