Periodic summation explained

s(t) can be made into a periodic function s _{P(t)} with period P by summing the translations of the function s(t) by integer multiples of P. This is called periodic summation: s _{P(t)} = \sum ^{inf ty} _{n =-inf ty} s(t + nP)

When

s _{P(t)} is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, S(f) \triangleq l{F}\{s(t)\}, at intervals of \tfrac{ 1 }{P} .^{[1] [2]} 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

R /(P Z ) then one can write:\varphi _{P} : R /(P Z ) \to R \varphi _{P(x)} = \sum _{\tau\in} s(\tau) ~ .

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

Notes and References

1. Book: Pinsky, Mark. Introduction to Fourier Analysis and Wavelets. 2001. Brooks/Cole. 978-0534376604.
2. Book: Zygmund, Antoni. Trigonometric Series . 2nd. Trigonometric Series. 1988. Cambridge University Press. 978-0521358859.