Fourier series

In mathematics, a Fourier series (/ˈfʊrieɪ, -iər/) is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, 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). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving the weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform. a n = 2 P ∫ P s ( x ) ⋅ cos ⁡ ( 2 π x n P )   d x b n = 2 P ∫ P s ( x ) ⋅ sin ⁡ ( 2 π x n P )   d x . {displaystyle {egin{aligned}a_{n}&={frac {2}{P}}int _{P}s(x)cdot cos left(2pi x{ frac {n}{P}} ight) dx\b_{n}&={frac {2}{P}}int _{P}s(x)cdot sin left(2pi x{ frac {n}{P}} ight) dx.end{aligned}}}     (Eq.1) s N ( x ) = a 0 2 + ∑ n = 1 N ( a n cos ⁡ ( 2 π n x P ) + b n sin ⁡ ( 2 π n x P ) ) . {displaystyle {egin{aligned}s_{N}(x)={frac {a_{0}}{2}}+sum _{n=1}^{N}left(a_{n}cos left({ frac {2pi nx}{P}} ight)+b_{n}sin left({ frac {2pi nx}{P}} ight) ight).end{aligned}}}     (Eq.2) s N ( x ) = A 0 2 + ∑ n = 1 N A n ⋅ cos ⁡ ( 2 π n x P − φ n ) . {displaystyle s_{N}(x)={frac {A_{0}}{2}}+sum _{n=1}^{N}A_{n}cdot cos left({ frac {2pi nx}{P}}-varphi _{n} ight).}     (Eq.3) s N ( x ) = ∑ n = − N N c n ⋅ e i 2 π n x P . {displaystyle s_{N}(x)=sum _{n=-N}^{N}c_{n}cdot e^{i{ frac {2pi nx}{P}}}.}     (Eq.4) s N ( x ) = ∑ n = − N N c n ⋅ e i 2 π n x P . {displaystyle s_{N}(x)=sum _{n=-N}^{N}c_{n}cdot e^{i{ frac {2pi nx}{P}}}.}     (Eq.5)The first four partial sums of the Fourier series for a square waveAnother visualisation of an approximation of a square wave by taking the first 1, 2, 3 and 4 terms of its Fourier series. (An interactive animation can be seen here)A visualisation of an approximation of a sawtooth wave of the same amplitude and frequency for comparisonExample of convergence to a somewhat arbitrary function. Note the development of the 'ringing' (Gibbs phenomenon) at the transitions to/from the vertical sections. s ( x ) = a 0 2 + ∑ n = 1 ∞ [ a n cos ⁡ ( n x ) + b n sin ⁡ ( n x ) ] = 2 π ∑ n = 1 ∞ ( − 1 ) n + 1 n sin ⁡ ( n x ) , f o r x − π ∉ 2 π Z . {displaystyle {egin{aligned}s(x)&={frac {a_{0}}{2}}+sum _{n=1}^{infty }left\&={frac {2}{pi }}sum _{n=1}^{infty }{frac {(-1)^{n+1}}{n}}sin(nx),quad mathrm {for} quad x-pi otin 2pi mathbb {Z} .end{aligned}}}     (Eq.7)Multiplying both sides by cos ⁡ ( 2 k + 1 ) π y 2 {displaystyle cos(2k+1){frac {pi y}{2}}} , and then integrating from y = − 1 {displaystyle y=-1} to y = + 1 {displaystyle y=+1} yields:Theorem. The trigonometric polynomial f N {displaystyle f_{N}} is the unique best trigonometric polynomial of degree N {displaystyle N} approximating f ( x ) {displaystyle f(x)} , in the sense that, for any trigonometric polynomial p ≠ f N {displaystyle p eq f_{N}} of degree N {displaystyle N} , we have In mathematics, a Fourier series (/ˈfʊrieɪ, -iər/) is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, 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). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving the weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform. The Fourier series is named in honour 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 (continuous) 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. 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, thin-walled shell theory, etc. Consider a real-valued function, s ( x ) , {displaystyle s(x),}   that is integrable on an interval of length P , {displaystyle P,}   which will be the period of the Fourier series.  Common examples of analysis intervals are: The analysis process determines the weights, indexed by integer n , {displaystyle n,} which is also the number of cycles of the n th {displaystyle n^{ ext{th}}} harmonic in the analysis interval. Therefore, the length of a cycle, in the units of x , {displaystyle x,} is P / n . {displaystyle P/n.}   And the corresponding harmonic frequency is n / P . {displaystyle n/P.}   sin ⁡ ( 2 π x n P ) {displaystyle sin left(2pi x{ frac {n}{P}} ight)} and cos ⁡ ( 2 π x n P ) {displaystyle cos left(2pi x{ frac {n}{P}} ight)} are n t h {displaystyle n^{th}} harmonics, and their amplitudes (weights) are found by integration over the interval of length P {displaystyle P} : The synthesis process (the actual Fourier series) is: In general, integer N {displaystyle N} is theoretically infinite. Even so, the series might not converge or exactly equate to s ( x ) {displaystyle s(x)} at all values of x {displaystyle x} (such as a single-point discontinuity) in the analysis interval.  For the 'well-behaved' functions typical of physical processes, equality is customarily assumed.