Definition:Sawtooth Wave/Inverse
Definition
An inverse sawtooth wave is a periodic real function $S: \R \to \R$ defined as follows:
- $\forall x \in \R: \map S x = \begin {cases}
-x & : x \in \openint {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x > +\lambda \end {cases}$
where $\lambda$ is a given real constant
Points of Discontinuity
The points $\paren {2 r + 1} \lambda$, for $r \in \Z$, are jump discontinuities.
The values $\map S {\paren {2 r + 1} \lambda}$ may or may not be explicitly defined.
It is a common approach to include one of the endpoints of the interval from $-\lambda$ to $\lambda$, in order to ensure that the domain of $S$ is simply defined.
For the sawtooth wave for example:
- $\forall x \in \R: \map S x = \begin {cases}
x & : x \in \hointr {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x \ge +\lambda \end {cases}$
and, for the inverse sawtooth wave:
- $\forall x \in \R: \map S x = \begin {cases}
-x & : x \in \hointr {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x \ge +\lambda \end {cases}$
Another approach is to make $\map S {\paren {2 r + 1} \lambda} = 0$ for all $r \in \Z$.
The precise treatment of the discontinuities is often irrelevant or immaterial.
Graph of Inverse Sawtooth Wave
The graph of $S$ is given below:
Also known as
An inverse sawtooth wave can also be referred to as a reverse sawtooth wave.
Also see
- Results about sawtooth waves can be found here.