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Discretization of a function
In mathematics, the discretization of a function is the operation $${\bot \! \bot \! \bot}_{T}$$ that assigns the generalized function $${\bot \! \bot \! \bot}_{T} f$$ defined by



( {\bot \! \bot \! \bot}_{T} f )(t) \,\stackrel{\mathrm{def}}{=}\, \sum_{k=-\infty}^\infty \, f(kT) \, \delta(t-kT) $$

to a smooth regular function $$f(t)$$ that is not growing faster than polynomials, where $$\delta(t)$$ is the Dirac delta and $$T$$ is a positive, real increment between consecutive samples $$f(kT)$$ of function $$f(t)$$. The generalized function $${\bot \! \bot \! \bot}_{T} f$$ is also called the discretization of $$f$$ with increments $$T$$ or discrete function of $$f$$ with increments $$T$$. Discretization is an operation that is closely related to periodization via the Discretization-Periodization theorem. Example: Discretizing the function that is constantly one yields the Dirac comb.

Periodization of a function
In mathematics, the periodization of a function or generalized function is the operation $${\triangle \! \triangle \! \triangle}_T$$ that assigns the (generalized) function $${\triangle \! \triangle \! \triangle}_T f$$ defined by



( {\triangle \! \triangle \! \triangle}_T f )(t) \,\stackrel{\mathrm{def}}{=}\, \sum_{k=-\infty}^\infty \, f(t-kT) $$

to a (generalized) function $$f(t)$$ that is of compact support or at least rapidly decreasing to zero as $$|t|$$ tends to infinity, where $$T$$ is a positive, real number determining the period of $${\triangle \! \triangle \! \triangle}_T f$$. The periodic function $${\triangle \! \triangle \! \triangle}_T f $$ is also called periodization of $$f$$, periodic function of $$f$$ or periodic continuation of function $$f$$ with period $$T$$. Periodization is an operation that is closely related to discretization via the Discretization-Periodization theorem. Example: Periodizing the Dirac delta yields the Dirac comb.

Dirac Comb Identity


{\bot \! \bot \! \bot}_{T} \, 1  \,\,\, \equiv \,\,\,   {\triangle \! \triangle \! \triangle}_T \, \delta $$

Poisson Summation Formula
Poisson Summation Formula - Symmetric Version. For appropriate functions $$f,\,$$ the Poisson summation formula may be stated as:

Poisson Summation Formula - Classical Version. With the substitution, $$g(xT)\ \stackrel{\text{def}}{=}\ f(x),\,$$ and the Fourier transform property,  $$\mathcal{F}\{g(x T)\}\ = \frac{1}{T} \cdot \hat g\left(\frac{\nu}{T}\right)$$  (for T > 0),  $$ becomes:

Poisson Summation Formula - General Version. With another definition, $$s(t+x)\ \stackrel{\text{def}}{=}\ g(x),\,$$  and the transform property  $$\mathcal{F}\{s(t+x)\}\ = \hat s(\nu)\cdot e^{i 2\pi \nu t},$$  $$ becomes a periodic summation (with period T) and its equivalent Fourier series:

Similarly, the periodic summation of a function's Fourier transform has this Fourier series equivalent:

where T represents the time interval at which a function s(t) is sampled, and 1/T is the rate of samples/sec.

Poisson Summation Formula in terms of Discretization and Periodization
Writing $$ and $$ in terms of discretization $${\bot \! \bot \! \bot}$$ and periodization $${\triangle \! \triangle \! \triangle}$$, it leads to the Discretization-Periodization Theorem on generalized functions:

where $$ becomes $$ and $$ becomes $$.