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.
Also see
- Sampling Function is Tempered Distribution
- 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 ะจ, whose name is itself pronounced shah.
Sources
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Frontispiece
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Chapter $4$: Notation for some useful Functions: Summary of special symbols: Table $4.1$ Special symbols
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Inside Back Cover