Periodic summation

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

f_{P}(x)=\sum _{{n=-\infty }}^{\infty }f(x+nP)=\sum _{{n=-\infty }}^{\infty }f(x-nP).

When  f_{P}  is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform of  f  at intervals of  \scriptstyle 1/P.[1][2]  That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of function  f,  is equivalent to a periodic summation of the Fourier transform of  f,,  which is known as a discrete-time Fourier transform.

Quotient space as domain

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

\varphi _{P}:{\mathbb {R}}/(P\cdot {\mathbb {Z}})\to {\mathbb {R}}
\varphi _{P}(x)=\sum _{{\tau \in x}}f(\tau )

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