Inmathematics, anyintegrable function${\displaystyle s(t)}$ can be made into aperiodic function${\displaystyle s_P(t)}$ with periodP by summing the translations of the function${\displaystyle s(t)}$ byinteger multiples ofP. This is calledperiodic summation:

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

When${\displaystyle s_P(t)}$ is represented as aFourier series, the Fourier coefficients are equal to the values of thecontinuous Fourier transform,${\displaystyle S(f)\triangleq \mathcal{F}\{s(t)\},}$ at intervals of${\displaystyle \frac{1}{P}}$.^[1][2] This follows easily from recognizing that the formula for finding then^th coefficient of the Fourier series for the periodic summation is identical to the formula for the value of the Fourier transform of the original function at${\displaystyle n/P.}$ The identity is also a form of thePoisson summation formula.

This implies that the periodic summation of anyband-limited function, such as thesinc function, is a sum of a finite number of sine waves, or even just a single sine wave or zero if the period is less than or equal to half the inverse of the upper frequency limit. A periodic summation of a function can be identically zero if the Fourier transform of the function is zero at all multiples of some frequency, but if all periodic summations (that is, with all periods) are zero then the function must be identically zero.

Similarly, a Fourier series whose coefficients are samples of${\displaystyle s(t)}$ at constant intervals (T) is equivalent to aperiodic summation of${\displaystyle S(f),}$ which is known as adiscrete-time Fourier transform.

The periodic summation of aDirac delta function is theDirac comb. Likewise, the periodic summation of anintegrable function is itsconvolution with the Dirac comb.

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

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

${\displaystyle {\phi }_P(x)=\sum _{\tau \in x}s(\tau ).}$

The arguments of${\displaystyle {\phi }_P}$ areequivalence classes ofreal numbers that share the samefractional part when divided by${\displaystyle P}$.

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

• Functions and mappings
• Signal processing

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