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:

:$s_P(t) = \sum_^\infty s(t + nP) = \sum_^\infty s(t - nP).$

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, $S(f) \triangleq \mathcal\,$ at intervals of $\tfrac$. 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 instead represented using the quotient space domain
$\mathbb/(P\mathbb)$ then one can write:

:$\varphi_P : \mathbb/(P\mathbb) \to \mathbb$
:$\varphi_P(x) = \sum_ s(\tau) ~ .$

The arguments of $\varphi_P$ 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

Functions and mappings
Signal processing