Forms of The Equation
For appropriate functions the Poisson summation formula may be stated as:
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where is the Fourier transform of ; that is
(Eq.1)
With the substitution, and the Fourier transform property, (for P > 0), Eq.1 becomes:
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(Stein & Weiss 1971).
(Eq.2)
With another definition, and the transform property Eq.2 becomes a periodic summation (with period P) and its equivalent Fourier series:
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(Pinsky 2002; Zygmund 1968).
(Eq.3)
Similarly, the periodic summation of a function's Fourier transform has this Fourier series equivalent:
-
(Eq.4)
where T represents the time interval at which a function s(t) is sampled, and 1/T is the rate of samples/sec.
Read more about this topic: Poisson Summation Formula
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