Periodic function

The repeatable part of the function or waveform is called a cycle.

[1] For example, the trigonometric functions, which repeat at intervals of

Any function that is not periodic is called aperiodic.

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

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, i.e. 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 periodicity 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.

This function repeats on intervals of length

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.

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.

The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

, the complex exponential is made up of cosine and sine waves.

This means that Euler's formula (above) has the property such that if

("Incommensurate" in this context means not real multiples of each other.)

Periodic functions can take on values many times.

Some periodic functions can be described by Fourier series.

For instance, for L2 functions, Carleson's theorem states that they have a pointwise (Lebesgue) almost everywhere convergent Fourier series.

-periodic function, the converse is not necessarily true.

[3] A further generalization appears in the context of Bloch's theorems 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 where

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

, and an antiperiodic function is the special case

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.

To this end you can use the notion of a quotient space: That is, each element in

is an equivalence class of real numbers that share the same fractional part.

Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F = 1⁄f [f1 f2 f3 ... fN] where all non-zero elements ≥1 and at least one of the elements of the set is 1.

If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.