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 $\mathbb{R}/(P\cdot\mathbb{Z})$ 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.