Definition:Sampling Function

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The sampling function is the distribution $\operatorname {III}_T: \map \DD \R \to \R$ defined as:

$\forall x \in \R: \map {\operatorname {III}_T } x := \ds \sum_{n \mathop \in \Z} \map \delta {x - T n}$


$T \in \R_{\ne 0}$ is a non-zero real number
$\delta$ denotes the Dirac delta distribution.

When $T = 1$, it is usually omitted:

$\forall x \in \R: \map {\operatorname {III} } x := \ds \sum_{n \mathop \in \Z} \map \delta {x - n}$

Graph of Sampling Function

The graph of the sampling function $\operatorname {III}: \map \DD \R \to \R$ is illustrated below:


It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \Z$.

$2$ Dimensional Form

Let $\operatorname {III}: \R \to \R$ denote the sampling function.

The $2$-dimensional form of $\operatorname {III}$ is defined and denoted:

$\forall x, y \in \R: \map {\operatorname { {}^2 III} } {x, y} := \map {\operatorname {III} } x \map {\operatorname {III} } y$

Also known as

The sampling function $\operatorname {III}$ can also be seen referred to as:

the replicating function
the Dirac comb.

It can be referred to and voiced as shah.

Also see

Linguistic Note

The name shah for the sampling function derives from its similarity in shape and appearance to the Russian ะจ, whose name is itself pronounced shah.