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.  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 $P$.

See also

• Dirac comb
• Circular convolution
• Discrete-time Fourier transform