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Concept
Poisson summation formula
Summary
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
Consider an aperiodic function with Fourier transform alternatively designated by and
The basic Poisson summation formula is:
Also consider periodic functions, where parameters and are in the same units as :
Then is a special case (P=1, x=0) of this generalization:
which is a Fourier series expansion with coefficients that are samples of function Similarly:
also known as the important Discrete-time Fourier transform.
The Poisson summation formula can also be proved quite conceptually using the compatibility of Pontryagin duality with short exact sequences such as
holds provided is a continuous integrable function which satisfies
for some and every Note that such is uniformly continuous, this together with the decay assumption on , show that the series defining converges uniformly to a continuous function. holds in the strong sense that both sides converge uniformly and absolutely to the same limit.
holds in a pointwise sense under the strictly weaker assumption that has bounded variation and
The Fourier series on the right-hand side of is then understood as a (conditionally convergent) limit of symmetric partial sums.
As shown above, holds under the much less restrictive assumption that is in , but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of In this case, one may extend the region where equality holds by considering summability methods such as Cesàro summability.
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Dirac comb
In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula for some given period . Here t is a real variable and the sum extends over all integers k. The Dirac delta function and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the s as the comb's teeth), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol , where the period is omitted, represents a Dirac comb of unit period.
Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
Periodic summation
In mathematics, any integrable function can be made into a periodic function with period P by summing the translations of the function by integer multiples of P. This is called periodic summation: When is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, at intervals of . That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of at constant intervals (T) is equivalent to a periodic summation of which is known as a discrete-time Fourier transform.