Periodic summation

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

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 $s_P(t)$ is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform $, S(f) \ \stackrel{\mathrm{def}}{=} \ \mathcal{F}\{s(t)\},$ 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 $s(t)$ at constant intervals (T) is equivalent to a periodic summation of $S(f),$ 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 ${\mathbb {R}}/(P{\mathbb {Z}})$ then one can write

\varphi _{P}:{\mathbb {R}}/(P{\mathbb {Z}})\to {\mathbb {R}}

\varphi_P(x) = \sum_{\tau\in x} s(\tau)

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

Citations

↑ Pinsky, Mark (2001). Introduction to Fourier Analysis and Wavelets. Brooks/Cole. ISBN 978-0534376604.
↑ Zygmund, Antoni (1988). Trigonometric series (2nd ed.). Cambridge University Press. ISBN 978-0521358859.

Periodic summation

Quotient space as domain

Citations

See also