Topological category

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

$f_P(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}f(x+nP)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}f(x-nP).$

When  $f_P$  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  $1/P.$  That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of function  $f,$  is equivalent to a periodic summation of the Fourier transform of  $f,$,  which is known as a discrete-time Fourier transform.

Quotient space as domain

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

${\phi }_P:\mathbb{ℝ}/(P\cdot \mathbb{\mathbb{Z}})\to \mathbb{ℝ}$

${\phi }_P(x)=\sum _{\tau \in x}f(\tau )$

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

Citations

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See also

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