Fourier series

A Fourier series (/ˈfʊrieɪ, -iər/) is a series expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. 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 Fourier series § Definition.

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, a more general tool that can even 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 the circle group, denoted by \(\mathbb{T}\) or \(S_1\). The Fourier transform is also part of Fourier analysis, but is defined for functions on \(\mathbb{R}^n\).

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 series expansion. 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.

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 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. 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.

Independently of Fourier, astronomer Friedrich Wilhelm Bessel introduced Fourier series to solve Kepler's equation. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.

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 and Bernhard Riemann 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, shell theory, etc.

Joseph Fourier wrote

, Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides (1807).

This immediately gives any coefficient a_k of the trigonometric 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 integral \[\begin{aligned} &\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy \\ &= \int_{-1}^1\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy \end{aligned}\] can be carried out term-by-term. But all terms involving \(\cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}\) for j ≠ k vanish when integrated from −1 to 1, leaving only the \(k^{\text{th}}\) term, which is 1.

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 (below) looks more complicated than the simple formula \(s(x)=\tfrac{x}{\pi}\), so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose sides measure \(\pi\) meters, with coordinates \((x,y) \in [0,\pi] \times [0,\pi]\). If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by \(y=\pi\), is maintained at the temperature gradient \(T(x,\pi)=x\) degrees Celsius, for \(x\) in \((0,\pi)\), then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by

\(T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.\)

Here, \(\sinh\) is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of the equation from Analysis § Example by \(\sinh(ny)/\sinh(n\pi)\). While our example function \(s(x)\) seems to have a needlessly complicated Fourier series, the heat distribution \(T(x,y)\) is nontrivial. The function \(T\) cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.

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.

The Fourier series of a complex-valued P-periodic function \(s(x)\), integrable over the interval \([0,P]\) on the real line, is defined as a trigonometric series of the form \[\sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x },\] such that the Fourier coefficients \(c_n\) are complex numbers defined by the integral \[c_n = \frac{1}{P}\int_0^P s(x)\ e^{-i 2\pi \tfrac{n}{P} x }\,dx.\] The series does not necessarily converge (in the pointwise sense) and, even if it does, it is not necessarily equal to \(s(x)\). Only when certain conditions are satisfied (e.g. if \(s(x)\) is continuously differentiable) does the Fourier series converge to \(s(x)\), i.e., \[s(x) = \sum_{n=-\infty}^\infty c_n e^{i 2\pi \tfrac{n}{P} x }.\] For functions satisfying the Dirichlet sufficiency conditions, pointwise convergence holds. However, these are not necessary conditions and there are many theorems about different types of convergence of Fourier series (e.g. uniform convergence or mean convergence). The definition naturally extends to the Fourier series of a (periodic) distribution \(s\) (also called Fourier-Schwartz series). Then the Fourier series converges to \(s(x)\) in the distribution sense.

The process of determining the Fourier coefficients of a given function or signal is called analysis, while forming the associated trigonometric series (or its various approximations) is called synthesis.

A Fourier series can be written in several equivalent forms, shown here as the \(N^\text{th}\) partial sums \(s_N(x)\) of the Fourier series of \(s(x)\):

The harmonics are indexed by an integer, \(n,\) which is also the number of cycles the corresponding sinusoids make in interval \(P\). Therefore, the sinusoids have:

• a wavelength equal to \(\tfrac{P}{n}\) in the same units as \(x\).
• a frequency equal to \(\tfrac{n}{P}\) in the reciprocal units of \(x\).

These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum. In the limit \(N\to\infty\), a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.

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 \(x\) represents frequency instead of time. In general, the coefficients are determined by analysis of a given function \(s(x)\) whose domain of definition is an interval of length \(P\).

The \(\tfrac{2}{P}\) scale factor follows from substituting Eq.1 into Eq.3 and utilizing the orthogonality of the trigonometric system. The equivalence of Eq.1 and Eq.2 follows from Euler's formula \[\cos x = \frac{e^{ix} + e^{-ix}}{2}, \quad \sin x = \frac{e^{ix} - e^{-ix}}{2i},\] resulting in:

\(c_n = \begin{cases} \tfrac{1}{2}(a_n +i b_n) & \text{if } n > 0,\\ a_n & \text{if } n = 0,\\ \tfrac{1}{2}(a_{-n} - i b_{-n}) & \text{if } n < 0,\\ \end{cases}\)

with \(c_{0}\) being the mean value of \(s\) on the interval \(P\). Conversely:

\(\begin{aligned} a_0 &= c_0 &\\ a_n &= c_n+c_{-n} \qquad &\textrm{for}~ n > 0 \\ b_n &= (c_n-c_{-n})/i \qquad &\textrm{for}~ n > 0 \end{aligned}\)

If the function \(s(x)\) is real-valued then the Fourier series can also be represented as

where \(A_{n}\) is the amplitude and \(\varphi_{n}\) is the phase shift of the \(n^{th}\) harmonic.

The equivalence of Eq.4 and Eq.1 follows from the trigonometric identity: \[\cos\left(2\pi \tfrac{n}{P}x-\varphi_n\right) = \cos(\varphi_n)\cos\left(2\pi \tfrac{n}{P} x\right) + \sin(\varphi_n)\sin\left(2\pi \tfrac{n}{P} x\right),\] which implies \[a_n = A_n \cos(\varphi_n)\quad \text{and}\quad b_n = A_n \sin(\varphi_n)\]

are the rectangular coordinates of a vector written in polar coordinates as \[A_n \angle \varphi_n = a_n + i b_{n}\] where \[A_n = \sqrt{a_n^2 + b_n^2}\quad \text{and}\quad \varphi_n = \operatorname{atan2}(b_n, a_n) = -\operatorname{Arg}(c_n)\]

An example of determining the parameter \(\varphi_n\) for one value of \(n\) is shown in Figure 2. It is the value of \(\varphi\) at the maximum correlation between \(s(x)\) and a cosine template, \(\cos(2\pi \tfrac{n}{P} x - \varphi)\). The blue graph is the cross-correlation function, also known as a matched filter:

\(\begin{aligned} \Chi(\varphi) &= \int_{P} s(x) \cdot \cos\left( 2\pi \tfrac{n}{P} x -\varphi \right)\, dx\quad \varphi \in \left[ 0, 2\pi \right]\\ &=\cos(\varphi) \underbrace{\int_{P} s(x) \cdot \cos\left( 2\pi \tfrac{n}{P} x\right) dx}_{X(0)} + \sin(\varphi) \underbrace{\int_{P} s(x) \cdot \sin\left( 2\pi \tfrac{n}{P} x\right) dx}_{ X(\pi/2) } \end{aligned}\)

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum: \[X'(\varphi) = \sin(\varphi)\cdot X(0) - \cos(\varphi)\cdot X(\pi/2) = 0, \quad \textrm{at}\ \varphi = \varphi_n.\] Hence \[\varphi_n \equiv \arctan(b_n/a_n) = \arctan(X(\pi/2)/X(0)).\]

The notation \(c_n\) is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (\(s,\) in this case), such as \(\widehat{s}(n)\) or \(S[n],\) and functional notation often replaces subscripting:

\(\begin{aligned} s(x) &= \sum_{n=-\infty}^\infty \widehat{s}(n)\cdot e^{i 2\pi \tfrac{n}{P} x} && \scriptstyle \text{common mathematics notation} \\ &= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} && \scriptstyle \text{common engineering notation} \end{aligned}\)

In engineering, particularly when the variable \(x\) represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:

\(S(f) \ \triangleq \ \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right),\)

where \(f\) represents a continuous frequency domain. When variable \(x\) has units of seconds, \(f\) has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of \(\tfrac{1}{P}\), which is called the fundamental frequency. \(s(x)\) can be recovered from this representation by an inverse Fourier transform:

\(\begin{aligned} \mathcal{F}^{-1}\{S(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right) e^{i 2 \pi f x}\,df, \\[6pt] &= \sum_{n=-\infty}^\infty S[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{P}\right) e^{i 2 \pi f x}\,df, \\[6pt] &= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} \ \ \triangleq \ s(x). \end{aligned}\)

The constructed function \(S(f)\) is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.

Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.

• \(s(x)\) designates a periodic function with period \(P.\)
• \(a_0, a_n, b_n\) designate the Fourier series coefficients (sine-cosine form) of the periodic function \(s(x).\)

This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:

• Complex conjugation is denoted by an asterisk.
• \(s(x),r(x)\) designate \(P\)-periodic functions or functions defined only for \(x \in [0,P].\)
• \(S[n], R[n]\) designate the Fourier series coefficients (exponential form) of \(s\) and \(r.\)

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:

\(\begin{array}{rlcccccccc} \mathsf{Time\ domain} & s & = & s_{\mathrm{RE}} & + & s_{\mathrm{RO}} & + & i\ s_{\mathrm{IE}} & + & i\ s_{\mathrm{IO}} \\ &\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\ \mathsf{Frequency\ domain} & S & = & S_\mathrm{RE} & + & i\ S_\mathrm{IO}\, & + & i\ S_\mathrm{IE} & + & S_\mathrm{RO} \end{array}\)

From this, various relationships are apparent, for example:

• The transform of a real-valued function \((s_\mathrm{RE}+s_\mathrm{RO})\) is the conjugate symmetric function \(S_\mathrm{RE}+i\ S_\mathrm{IO}.\) Conversely, a conjugate symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function \((i\ s_\mathrm{IE}+i\ s_\mathrm{IO})\) is the conjugate antisymmetric function \(S_\mathrm{RO}+i\ S_\mathrm{IE},\) and the converse is true.
• The transform of a conjugate symmetric function \((s_\mathrm{RE}+i\ s_\mathrm{IO})\) is the real-valued function \(S_\mathrm{RE}+S_\mathrm{RO},\) and the converse is true.
• The transform of a conjugate antisymmetric function \((s_\mathrm{RO}+i\ s_\mathrm{IE})\) is the imaginary-valued function \(i\ S_\mathrm{IE}+i\ S_\mathrm{IO},\) and the converse is true.

If \(s\) belongs to \(L^2(P)\) (periodic over an interval of length \(P\)) then: \[\frac{1}{P}\int_{P} |s(x)|^2 \, dx = \sum_{n=-\infty}^\infty \Bigl|S[n]\Bigr|^2.\]