Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

An illustration of a periodic function with period ${\displaystyle P.}$

Definition

A function f is said to be periodic if, for some nonzero constant P, it is the case that

${\displaystyle f(x+P)=f(x)}$

for all values of x in the domain. A nonzero constant P for which this is the case is called a period of the function. If there exists a least positive[1] constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods of the function.

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic tessellations of the plane. A sequence can also be viewed as a function defined on the natural numbers, and for a periodic sequence these notions are defined accordingly.

Examples

A graph of the sine function, showing two complete periods

For example, the sine function is periodic with period ${\displaystyle 2\pi }$ , since

${\displaystyle \sin(x+2\pi )=\sin x}$

for all values of ${\displaystyle x}$ . This function repeats on intervals of length ${\displaystyle 2\pi }$  (see the graph to the right).

Everyday examples are seen when the variable is time; for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.

For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function ${\displaystyle f}$  that gives the "fractional part" of its argument. Its period is 1. In particular,

${\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}$

The graph of the function ${\displaystyle f}$  is the sawtooth wave.

A plot of ${\displaystyle f(x)=\sin(x)}$  and ${\displaystyle g(x)=\cos(x)}$ ; both functions are periodic with period 2π.

The trigonometric functions sine and cosine are common periodic functions, with period 2π (see the figure on the right). The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.

Properties

If a function ${\displaystyle f}$  is periodic with period ${\displaystyle P}$ , then for all ${\displaystyle x}$  in the domain of ${\displaystyle f}$  and all positive integers ${\displaystyle n}$ ,

${\displaystyle f(x+nP)=f(x)}$

If ${\displaystyle f(x)}$  is a function with period ${\displaystyle P}$ , then ${\displaystyle f(ax)}$ , where ${\displaystyle a}$  is a non-zero real number, is periodic with period ${\displaystyle {\cfrac {P}{|a|}}}$ .

For example, ${\displaystyle f(x)=\sin(x)}$  has period ${\displaystyle 2\pi }$  therefore ${\displaystyle \sin(5x)}$  will have period ${\displaystyle {\cfrac {2\pi }{5}}}$ .

Double-periodic functions

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.)

Complex example

Using complex variables we have the common period function:

${\displaystyle e^{ikx}=\cos kx+i\,\sin kx.}$

Since the cosine and sine functions are both periodic with period 2π, and the complex exponential above is made up of cosine and sine waves, the above (actually Euler's formula) has the following property. If L is the period of the function then

${\displaystyle L=2\pi /k}$

Generalizations

Antiperiodic functions

One common subset of periodic functions is that of antiperiodic functions. This is a function f such that f(x + P) = −f(x) for all x. (Thus, a P-antiperiodic function is a 2P-periodic function.) For example, the sine or cosine function is π-antiperiodic and 2π-periodic. While a P-antiperiodic function is a 2P-periodic function, the inverse is not necessarily true.

Bloch-periodic functions

A further generalization appears in the context of Bloch waves and Floquet theory, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form:

${\displaystyle f(x+P)=e^{ikP}f(x)}$

where k is a real or complex number (the Bloch wavevector or Floquet exponent). Functions of this form are sometimes called Bloch-periodic in this context. A periodic function is the special case k = 0, and an antiperiodic function is the special case k = π/P.

Quotient spaces as domain

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a quotient space:

${\displaystyle {\mathbb {R} /\mathbb {Z} }=\{x+\mathbb {Z} :x\in \mathbb {R} \}=\{\{y:y\in \mathbb {R} \land y-x\in \mathbb {Z} \}:x\in \mathbb {R} \}}$ .

That is, each element in ${\displaystyle {\mathbb {R} /\mathbb {Z} }}$  is an equivalence class of real numbers that share the same fractional part. Thus a function like ${\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }$  is a representation of a 1-periodic function.

Calculating period

Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F = [f1 f2 f3 ... fN]/f where all non-zero elements >= 1 and at least one of the elements of the set is 1. To find the period, T, first find the Least Common Denominator of all the elements in the set. Period can be found as T = LCD/f. Consider that for a simple sinusoid, T = 1/f. Therefore, the LCD can be seen as a periodicity multiplier.

For set representing all notes of Western Major Scale: [1,9/8,5/4,4/3,3/2,5/3,15/8] the LCD is 24 therefore T = 24/f. For set representing all notes of a major triad: [1,5/4,3/2] the LCD is 4 therefore T = 4/f. For set representing all notes of a minor triad: [1 6/5 3/2] the LCD is 10 therefore T = 10/f.

If no Least Common Denominator exists, such that if one of the above elements were irrational, then the wave would not be periodic. [2]