Definition:Even Impulse Pair Function
Jump to navigation
Jump to search
Definition
The even impulse pair function is the real function $\operatorname {II}: \R \to \R$ defined as:
- $\forall x \in \R: \map {\operatorname {II} } x := \dfrac 1 2 \map \delta {x + \dfrac 1 2} + \dfrac 1 2 \map \delta {x - \dfrac 1 2}$
where $\delta$ denotes the Dirac delta function.
Graph of Even Impulse Pair Function
The graph of the even impulse pair function is illustrated below:
It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \set {-\dfrac 1 2, \dfrac 1 2}$.
$2$ Dimensional Form
Let $\operatorname {II}: \R \to \R$ denote the even impulse pair function.
The $2$-dimensional form of $\operatorname {II}$ is defined and denoted:
- $\forall x, y \in \R: \map {\operatorname { {}^2 II} } {x, y} := \map {\operatorname {II} } x \map {\operatorname {II} } y$
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