Periodic summation

A Fourier transform and 3 variations caused by periodic sampling (at interval T) and/or periodic summation (at interval P) of the underlying time-domain function.

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.

Quotient space as domain

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

${\displaystyle \varphi _{P}:\mathbb {R} /(P\mathbb {Z} )\to \mathbb {R} }$

${\displaystyle \varphi _{P}(x)=\sum _{\tau \in x}s(\tau )~.}$

The arguments of ${\displaystyle \varphi _{P}}$ are equivalence classes of real numbers that share the same fractional part when divided by ${\displaystyle P}$.

Citations

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

See also

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