A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series.[1] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.
The study of the convergence of Fourier series focus on the behaviors of the partial sums, which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of a square wave.
Fourier series are closely related to the Fourier transform, which can be used to find the frequency information for functions that are not periodic. Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on a circle, usually denoted as
T
S1
Rn
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-valued functions of real arguments, and used the sine and cosine functions in the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.
A Fourier series is a continuous, periodic function created by a summation of harmonically related sinusoidal functions. It has several different, but equivalent, forms, shown here as partial sums. But in theory
N → infty.
P,
s\scriptscriptstyle(x)
The harmonics are indexed by an integer,
n,
P
\tfrac{P}{n}
x
\tfrac{n}{P}
x
Clearly these series can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the infinite number of terms. The amplitude-phase form is particularly useful for its insight into the rationale for the series coefficients. (see) The exponential form is most easily generalized for complex-valued functions. (see)
The equivalence of these forms requires certain relationships among the coefficients. For instance, the trigonometric identity:means that:
Therefore
An
Bn
Dn
\varphin.
The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform. In the latter case, the exponential form of Fourier series synthesizes a discrete-time Fourier transform where variable
x
But typically the coefficients are determined by frequency/harmonic analysis of a given real-valued function
s(x),
x
The objective is for
s\scriptstyle{infty
s(x)
x
P.
The notation
\intP
[-P/2,P/2]
[0,P]
P\triangleq2\pi
s(x)
P
s\scriptstyle{infty
\tfrac{2}{P}
s(x)=\cos\left(2\pi\tfrac{k}{P}x\right).
n=k
Ak=1
Bk=0.
Ak=
2 | |
P |
\underbrace{\intP\cos2\left(2\pi\tfrac{k}{P}x\right)dx}P/2=1,
Another applicable identity is Euler's formula:
\begin{align} \cos\left(2\pi\tfrac{n}{P}x-\varphin\right)&{}\equiv\tfrac{1}{2}e{P}x-\varphin\right)}+\tfrac{1}{2}e-i{P}x-\varphin\right)}\\[6pt] &=\left(\tfrac{1}{2}
-i\varphin | |
e |
\right) ⋅ ei{P}x}+\left(\tfrac{1}{2}
-i\varphin | |
e |
\right)* ⋅ ei{P}x} \end{align}
(Note: the ∗ denotes complex conjugation.)
Substituting this into and comparison with ultimately reveals:
Conversely:
Substituting into also reveals:[2]
and also apply when
s(x)
\operatorname{Re}(sN(x))
\operatorname{Im}(sN(x))
sN(x)=\operatorname{Re}(sN(x))+i \operatorname{Im}(sN(x)).
The coefficients
Dn
\varphin
s(x)
\tfrac{n}{P}
f,
[x0,x0+P],
is essentially a matched filter, with template
\cos(2\pifx)
\Chif(\tau)
(D)
f
s(x)
\tau
(\varphi)
s(x)
f
4th
\begin{align} \Chin(\varphi)&=\tfrac{2}{P}\intPs(x) ⋅ \cos\left(2\pi\tfrac{n}{P}x-\varphi\right)dx ; \varphi\in[0,2\pi]\\ &=\cos(\varphi) ⋅ \underbrace{\tfrac{2}{P}\intPs(x) ⋅ \cos\left(2\pi\tfrac{n}{P}x\right)dx}A+\sin(\varphi) ⋅ \underbrace{\tfrac{2}{P}\intPs(x) ⋅ \sin\left(2\pi\tfrac{n}{P}x\right)dx}B\\ &=\cos(\varphi) ⋅ A+\sin(\varphi) ⋅ B \end{align}
The derivative of
\Chin(\varphi)
\Chi'n(\varphi)=\sin(\varphi) ⋅ A-\cos(\varphi) ⋅ B=0 \longrightarrow \tan(\varphi)=
B | |
A |
\longrightarrow \varphi=\arctan(B,A)
Therefore, computing
An
Bn
\varphin
\begin{align} Dn\triangleq\Chin(\varphin) &=\cos(\varphin) ⋅ An+\sin(\varphin) ⋅ Bn\\ &=
An | ||||||
|
The notation
Cn
\widehat{s}(n)
S[n]
\begin{align} s(x)&=
infty | |
\sum | |
n=-infty |
\widehat{s}(n) ⋅ ei{P}x}&&\scriptstylecommonmathematicsnotation\\ &=
infty | |
\sum | |
n=-infty |
S[n] ⋅ ei{P}x}&&\scriptstylecommonengineeringnotation \end{align}
In engineering, particularly when the variable
x
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
S(f) \triangleq
infty | |
\sum | |
n=-infty |
S[n] ⋅ \delta\left(f-
n | |
P |
\right),
where
f
x
f
\tfrac{1}{P}
sinfty(x)
\begin{align} l{F}-1\{S(f)\}&=
infty | |
\int | |
-infty |
\left(
infty | |
\sum | |
n=-infty |
S[n] ⋅ \delta\left(f-
n | |
P |
\right)\right)eidf,\\[6pt] &=
infty | |
\sum | |
n=-infty |
S[n] ⋅
infty | ||
\int | \delta\left(f- | |
-infty |
n | |
P |
\right)eidf,\\[6pt] &=
infty | |
\sum | |
n=-infty |
S[n] ⋅ ei{P}x} \triangleq sinfty(x). \end{align}
The constructed function
S(f)
]]
Consider a sawtooth function:
s(x)=
x | |
\pi |
, for-\pi<x<\pi,
s(x+2\pik)=s(x), for-\pi<x<\piandk\inZ.
\begin{align} An&=
1 | |
\pi |
\pi | |
\int | |
-\pi |
s(x)\cos(nx)dx=0, n\ge0.\\[4pt] Bn&=
1 | |
\pi |
\pi | |
\int | |
-\pi |
s(x)\sin(nx)dx\\[4pt] &=-
2 | |
\pin |
\cos(n\pi)+
2 | |
\pi2n2 |
\sin(n\pi)\\[4pt] &=
2(-1)n+1 | |
\pin |
, n\ge1.\end{align}
s(x)
x
s
When
x=\pi
x=\pi
This example leads to a solution of the Basel problem.
See main article: Convergence of Fourier series. A proof that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions) is overviewed in .
In engineering applications, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are better-behaved than functions encountered in other disciplines. In particular, if
s
s(x)
s
s(x)
[x0,x0+P]
The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous[4] and later generalized to any piecewise-smooth) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy.[5] Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles.
The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.
From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet[6] and Bernhard Riemann[7] [8] expressed Fourier's results with greater precision and formality.
Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics,[9] shell theory,[10] etc.
Joseph Fourier wrote:
This immediately gives any coefficient ak of the trigonometrical series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integralcan be carried out term-by-term. But all terms involving
\cos(2j+1) | \piy | \cos(2k+1) |
2 |
\piy | |
2 |
kth
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
The Fourier series expansion of the sawtooth function (above) looks more complicated than the simple formula
s(x)=\tfrac{x}{\pi}
\pi
(x,y)\in[0,\pi] x [0,\pi]
y=\pi
T(x,\pi)=x
x
(0,\pi)
T(x,y)=
infty | |
2\sum | |
n=1 |
(-1)n+1 | |
n |
\sin(nx){\sinh(ny)\over\sinh(n\pi)}.
\sinh(ny)/\sinh(n\pi)
s(x)
T(x,y)
T
Another application is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.
Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.
s(x)
P
A0,An,Bn
s(x)
Time domain s(x) | Plot | Frequency domain (sine-cosine form) \begin{align}&A0\ &An forn\ge1\ &Bn forn\ge1\end{align} | Remarks | Reference | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
s(x)=A\left | \sin\left(\fracx\right)\right | \quad \text 0 \le x < P | \begin{align} A0=&
\\ An=&\begin{cases}
& neven\\ 0& nodd \end{cases}\\ Bn=&0\\ \end{align} | Full-wave rectified sine | [11] | ||||||||||||||
s(x)=\begin{cases} A\sin\left(
x\right)& for0\lex<P/2\\ 0& forP/2\lex<P\\ \end{cases} | \begin{align} A0=&
\\ An=&\begin{cases}
& neven\\ 0& nodd \end{cases}\\ Bn=&\begin{cases}
& n=1\\ 0& n>1 \end{cases}\\ \end{align} | Half-wave rectified sine | |||||||||||||||||
s(x)=\begin{cases} A& for0\lex<D ⋅ P\\ 0& forD ⋅ P\lex<P\\ \end{cases} | \begin{align} A0=&AD\\ An=&
\sin\left(2\pinD\right)\\ Bn=&
\left(\sin\left(\pinD\right)\right)2\\ \end{align} | 0\leD\le1 | |||||||||||||||||
for0\lex<P | \begin{align} A0=&
\\ An=&0\\ Bn=&
\\ \end{align} | ||||||||||||||||||
for0\lex<P | \begin{align} A0=&
\\ An=&0\\ Bn=&
\\ \end{align} | ||||||||||||||||||
\left(x-
\right)2 for0\lex<P | \begin{align} A0=&
\\ An=&
\\ Bn=&0\\ \end{align} |
This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:
s(x),r(x)
P
x\in[0,P].
S[n],R[n]
s
r.
Property | Time domain | Frequency domain (exponential form) | Remarks | Reference | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Linearity | a ⋅ s(x)+b ⋅ r(x) | a ⋅ S[n]+b ⋅ R[n] | a,b\inC | |||||||||||
Time reversal / Frequency reversal | s(-x) | S[-n] | ||||||||||||
Time conjugation | s*(x) | S*[-n] | ||||||||||||
Time reversal & conjugation | s*(-x) | S*[n] | ||||||||||||
Real part in time | \operatorname{Re}{(s(x))} |
(S[n]+S*[-n]) | ||||||||||||
Imaginary part in time | \operatorname{Im}{(s(x))} |
(S[n]-S*[-n]) | ||||||||||||
Real part in frequency |
(s(x)+s*(-x)) | \operatorname{Re}{(S[n])} | ||||||||||||
Imaginary part in frequency |
(s(x)-s*(-x)) | \operatorname{Im}{(S[n])} | ||||||||||||
Shift in time / Modulation in frequency | s(x-x0) | S[n] ⋅
{P}n} | x0\inR | [12] | ||||||||||
Shift in frequency / Modulation in time | s(x) ⋅
| S[n-n0] | n0\inZ |
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:[13]
\begin{array}{rccccccccc} Timedomain&s&=&
s | |
RE |
&+&
s | |
RO |
&+&i
s | |
IE |
&+&
\underbrace{i s | |
IO |
From this, various relationships are apparent, for example:
If
S
See main article: Parseval's theorem. If
s
L2(P)
P
See main article: Plancherel theorem. If
c0,c\pm,c\pm,\ldots
s\inL2(P)
S[n]=cn
n
Given
P
s | |
P |
r | |
P |
S[n]
R[n],
n\inZ,
P
S
R
P
\left\{cn\right\}n
c0(Z)
L1([0,2\pi])
\ell2(Z)
We say that
s
Ck(T)
s
R
k
kth
s\inC1(T)
\widehat{s'}[n]
s'
\widehat{s}[n]
s
\widehat{s'}[n]=in\widehat{s}[n]
s\inCk(T)
\widehat{s(k)
k\geq1
\widehat{s(k)
n\toinfty
|n|k\widehat{s}[n]
k\geq1
See main article: Compact group, Lie group and Peter–Weyl theorem.
One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L2(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the case.
An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups.
See main article: Laplace operator and Riemannian manifold.
If the domain is not a group, then there is no intrinsically defined convolution. However, if
X
X
X
L2(X)
X
[-\pi,\pi]
X
See main article: Pontryagin duality.
The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.
This generalizes the Fourier transform to
L1(G)
L2(G)
G
G
[-\pi,\pi]
G
R
We can also define the Fourier series for functions of two variables
x
y
[-\pi,\pi] x [-\pi,\pi]
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the JPEG image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.[15]
A three-dimensional Bravais lattice is defined as the set of vectors of the form:where
ni
ai
f(r)
R
f(r)=f(R+r)
r
ai\triangleq|ai|,
ai
ai
\hat{ai
ai
Thus we can define a new function,
This new function,
g(x1,x2,x3)
a1
a2
a3
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers
m1,m2,m3
g
\left[0,a1\right]
x1
And then we can write:
Further defining:
We can write
g
Finally applying the same for the third coordinate, we define:
We write
g
Re-arranging:
Now, every reciprocal lattice vector can be written (but does not mean that it is the only way of writing) as
G=m1g1+m2g2+m3g3
mi
gi
gi |
⋅
aj=2\pi\delta | |
ij |
\deltaij=1
i=j
\deltaij=0
i ≠ j
G
r
So it is clear that in our expansion of
g(x1,x2,x3)=f(r)
where
Assumingwe can solve this system of three linear equations for
x
y
z
x1
x2
x3
x
y
z
x1
x2
x3
(it may be advantageous for the sake of simplifying calculations, to work in such a rectangular coordinate system, in which it just so happens that
a1
a2
a3
a1
a2
a3
We can write now
h(G)
x1
x2
x3
dr
dxdydz
C
a1 ⋅ (a2 x a3)
See main article: Hilbert space.
In the language of Hilbert spaces, the set of functions
inx | |
\left\{e | |
n=e |
:n\in\Z\right\}
L2([-\pi,\pi])
[-\pi,\pi]
f
g
\langlef,g\rangle \triangleq
1 | |
2\pi |
\pi | |
\int | |
-\pi |
f(x)g*(x)dx,
g*(x)
g(x).
infty | |
f=\sum | |
n=-infty |
\langlef,en\rangleen.
1
L2([-\pi,\pi])
1
\sqrt{2}\cos(nx)
\sqrt{2}\sin(nx)
See main article: Convergence of Fourier series.
These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.[16] [17] [18] [19]
The earlier :
s | |
N |
(x)=
N | |
\sum | |
n=-N |
S[n] ei{P}x},
N
p | |
N |
N | |
(x)=\sum | |
n=-N |
p[n] ei{P}x}.
Parseval's theorem implies that:
See also: Gibbs phenomenon. Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
We have already mentioned that if
s
(i ⋅ n)S[n]
nth
s'
sinfty
s
s
This result can be proven easily if
s
C2
n2S[n]
n → infty
s
s
\alpha>1/2
\supx|s(x)-
s | |
N |
(x)|\le\sum|n||S[n]|
proves uniform convergence.
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at
x
s
x
L2
Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact.
In 1922, Andrey Kolmogorov published an article titled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere.[20]