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Periodic summation

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Sum of a function's values every _P_ offsets
A Fourier transform and 3 variations caused by periodic sampling (at interval T) and/or periodic summation (at interval P) of the underlying time-domain function.

In mathematics, any integrable function s ( t ) {\displaystyle s(t)} can be made into a periodic function s P ( t ) {\displaystyle s_{P}(t)} with period P by summing the translations of the function s ( t ) {\displaystyle s(t)} by integer multiples of P. This is called periodic summation:

s P ( t ) = ∑ n = − ∞ ∞ s ( t + n P ) {\displaystyle s_{P}(t)=\sum _{n=-\infty }^{\infty }s(t+nP)}


When s P ( t ) {\displaystyle s_{P}(t)} is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, S ( f ) ≜ F { s ( t ) } , {\displaystyle S(f)\triangleq {\mathcal {F}}\{s(t)\},} at intervals of 1 P {\displaystyle {\tfrac {1}{P}}} . That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of s ( t ) {\displaystyle s(t)} at constant intervals (T) is equivalent to a periodic summation of S ( f ) , {\displaystyle S(f),} which is known as a discrete-time Fourier transform.

The periodic summation of a Dirac delta function is the Dirac comb. Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb.

Quotient space as domain

If a periodic function is instead represented using the quotient space domain R / ( P Z ) {\displaystyle \mathbb {R} /(P\mathbb {Z} )} then one can write:

φ P : R / ( P Z ) → R {\displaystyle \varphi _{P}:\mathbb {R} /(P\mathbb {Z} )\to \mathbb {R} }
φ P ( x ) = ∑ τ ∈ x s ( τ )   . {\displaystyle \varphi _{P}(x)=\sum _{\tau \in x}s(\tau )~.}

The arguments of φ P {\displaystyle \varphi _{P}} are equivalence classes of real numbers that share the same fractional part when divided by P {\displaystyle P} .

Citations

  1. Pinsky, Mark (2001). Introduction to Fourier Analysis and Wavelets. Brooks/Cole. ISBN 978-0534376604.
  2. Zygmund, Antoni (1988). Trigonometric Series (2nd ed.). Cambridge University Press. ISBN 978-0521358859.

See also

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