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:
“No rent-roll nor army-list can dignify skulking and dissimulation: and the first point of courtesy must always be truth, as really all the forms of good-breeding point that way.”
—Ralph Waldo Emerson (1803–1882)
“Literary works cannot be taken over like factories, or literary forms of expression like industrial methods. Realist writing, of which history offers many widely varying examples, is likewise conditioned by the question of how, when and for what class it is made use of.”
—Bertolt Brecht (1898–1956)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)