Periodic summation

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\mathbb{Z}) then one can write

\varphi_P : \mathbb{R}/(P\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