Periodic summation

In mathematics, any integrable function ${\displaystyle s(t)}$ can be made into a periodic function ${\displaystyle s_{P}(t)}$ with period P by summing the translations of the function ${\displaystyle s(t)}$ by integer multiples of P. This is called periodic summation:

${\displaystyle s_{P}(t)=\sum _{n=-\infty }^{\infty }s(t+nP)}$

When ${\displaystyle s_{P}(t)}$ is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, ${\displaystyle S(f)\triangleq {\mathcal {F}}\{s(t)\},}$ at intervals of ${\displaystyle {\tfrac {1}{P}}}$.^[1][2] That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of ${\displaystyle s(t)}$ at constant intervals (T) is equivalent to a periodic summation of ${\displaystyle S(f),}$ 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.