Discretization of a function [edit]
In mathematics, the discretization of a function is the
operation
that assigns the
generalized function defined by
to a smooth regular function that is not growing faster than polynomials,
where is the Dirac delta
and is a positive, real increment between consecutive samples of
function . The generalized function is also called the discretization of
with increments or discrete function of with increments . Discretization is an operation that is closely related to periodization via the Discretization-Periodization theorem.
Example: Discretizing the function that is constantly one yields the Dirac comb.
Periodization of a function[edit]
In mathematics, the periodization of a function or generalized function is the
operation
that assigns the (generalized) function
defined by
to a (generalized) function that is of compact support or at least rapidly decreasing to zero as tends to infinity,
where is a positive, real number determining the period of .
The periodic function is also called
periodization of ,
periodic function of or
periodic continuation of function with period .
Periodization is an operation that is closely related to discretization via the Discretization-Periodization theorem.
Example: Periodizing the Dirac delta yields the Dirac comb.
Dirac Comb Identity[edit]
Poisson Summation Formula[edit]
Poisson Summation Formula - Symmetric Version. For appropriate functions the Poisson summation formula may be stated as:
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where is the Fourier transform of ; that is
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(Eq.1)
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Poisson Summation Formula - Classical Version. With the substitution, and the Fourier transform property, (for T > 0), Eq.1 becomes:
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( Stein & Weiss 1971) harv error: no target: CITEREFSteinWeiss1971 (help).
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(Eq.2)
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Poisson Summation Formula - General Version. With another definition, and the transform property Eq.2 becomes a periodic summation (with period T) and its equivalent Fourier series:
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( Pinsky 2002 harvnb error: no target: CITEREFPinsky2002 (help); Zygmund 1968 harvnb error: no target: CITEREFZygmund1968 (help)).
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(Eq.3)
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Similarly, the periodic summation of a function's Fourier transform has this Fourier series equivalent:
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( Gasquet & Witomski 1999 harvnb error: no target: CITEREFGasquetWitomski1999 (help)).
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(Eq.4)
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where T represents the time interval at which a function s(t) is sampled, and 1/T is the rate of samples/sec.
Poisson Summation Formula in terms of Discretization and Periodization[edit]
Writing Eq.3 and Eq.4 in terms of
discretization and
periodization ,
it leads to the Discretization-Periodization Theorem on generalized functions:
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i.e. the Fourier transform of a periodization of corresponds to a discretization of its spectrum
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(Eq.5)
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i.e. the Fourier transform of a discretization of corresponds to a periodization of its spectrum
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(Eq.6)
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where Eq.3 becomes Eq.5 and Eq.4 becomes Eq.6.