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Periodic summation

Summary

In signal processing, any periodic function ${\displaystyle s_{P}(t)}$ with period P can be represented by a summation of an infinite number of instances of an aperiodic function ${\displaystyle s(t)}$, that are offset by integer multiples of P. This representation is called periodic summation:

${\displaystyle s_{P}(t)=\sum _{n=-\infty }^{\infty }s(t+nP)=\sum _{n=-\infty }^{\infty }s(t-nP).}$
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 ${\displaystyle s_{P}(t)}$ is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, ${\displaystyle S(f)\triangleq {\mathcal {F}}\{s(t)\},}$ 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 ${\displaystyle s(t)}$ at constant intervals (T) is equivalent to a periodic summation of ${\displaystyle S(f),}$ 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 represented using the quotient space domain ${\displaystyle \mathbb {R} /(P\mathbb {Z} )}$ then one can write

${\displaystyle \varphi _{P}:\mathbb {R} /(P\mathbb {Z} )\to \mathbb {R} }$
${\displaystyle \varphi _{P}(x)=\sum _{\tau \in x}s(\tau )}$

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

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.