en>Mark viking: Added a description of a topological category in terms of the grounding functor

2013-10-01T20:28:34Z

Added a description of a topological category in terms of the grounding functor

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In [[signal processing]], any [[periodic function]]  <math>f_P</math>  with period '''P''' can be represented by a summation of an infinite number of instances of an aperiodic function,  <math>f</math> , that are offset by integer multiples of '''P'''.  This representation is called '''periodic summation:'''

:<math>f_P(x) = \sum_{n=-\infty}^\infty f(x + nP) = \sum_{n=-\infty}^\infty f(x - nP).</math>

When  <math>f_P</math>  is alternatively represented as a complex [[Fourier series]], the Fourier coefficients are proportional to the values (or "samples") of the [[continuous Fourier transform]] of  <math>f</math>  at intervals of  <math>\scriptstyle 1/P.</math><ref>{{cite book|last=Pinsky|first=Mark|title=Introduction to Fourier Analysis and Wavelets|year=2001|publisher=Brooks/Cole|isbn=978-0534376604}}</ref><ref>{{cite book|last=Zygmund|first=Antoni|title=Trigonometric series (2nd ed.)|year=1988|publisher=Cambridge University Press|isbn=978-0521358859}}</ref>  That identity is a form of the [[Poisson summation formula]]. Similarly, a Fourier series whose coefficients are samples of function  <math>f,</math>  is equivalent to a periodic summation of the Fourier transform of  <math>f,</math>,  which is known as a [[discrete-time Fourier transform]].

== Quotient space as domain ==

If a periodic function is represented using the [[Quotient space (linear algebra)|quotient space]] [[Domain of a function|domain]]
<math>\mathbb{R}/(P\cdot\mathbb{Z})</math> then one can write

:<math>\varphi_P : \mathbb{R}/(P\cdot\mathbb{Z}) \to \mathbb{R}</math>
:<math>\varphi_P(x) = \sum_{\tau\in x} f(\tau)</math>

instead. The arguments of <math>\varphi_P</math> are [[equivalence class]]es of [[real number]]s that share the same [[fractional part]] when divided by <math>P</math>.

== Citations ==
{{reflist}}

== See also ==
*[[Dirac comb]]
*[[Circular convolution]]
*[[Discrete-time Fourier transform]]

[[Category:Functions and mappings]] 
[[Category:Signal processing]]

Topological category - Revision history

en>Mark viking: Added a description of a topological category in terms of the grounding functor