Forms of The Equation
For appropriate functions the Poisson summation formula may be stated as:
-
where is the Fourier transform of ; that is
(Eq.1)
With the substitution, and the Fourier transform property, (for P > 0), Eq.1 becomes:
-
(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:
-
(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
Famous quotes containing the words forms of, forms and/or equation:
“I am prisoner of a gaudy and unlivable present, where all forms of human society have reached an extreme of their cycle and there is no imagining what new forms they may assume.”
—Italo Calvino (1923–1985)
“I had a glimpse through curtain laces
Of youthful forms and youthful faces.”
—Robert Frost (1874–1963)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)