Definition:Filtering Function

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Definition

The filtering function is the real function $\operatorname {sinc}: \R \to \R$ defined as:

$\forall x \in \R: \map {\operatorname {sinc} } x := \dfrac {\sin \pi x} {\pi x}$

where $\sin$ denotes the (real) sine function.


Graph of Filtering Function

The graph of the filtering function is illustrated below:


Filtering-function.png


$2$ Dimensional Form

Let $\operatorname {sinc}: \R \to \R$ denote the filtering function.

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

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


Also known as

The filtering function is also known as the interpolating function.

The filtering function of $x$ is often voiced sinc $x$, exactly as written.


Warning

The filtering function is also known as the interpolating function.

The filtering function $\map {\operatorname {sinc} } x$ is not the same as the real function $f$ defined as $\forall x \in \R: \map f x = \dfrac {\sin x} x$, which is depicted below:


Sine-of-x-over-x.png


Sources

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