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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:

$\sum _{n=-\infty }^{\infty }f(n)=\sum _{k=-\infty }^{\infty }{\hat {f}}\left(k\right),$     where ${\hat {f}}$  is the Fourier transform of $f\,$ ;  that is   ${\hat {f}}(\nu )={\mathcal {F}}\{f(x)\}.$

(Eq.1)

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

$\sum _{n=-\infty }^{\infty }g(nT)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{\hat {g}}\left({\frac {k}{T}}\right)$     (Stein & Weiss 1971).

(Eq.2)

Poisson Summation Formula - General Version. With another definition,  $s(t+x)\ {\stackrel {\text{def}}{=}}\ g(x),\,$  and the transform property  ${\mathcal {F}}\{s(t+x)\}\ ={\hat {s}}(\nu )\cdot e^{i2\pi \nu t},$  Eq.2 becomes a periodic summation (with period T) and its equivalent Fourier series:

$\underbrace {\sum _{n=-\infty }^{\infty }s(t+nT)} _{periodization}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{\hat {s}}\left({\frac {k}{T}}\right)\ e^{i2\pi {\frac {k}{T}}t}\equiv {\mathcal {F}}^{-1}\underbrace {\left\{{\frac {1}{T}}\ \sum _{k=-\infty }^{\infty }{\hat {s}}({\frac {k}{T}})\ \delta (\nu -{\frac {k}{T}})\right\}} _{discretization}$     (Pinsky 2002; Zygmund 1968).

(Eq.3)

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

$\underbrace {\sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)} _{periodization}=T\sum _{n=-\infty }^{\infty }s(nT)\ e^{-i2\pi nT\nu }\ \ \equiv {\mathcal {F}}\underbrace {\left\{T\sum _{n=-\infty }^{\infty }s(nT)\ \delta (t-nT)\right\}} _{discretization}$     (Gasquet & Witomski 1999).

(Eq.4)

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 Eq.3 and Eq.4 in terms of discretization ${\bot \!\bot \!\bot }$ and periodization ${\triangle \!\triangle \!\triangle }$, it leads to the Discretization-Periodization Theorem on generalized functions:

${\mathcal {F}}{\left\{({\triangle \!\triangle \!\triangle }_{T}s)(t)\right\}}={\frac {1}{T}}\ ({\bot \!\bot \!\bot }_{\frac {1}{T}}{\hat {s}})(\nu )$     i.e. the Fourier transform of a periodization of $s(t)$ corresponds to a discretization of its spectrum ${\hat {s}}(\nu )$

(Eq.5)

${\frac {1}{T}}\ ({\triangle \!\triangle \!\triangle }_{\frac {1}{T}}{\hat {s}})(\nu )={\mathcal {F}}{\left\{({\bot \!\bot \!\bot }_{T}s)(t)\right\}}$     i.e. the Fourier transform of a discretization of $s(t)$ corresponds to a periodization of its spectrum ${\hat {s}}(\nu )$

(Eq.6)

where Eq.3 becomes Eq.5 and Eq.4 becomes Eq.6.