# Definition:Sampling Function

(Redirected from Definition:Dirac Comb)

## Definition

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

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

where $\delta$ denotes the Dirac delta distribution.

Whenever $T = 1$, it can be omitted.

### Graph of Sampling Function

The graph of the sampling function 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 or the Dirac comb.

It can referred to and voiced as shah.

## Also see

• Results about the sampling function can be found here.

## Linguistic Note

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