Definition:Sawtooth Wave

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Definition

A 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 Sawtooth Wave

The graph of $S$ is given below:

Sawtooth-wave.png


Inverse Sawtooth Wave

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


Also known as

A sawtooth wave can also be referred to as a saw wave.


Also see

  • Results about sawtooth waves can be found here.