Definition:Periodic Function/Real

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Definition

Let $f: \R \to \R$ be a real function.


Then $f$ is periodic if and only if:

$\exists L \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + L}$


Period

The period of $f$ is the smallest value $L \in \R_{>0}$ such that:

$\forall x \in \R: \map f x = \map f {x + L}$


Frequency

Let $f: \R \to \R$ be a periodic real function.

The frequency $\nu$ of $f$ is the reciprocal of the period $L$ of $f$:

$\nu = \dfrac 1 L$


Amplitude

The amplitude of $f$ is the maximum absolute difference of the value of $f$ from a reference level.


Phase

For a particular value of the independent variable $x$, the phase is the part or fraction of the period of $f$ through which $x$ has advanced, as measured from some arbitrary origin.


Also defined as

In some circumstances it can make the algebra simpler to define the period as being $2$ times a specified constant.

Hence some sources define a periodic real function as one such that:

$\exists L \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + 2 L}$


Examples

Sawtooth Function

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x - \floor x$

where $\floor x$ denotes the floor function.


Sawtooth-Function.png


$f$ is periodic with period $1$.


Also see

  • Results about periodic real functions can be found here.


Sources

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