Periodic summation

In signal processing, any periodic function  f_T  with period  T,  can be represented by a summation of an infinite number of instances of an aperiodic function,  f,  that are offset by integer multiples of  T.  This representation is called periodic summation:

f_T(t) = \sum_{n=-\infty}^\infty f(t %2B nT) = \sum_{n=-\infty}^\infty f(t - nT).

When  f_T  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/T.[1]  That identity is known as the Poisson summation formula.

Quotient space as domain

If a periodic function is represented using the quotient space domain \mathbb{R}/(T\cdot\mathbb{Z}) then one can write

\varphi_T�: \mathbb{R}/(T\cdot\mathbb{Z}) \to \mathbb{R}
\varphi_T(t) = \sum_{\tau\in t} f(\tau)

instead. The arguments of \varphi_T are equivalence classes of real numbers that share the same fractional part when divided by T.

Notes

  1. ^ The form of the transform assumed here is F(\nu)\ \stackrel{\mathrm{def}}{=}\int_{-\infty}^{\infty} f(t)\ e^{-i2\pi\nu t}\, dt.

See also