A periodic function repeats its values at set intervals, called periods. A “function” is just a type of equation where every input (e.g. the x-value) results in a unique output (e.g. the y-value).
More formally, we say that this type of function has a positive constant “k” where any input(x):
f(x + k) – f(x).
A periodic function is sometimes called fully periodic, purely periodic, or strictly periodic (Depner & Rasmussen, 2017). This broad class of functions, which can all be represented by a Fourier series, also includes (mathematically speaking) almost-periodic functions.
The period, P, is the length of one complete cycle. It is defined as the smallest value for which the above notation holds true. The graph repeats itself after P units. You can think of a period as a repeating interval on a graph: it’s the area you can cut and paste over and over again to make a full graph of the function. To put that another way, a graph with period P stays the same if you shift it along the x-axis to the left or right.
The period (P) must be greater than zero; In other words, you can’t have a negative period.
Examples of Periodic Functions
Trigonometric functions are all periodic. The sine function and cosine function are two well known examples.
The constant function is not a periodic function because—although it repeats—the periods are all equal to zero. It is an example of an aperiodic function (“aperiodic” means any function that isn’t periodic).
Real Life Examples
- Motion of a Ferris wheel.
- Musical sounds—it’s what makes them different them from random sounds (Hall, n.d.).
- The number of hours of sunlight over the course of one year.
- Flickering of a fluorescent light.
An aperiodic function (or non periodic function) is any function that isn’t periodic (Depner & Rasmussen, 2017). As periodic functions repeat their values at set periods, you could also think of a non periodic function as one that doesn’t have repeating intervals.
Although an aperiodic function isn’t not periodic in nature, there is a very close relationship: mathematically, you can think of them as periodic functions with a period of infinity (Adams, 2020).
“The transition from a periodic function to an aperiodic function is accomplished by allowing the fundamental period T to increase without limit. In other words, if T becomes infinite, the function never repeats itself and, therefore, the function is aperiodic” ~ Caggiano (1996)
Aperiodic Function Subclasses
Two important subclasses of aperiodic functions are almost periodic and quasiperiodic functions.
At first, it might seem that subclasses of “non periodic functions” isn’t useful at all. But the opposite is true: many of these functions have very close relationships with periodic functions, mathematically speaking. What is considered “close” differs from author to author, but in general they are connected by their periodic nature:
- Almost-periodic function, although not periodic themselves, can be represented by a sum of two or more periodic functions.
- Quasiperiodic functions are a combination of periodic functions of different frequencies that never completely match up.
Almost-periodic functions, are an important class of aperiodic functions. They can be represented by a sum of two or more periodic functions. In other words, they can be formed by summing two or more harmonic parts. Two of the parts must be frequencies that aren’t rational multiples of one another (Depner & Rasmussen, 2017).
As an example, the following almost periodic function has two distinct harmonic parts:
f(t) = 6 sin(4t) + 14 cos(6√4t).
Quasi-periodic functions are a special case of almost periodic functions. They are a not periodic; They are a combination of periodic functions of different frequencies that never match exactly.
Perhaps the simplest way to create one is just to add two periodic functions: one with a rational period and one with an irrational period (Ong, 2020). Fourier transforms of quasi-periodic functions are discrete sets of delta functions; they can always be expressed as a series of sines and cosines with non matching lengths—or with an amount of arithmetically independent basis vectors that exceed the number of independent variables (Cahn, 2001).
There are several ways to define quasiperiodic functions mathematically. One fairly straightforward way (Jorba & Simo, 1984):
“A function f is a quasiperiodic function with basic frequencies ω1, …, ωr if f(t) = F(θ1,…, θr) where F is 2π periodic in all its arguments and θj = ωjt for j = 1, …,r”
Desmos Graphing Calculator.
Adams, M. (2020). Continuous-Time Signals and Systems (Edition 2.0).
Caggiano, D. (1996). Comparison of Different Signal Processing Algorithms to Extract the Respiration Waveform from the ECG. Retrieved November 13, 2020 from: http://archives.njit.edu/vol01/etd/1990s/1996/njit-etd1996-014/njit-etd1996-014.pdf
Cahn, J. (2001). Quasicrystals. Journal of Research of the National Institute of Standards and Technology. 106, 975–982.
Depner, J. & Rasmussen, T. (2017). Hydrodynamics of Time-Periodic Groundwater Flow: Diffusion Waves in Porous Media, Geophysical Monograph 224. American Geophysical Union.
Dua, R. (2014). Experimentation of Transforms. Retrieved November 13, 2020 from: http://web.mst.edu/~rdua/Digital%20Signal%20Processing_files/Sample%202.pdf
Hall, R. Sounding Number. Retrieved November 29, 2019 from: http://people.sju.edu/~rhall/SoundingNumber/periodicfunctions.pdf
Jorba, A. & Simo, C. (1984). On Quasiperiodic Perturbations of Elliptic Equilibrium Points. Retrieved November 13, 2020 from: https://upcommons.upc.edu/bitstream/handle/2117/901/9501jorba.pdf
Chapter 19: Trigonometry: Introducing Periodic Functions. Retrieved November 29, 2019 from: http://www.math.harvard.edu/archive/xb_spring_06/files/chap19-20.pdf
Ong, D. (2020). Quasiperiodic music. Retrieved November 13, 2020 from: https://export.arxiv.org/ftp/arxiv/papers/2009/2009.04667.pdf
Periodic Functions. Article posted on the Oregon State website. Retrieved November 29, 2019 from: https://oregonstate.edu/instruct/mth251/cq/Glossary/gloss.periodic.html