# Definition:Sampling Function

## Definition

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

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

where:

$T \in \R_{\ne 0}$ 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.

## 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.