Periodic summation

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In signal processing, any periodic function  with period P, can be represented by a summation of an infinite number of instances of an aperiodic function that are offset by integer multiples of P.  This representation is called periodic summation:

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.

When    is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform  at intervals of 1/P.[1][2]  That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of    at constant intervals (T) is equivalent to a periodic summation of    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[edit]

If a periodic function is represented using the quotient space domain then one can write

instead. The arguments of are equivalence classes of real numbers that share the same fractional part when divided by .

Citations[edit]

  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[edit]