Periodic summation

In signal processing, any periodic function${\displaystyle ,s_{P}(t)}$  with period P, can be represented by a summation of an infinite number of instances of an aperiodic function${\displaystyle ,s(t),}$ that are offset by integer multiples of P.  This representation is called periodic summation:

When  ${\displaystyle s_{P}(t)}$  is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform${\displaystyle ,S(f)\ {\stackrel {\mathrm {def} }{=}}\ {\mathcal {F}}\{s(t)\},}$  at intervals of 1/P.  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.

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