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In signal processing, any periodic function s P {\displaystyle s_{P}} with period P can be represented by a summation of an infinite number of instances of an aperiodic function s {\displaystyle s} , that are offset by integer multiples of P. This representation is called periodic summation:
When s P {\displaystyle s_{P}} is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values of the continuous Fourier transform, S ≜ F { s } , {\displaystyle S\triangleq {\mathcal {F}}\{s\},} at intervals of 1 P {\displaystyle {\tfrac {1}{P}}}. That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of s {\displaystyle s} at constant intervals is equivalent to a periodic summation of S , {\displaystyle S,} 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.