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, $S(f)\ \triangleq \ {\mathcal {F}}\{s(t)\},$  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