A Fourier series () is a summation of
harmonically
In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However, ...
related
sinusoidal functions, also known as components or harmonics. The result of the summation is a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see
Pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
and
Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized''
by a Fourier series. Such is the case of a
discrete-time Fourier transform.
Convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in th ...
means that as more and more components from the series are summed, each successive ''partial Fourier series'' sum will better
approximate
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
the function, and will equal the function with a
potentially infinite number of components. The
mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
s for this may be collectively referred to as the ''Fourier Theorem'' (see ). The figures below illustrate some partial Fourier series results for the components of a
square wave.
File:SquareWaveFourierArrows,rotated,nocaption 20fps.gif, A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).
File:Fourier Series.svg, The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums ''converge to'' (become more and more like) the square wave.
File:Fourier series and transform.gif, Function (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform is a frequency-domain representation that reveals the amplitudes of the summed sine waves.
Another analysis technique (not covered here), suitable for both periodic and non-periodic functions, is the
Fourier transform, which provides a frequency-continuum of component information. But when applied to a periodic function all components have zero amplitude, except at the harmonic frequencies. The inverse Fourier transform is a synthesis process (like the Fourier series), which converts the component information (often known as the
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
representation) back into its
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
representation.
Since
Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued functions of real arguments, and used the
sine and cosine functions as the
basis set for the decomposition. Many other
Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized i ...
have since been defined, extending his initial idea to many applications and birthing an
area of mathematics called
Fourier analysis.
Analysis process
This section describes the analysis process that derives the parameters of a Fourier series that approximates a known function,
An example of synthesizing an unknown function from known parameters is
discrete-time Fourier transform.
Common forms
The Fourier series can be represented in different forms. The ''amplitude-phase'' form, ''sine-cosine'' form, and ''exponential'' form are commonly used and are expressed here for a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
. (See and for alternative forms).
The number of terms summed,
, is a potentially infinite integer. Even so, the series might not
converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
or exactly equate to
at all values of
(
such as a single-point discontinuity) in the analysis interval. For the
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
functions typical of physical processes, equality is customarily assumed, and the
Dirichlet conditions
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
provide sufficient conditions.
The integer index,
, is also the number of cycles the
harmonic makes in the function's period
. Therefore:
* The
harmonic's
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
is
and in units of
.
* The
harmonic's
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
is
and in reciprocal units of
.
Amplitude-phase form
The Fourier series in
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
-
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
form is:
*Its
harmonic is
.
*
is the
harmonic's
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and
is its
phase shift
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
.
*The
fundamental frequency
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
of
is the term for when
equals 1, and can be referred to as the
harmonic.
*
is sometimes called the
harmonic or
DC component
DC, D.C., D/C, Dc, or dc may refer to:
Places
* Washington, D.C. (District of Columbia), the capital and the federal territory of the United States
* Bogotá, Distrito Capital, the capital city of Colombia
* Dubai City, as distinct from th ...
. It is the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
value of
.
Clearly can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing, for those not yet familiar with this concept, is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the potentially infinite number of terms (
).
The coefficients
and
can be understood and derived in terms of the
cross-correlation between
and a sinusoid at frequency
. For a general frequency
and an analysis interval
the cross-correlation function:
is essentially a
matched filter, with ''template''
. The
maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
of
is a measure of the amplitude
of frequency
in the function
, and the value of
at the maximum determines the phase
of that frequency. Figure 2 is an example, where
is a square wave (not shown), and frequency
is the
harmonic. It is also an example of deriving the maximum from just two samples, instead of searching the entire function. That is made possible by a
trigonometric identity
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
:
Combining this with gives:
which introduces the definitions of
and
. And we note for later reference that
and
can be simplified:
The derivative of
is zero at the phase of maximum correlation.
And the correlation peak value is:
:
Therefore
and
are the
Cartesian coordinates of a vector with
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
and
Sine-cosine form
Substituting into gives:
:
In terms of the readily computed quantities,
and
, recall that:
:
:
:
Therefore an alternative form of the Fourier series, using the Cartesian coordinates, is the sine-cosine form:
Exponential form
Another applicable identity is
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
:
:
(Note: the ∗ denotes
complex conjugation.)
Therefore, with definitions:
:
the final result is:
This is the customary form for generalizing to . Negative values of
correspond to
negative frequency The concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; it can be as simple as a wheel rotating clockwise or counterclockwise.
The rate is expressed in units such as revolutions (a.k.a. ''c ...
(explained in ).
Example
![Periodic identity function](https://upload.wikimedia.org/wikipedia/commons/e/e8/Periodic_identity_function.gif)
Consider a sawtooth function:
:
:
In this case, the Fourier coefficients are given by
:
It can be shown that the Fourier series converges to
at every point
where
is differentiable, and therefore:
When
, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ''s'' at
. This is a particular instance of the
Dirichlet theorem for Fourier series.
This example leads to a solution of the
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
.
Convergence
A proof that a Fourier series is a valid representation of any periodic function (that satisfies the
Dirichlet conditions
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
) is overviewed in .
In
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
applications, the Fourier series is generally presumed to converge almost everywhere (the exceptions being at discrete discontinuities) since the functions encountered in engineering are better-behaved than the functions that mathematicians can provide as counter-examples to this presumption. In particular, if
is continuous and the derivative of
(which may not exist everywhere) is square integrable, then the Fourier series of
converges absolutely and uniformly to
. If a function is
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
on the interval