# Fourier series A **Fourier series** is the expansion of a periodic function as a sum of [[Sine_wave|sinusoids]] whose frequencies are whole-number multiples of the function's repetition rate. A [[Signal|signal]] that repeats every T seconds contains only the fundamental frequency 1/T and its harmonics 2/T, 3/T and so on, and the series assigns each harmonic a complex coefficient that fixes its amplitude and phase. The coefficients are found by integrating the function against each harmonic over one period, and plotted against harmonic number they form the signal's line spectrum, the discrete [[Frequency_domain|frequency-domain]] picture that the [[Fourier_transform]] extends to signals that never repeat. Joseph Fourier introduced the series to solve the equation of heat conduction, and his claim that an arbitrary periodic function could be written this way was resisted for decades; the questions it raised shaped [[Mathematical_analysis|analysis]] for a century. [[Harmonic_analysis]] generalises the series to functions on groups and manifolds. The primary microsim on this page, *Fourier series*, builds a square, sawtooth or triangle wave from a chain of rotating vectors, one circle per harmonic, and lets the reader raise the number of harmonics from 1 to 50 while watching the traced wave sharpen, the mean-squared error fall, and a jump keep its overshoot of about 9 % no matter how many circles are added. ## History ### Beginnings The idea that a vibrating string moves as a superposition of sinusoidal modes predates Fourier. In the middle of the eighteenth century Jean le Rond d'Alembert and Leonhard Euler solved the [[Wave_equation|wave equation]] for the string, and Daniel Bernoulli argued that the general motion is a sum of harmonic modes, each a standing [[Sine_wave|sine wave]] with a whole number of half-wavelengths along the string.[^cn-string] Euler and d'Alembert objected that smooth sinusoids could not add up to a plucked string's cornered shape, and the question was left open. ### Fourier's motivation Fourier's problem was heat, not sound. Treating [[Thermal_conduction|conduction]] in a solid, he derived the [[Partial_differential_equation|partial differential equation]] now called the heat equation and solved it by separating the variables, which turns the spatial part into sinusoids and the time part into decaying exponentials. Matching an arbitrary initial temperature profile then requires exactly the expansion the string theorists had argued over; Fourier asserted that any profile could be expanded, gave the coefficient integrals, and applied them to profiles with corners and jumps. He presented the work to the Institut de France in a memoir of December 21, 1807, examined by Lagrange and Laplace among others; the memoir was not printed at the time, and the mature treatment appeared as *Théorie analytique de la chaleur* in 1822.[^gg1972][^fourier1822] Rigour came later: Peter Gustav Lejeune Dirichlet published the first convergence proof in 1829, for bounded functions with finitely many extrema and jumps.[^dirichlet1829] ### Other applications Beyond heat and strings, the series became the standard description of any repeating quantity. In [[Acoustics|acoustics]] the harmonic content of a sustained tone is its Fourier series; in [[Electrical_engineering|electrical engineering]] a periodic voltage passed through a [[Linear_time-invariant_system|linear time-invariant]] circuit is handled one harmonic at a time.[^j118] Voiced speech is modelled as a glottal pulse train filtered by the vocal tract, with lines at multiples of the pitch: 110 Hz gives a period of about 9.1 ms and lines every 110 Hz.[^j130] In [[Quantum_mechanics|quantum mechanics]] the electron states of a crystal obey [[Bloch's_theorem|Bloch's theorem]], a Fourier expansion over the lattice, and in [[Image_compression|image compression]] the [[Discrete_cosine_transform|discrete cosine transform]] describes blocks of pixels by a few low harmonics. ## Definition ### Synthesis Let x(t) be periodic with period T, fundamental frequency f0 = 1/T and [[Angular_frequency|angular frequency]] ω0 = 2π/T. Its Fourier series in exponential form is `x(t) = Σ_k c_k exp(j 2π k t / T)`, summed over all integers k.[^j104] Each term is a phasor spinning at k times the fundamental rate, and a real [[Signal|signal]] pairs the terms k and −k into a real sinusoid, so the series can also be written `x(t) = a_0 + Σ_{k≥1} [a_k cos(k ω0 t) + b_k sin(k ω0 t)]`. The microsim performs synthesis in the sine form: every target it offers is an odd function of time, so the cosine coefficients vanish and the partial sum it draws is `S_N(t) = Σ_{k=1}^{N} b_k sin(k ω0 t)`, the general series with `a_k = 0` and `c_k = b_k/(2j)` for k ≥ 1. ### Analysis The coefficients follow from the [[Orthogonality|orthogonality]] of the harmonics, whose products integrate to zero over a period unless the indices match. Multiplying the series by `exp(−j 2π k t / T)` and integrating therefore isolates one coefficient, `c_k = (1/T) ∫_T x(t) exp(−j 2π k t / T) dt`.[^j104] This step is analysis, and a_0 is the mean of x over a period. For a sampled signal the integral becomes a finite sum, the [[Discrete_Fourier_transform|discrete Fourier transform]]. #### Example For a square wave that switches between −1 and +1 with half a period at each level, the analysis integral gives `c_k = 2/(jπk)` for odd k and zero for even k.[^j104] In sine form that is `b_k = 4/(πk)` for odd k,[^j145] the law the microsim prints as its coefficient rule for the square wave; the first coefficient, 4/π ≈ 1.273, is larger than the wave itself, and the sketch's diagnostics show it as b_1. The even harmonics vanish because shifting the square wave by half a period changes its sign, which no even harmonic can do. ### Amplitude-phase form Combining the terms for k and −k of a real signal gives `x(t) = a_0 + Σ_{k≥1} A_k cos(k ω0 t + φ_k)`, with amplitude `A_k = 2|c_k|` and phase φ_k equal to the angle of c_k. This is the form the microsim's geometry realises: each harmonic is a vector of length A_k rotating at k ω0, and the radii of its nested circles are the amplitude spectrum. ### Common notations Mathematics texts usually write the series over [−π, π] with i for the imaginary unit and a_0/2 as the constant term; engineering texts prefer a period T, the variable f0 or ω0 and the letter j, since i is reserved for current.[^s29] Conventions also differ on whether N counts terms or the highest harmonic index: the sketch uses the index, so at N = 5 the square wave has three active circles. ## Table of common Fourier series The table lists the sine-series coefficients of the waveforms the microsim can synthesise, together with two classic waveforms from the textbook literature, all of unit amplitude and written with ω0 = 2π/T. | Waveform | Coefficients | Harmonics present | Decay | |---|---|---|---| | Square wave, ±1 | `b_k = 4/(πk)` | odd k only | 1/k | | Sawtooth, ramp from −1 to +1 | `b_k = (2/π)(−1)^(k+1)/k` | all k, alternating sign | 1/k | | Triangle wave, ±1 | `b_k = (8/π²)(−1)^((k−1)/2)/k²` | odd k only | 1/k² | | Half-wave rectified sine | `a_0 = 1/π`, `b_1 = 1/2`, `a_k = −(2/π)/(k² − 1)` for even k | 1 and even k | 1/k² | | Rectangular pulse train, height A, width Δ | `c_k = A exp(−jπkΔ/T) sin(πkΔ/T)/(πk)`, `c_0 = AΔ/T` | all k | 1/k envelope | The square and triangle rows are the closed forms `(4A/π) Σ sin(k ω0 t)/k` and `(8A/π²) Σ ± sin(k ω0 t)/k²` of the instrumentation literature,[^dd41] the half-wave rectified sine is the worked example of the signals textbook,[^j110] and the pulse train's coefficients depend on the period only through the duty ratio Δ/T.[^j107] The decay column matters most in practice: a discontinuous waveform has coefficients falling only as 1/k and needs many harmonics, while a continuous waveform with corners has coefficients falling as 1/k² and needs a handful.[^j110] In the sketch the square wave's circles shrink slowly and keep mattering out to N = 50, while the triangle's collapse so fast that the trace looks finished after three or four. A pulse train whose period grows without bound is the bridge to the [[Fourier_transform]]: the lines crowd together while their heights follow a fixed sinc-shaped envelope.[^j120] ## Table of basic transformation rules An operation on a periodic signal has a corresponding operation on its coefficients. The rules below are standard consequences of the analysis integral; d_k denotes the coefficients of a second signal y(t) of the same period. | Operation on x(t) | Effect on c_k | |---|---| | Linear combination `a x(t) + b y(t)` | `a c_k + b d_k` | | Time shift `x(t − t0)` | `c_k exp(−j 2π k t0/T)` | | Time reversal `x(−t)` | `c_{−k}` | | Derivative `dx/dt` | `(j 2π k/T) c_k` | | Product `x(t) y(t)` | `Σ_m c_m d_{k−m}` | | Periodic convolution `∫_T x(τ) y(t − τ) dτ` | `T c_k d_k` | | Filtering by an LTI system with response H(f) | `c_k H(k/T)` | The last rule is the one engineering leans on. A [[Linear_time-invariant_system|linear time-invariant]] system with [[Frequency_response|frequency response]] H(f) turns a periodic input into the periodic output `y(t) = Σ_k c_k H(k/T) exp(j 2π k t/T)`: every harmonic passes through with its own gain and phase and nothing new is created.[^j118] A first-order [[Low-pass_filter|RC low-pass]] filter, `H(f) = 1/(1 + j 2π f R C)`, attenuates the high harmonics of a pulse train and rounds its edges, but since H(0) = 1 the mean of the output equals the mean of the input for every cutoff.[^j118][^j145] The shift rule explains why the sketch's line spectrum shows only magnitudes: sliding the waveform in time rotates the phase of every coefficient but leaves |c_k| alone. ## Properties ### Symmetry relations A real signal has `c_{−k} = c_k*`, so its amplitude spectrum is even in k and its phase spectrum odd. An even signal has real coefficients and an odd signal imaginary ones, meaning cosines only or sines only, and a signal with half-wave symmetry, `x(t + T/2) = −x(t)`, has no even harmonics. The microsim's waveforms are all odd, so they need only b_k; the square and triangle also have half-wave symmetry, which is why their even circles are absent. ### Riemann–Lebesgue lemma For any integrable periodic function the coefficients tend to zero as |k| grows. The lemma fixes only the fact of decay; the rate is set by smoothness, each integrable derivative buying one more power of 1/k, which is the pattern of the table above and, in the sketch, the shrinking of the outer circles. ### Parseval's theorem The average power of a periodic signal equals the sum of the powers of its harmonics: `(1/T) ∫_T |x(t)|² dt = Σ_k |c_k|²`.[^j104] The theorem turns the error of a truncated series into a tail sum: stopping at harmonic K leaves the mean-square error `rms²(ε_K) = 2 Σ_{k>K} |c_k|²`, the power in the discarded harmonics.[^j110] For the half-wave rectified sine, whose rms value is A/2, stopping at K = 4 leaves an rms error of about 3 % of the signal, while a 99-term square-wave approximation is still worse, in this measure, than a 10-term half-wave one.[^j110][^j145] The same accounting gives the square wave's harmonic [[Distortion|distortion]]: 8/π² of its power is in the fundamental, so about a fifth, 20 % by the textbook's count, lies in the harmonics.[^j145] The microsim's MSE readout is the tail power computed numerically; for the square wave it falls from 0.06707 at N = 5 to 0.00914 at N = 50 in the sketch. ### Plancherel's theorem Parseval's identity is one instance of a more general statement: the map from a square-integrable periodic function to its coefficient [[Sequence|sequence]] preserves inner products and is onto, so every square-summable sequence is the spectrum of some finite-[[Energy|energy]] function, and the two descriptions are one space in two coordinate systems. ### Convolution theorems Two theorems pair multiplication and [[Convolution|convolution]]. The periodic convolution of two signals of period T has coefficients `T c_k d_k`, and the product of the two signals has coefficients equal to the discrete convolution of the two coefficient sequences; both are the periodic form of the convolution theorem of the Fourier transform.[^tm83] The filtering rule above is the first theorem in disguise, since an LTI system convolves its input with its [[Impulse_response|impulse response]]; the second is why [[Amplitude_modulation|amplitude modulation]] shifts a whole line spectrum. ### Derivative property Differentiating a periodic signal multiplies c_k by `j 2π k/T`. The rule is what made Fourier's method work on the heat equation, and it also governs decay: the derivative of a square wave is a train of impulses, whose coefficients do not decay at all, one power of k above the square wave's own 1/k law. ### Compact groups The harmonics `exp(j 2π k t/T)` are the characters of the circle, the [[Group_theory|group]] of plane rotations, and a Fourier series is an expansion in those characters. On any compact group the Peter–Weyl theorem replaces the characters by the matrix entries of the irreducible representations, which again form an orthogonal basis for square-integrable functions on the group. ### Riemannian manifolds On a curved space the harmonics are replaced by the eigenfunctions of the Laplace–Beltrami operator, the natural second-derivative operator of a [[Manifold|manifold]]: sinusoids on the circle, spherical harmonics on the sphere, and on a compact manifold in general a discrete orthogonal basis in which the eigenvalue plays the part of the squared frequency. ### Locally compact Abelian groups Pontryagin duality places the series and the transform under one roof. Every locally compact abelian group has a dual group of characters: the dual of the circle is the integers, giving the series; the dual of the real line is itself, giving the transform; and the dual of the integers modulo N is again the integers modulo N, giving the [[Discrete_Fourier_transform|discrete Fourier transform]]. ## Extensions ### Fourier-Stieltjes series A periodic measure, rather than a function, also has Fourier coefficients, obtained by integrating the harmonic against the measure. The basic example is a train of unit impulses spaced T apart, built from the [[Dirac_delta_function|Dirac delta function]]: every coefficient is 1/T, the series does not converge as a function, and yet the object is exactly what [[Sampling_(signal_processing)|sampling]] theory needs, since multiplying a signal by this train is sampling it. ### Fourier series on a square A function periodic in two variables expands in products `exp(j 2π (k x/T1 + l y/T2))`, one coefficient per pair (k, l). The coefficient array is the two-dimensional line spectrum of a periodic image and the basis of frequency-domain [[Digital_image_processing|digital image processing]]; block-based [[Image_compression|image compression]] uses the cosine-only version, the [[Discrete_cosine_transform|discrete cosine transform]], and keeps only the low-order coefficients. ### Fourier series of a Bravais-lattice-periodic function A function with the periodicity of a crystal lattice expands in plane waves `exp(j G·r)`, where G runs over the reciprocal lattice, the wave vectors with a whole number of cycles across every lattice translation. The electron density of a crystal is such a function, and its coefficients are what [[X-ray_crystallography|X-ray crystallography]] measures as diffraction intensities; [[Bloch's_theorem|Bloch's theorem]] writes the electron wavefunctions as plane waves modulated by lattice-periodic functions of the same kind. ### Hilbert space In the language of [[Functional_analysis|functional analysis]] the normalised harmonics `exp(j 2π k t/T)/√T` form an orthonormal basis of the space of square-integrable functions on one period, and the coefficient integral is the inner product of the signal with a basis vector. Parseval's theorem is then the theorem of Pythagoras, and the truncated series is an orthogonal projection onto the span of the first harmonics. ## Fourier theorem proving convergence of Fourier series ### Least squares property Among all trigonometric polynomials of degree at most K, the partial Fourier sum has the smallest mean-square error, and that error is the tail power `2 Σ_{k>K} |c_k|²` of Parseval's theorem.[^j110] Fitting sinusoids by least squares to unevenly spaced data is a separate technique, [[Least-squares_spectral_analysis|least-squares spectral analysis]], which reduces to the Fourier coefficients when the samples are uniform and span whole periods. ### Convergence theorems Dirichlet's theorem of 1829 gives pointwise convergence for piecewise smooth functions: the series converges to the function wherever it is continuous, and at a jump to the midpoint of the two one-sided limits.[^dirichlet1829][^j116] Lipót Fejér showed in 1900 that averaging the partial sums restores convergence for every continuous function.[^fejer1900] For functions that are merely square-integrable, convergence holds in the mean-square sense, which is the sense in which engineering works: the error power goes to zero even where the pointwise error does not.[^j116] Lennart Carleson proved in 1966 that the series of a square-integrable function also converges pointwise almost everywhere.[^carleson1966] ### Divergence The first counterexample was a continuous function whose series diverges at a point, constructed by Paul du Bois-Reymond in 1873, and in 1923 Andrey Kolmogorov exhibited an integrable function whose series diverges almost everywhere.[^cn-dbr][^kolmogorov1923] Engineering meets a milder failure every day: at a jump the partial sums overshoot by about 9 % of the jump height, an amount that does not shrink as harmonics are added, only narrows.[^j116] Henry Wilbraham described the effect in 1848 and [[Josiah_Willard_Gibbs|J. Willard Gibbs]] rediscovered it in letters to *Nature* in 1898 and 1899, after which Maxime Bôcher named it the Gibbs phenomenon in 1906.[^wilbraham1848][^gibbs1899][^bocher1906] For the ±1 square wave the overshoot is about 0.18 above +1, and a series truncated at 200 harmonics still rings at every edge.[^j116][^dd41] In the microsim the marker at the peak of the partial sum reads about 9 % of the jump on the square wave from N = 5 upward and never falls. ## Microsims The *Fourier series* sketch draws a periodic wave three ways at once. The left pane chains one rotating vector per harmonic, tip to tail, the k-th vector having length |b_k| and turning at k times the fundamental, and the tip of the last vector traces S_N(t). The top-right pane plots the partial sum over the exact target across two periods, with the Gibbs marker on discontinuous targets; the middle pane is the line spectrum, |b_k| against k for the first 24 harmonics, with the terms in the chain drawn bright; and the diagnostics panel prints the waveform, N with the number of active circles, the coefficient law, b_1 and the live MSE over one period. The controls are the slider *number of harmonics N* (1–50, default 5), the slider *animation speed* (0–3×, playback only), the button *Waveform* cycling Square, Sawtooth and Triangle, and the button *Animation* to play or pause; the arrow keys step N, the W key cycles the waveform and the space bar pauses. The same sums evaluated on a grid of samples are what a [[Discrete_Fourier_transform|discrete Fourier transform]] computes. A three.js companion on this page renders the same decomposition in depth, with the harmonics laid out as ribbons along a frequency axis. *Try:* Set the waveform to Square and slide N from 5 to 50: watch the MSE readout fall from 0.06707 to 0.00914 while the Gibbs marker stays near 9 % of the jump. Then switch to Triangle at the same N and watch the marker vanish and the MSE readout round to 0.00000, because the circle radii now fall as 1/k². ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Fourier_series) : [Wikitube](https://en.wikitube.io/wiki/Fourier_series) Skeleton mirrored at revision 1373499036. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Fourier_transform]] - [[Signal]] - [[Discrete_Fourier_transform]] - [[Discrete-time_Fourier_transform]] - [[Fast_Fourier_transform]] - [[Fourier_analysis]] - [[Harmonic_analysis]] - [[Discrete_cosine_transform]] - [[PORTAL_Signal_Processing]] ## Notes The sketch is ILLUSTRATIVE in two respects: the fundamental is drawn as 1 × ω0 with no unit, so the animation shows shape rather than a frequency in hertz, and the speed slider changes only the playback rate. Its Gibbs marker measures the peak of the drawn partial sum, so on the sawtooth, whose overshoot builds up slowly with N, it reads well below 9 % at small N. Page numbers in the references are PDF pages of the open editions. ## References [^j104]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 104–106 (PDF pages): the exponential Fourier series, the coefficient integral, the square-wave coefficients and Parseval's theorem. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j107]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 107–109 (PDF pages): the periodic pulse train's coefficients and their dependence on the duty ratio. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j110]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 110–113 (PDF pages): the half-wave rectified sine, the truncation-error formula, the 1/k and 1/k² decay rates, the 3 % error at K = 4 and the comparison of a 99-term square wave with a 10-term half-wave. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j116]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 116–117 (PDF pages): the 9 % Gibbs overshoot, mean-square convergence and the value of the series at a jump. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j118]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 118–119 (PDF pages): the response of a linear time-invariant filter to a periodic input and the RC low-pass response. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j120]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 120–121 (PDF pages): the Fourier transform as the limit of the line spectrum as the period grows. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j130]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, p. 130 (PDF page): the glottal pulse-train model of voiced speech and the 110 Hz pitch example. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^j145]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, pp. 145–146 (PDF pages): the sine-form square-wave coefficients, the rms value of the half-wave rectified sine, the 20 % square-wave distortion figure and the fixed output mean of an RC-filtered pulse train. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. [^dd41]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 2020, pp. 41–44 (PDF pages): the square-wave and triangle-wave series and the ringing of a 200-harmonic square wave. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^tm83]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026, pp. 83–84 (PDF pages): the convolution theorem. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^s29]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, p. 29 (PDF page): the engineering convention of j for the imaginary unit. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^fourier1822]: Fourier, J. *Théorie analytique de la chaleur*. Paris: Firmin Didot, 1822. [^gg1972]: Grattan-Guinness, I.; Ravetz, J. R. *Joseph Fourier 1768–1830: A Survey of His Life and Work, Based on a Critical Edition of His Monograph on the Propagation of Heat, Presented to the Institut de France in 1807*. Cambridge, Massachusetts: MIT Press, 1972. [^cn-string]: Citation needed: the vibrating-string memoirs of d'Alembert (1747), Euler (1748) and Daniel Bernoulli (1753) in the Berlin academy volumes; a history of mathematics giving the memoir titles and page numbers would settle the dates and the wording of each position. [^dirichlet1829]: Lejeune Dirichlet, P. G. "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". *Journal für die reine und angewandte Mathematik* 4 (1829), pp. 157–169. [^fejer1900]: Fejér, L. "Sur les fonctions bornées et intégrables". *Comptes rendus hebdomadaires des séances de l'Académie des sciences* 131 (1900), pp. 984–987. [^carleson1966]: Carleson, L. "On convergence and growth of partial sums of Fourier series". *Acta Mathematica* 116 (1966), pp. 135–157. [^kolmogorov1923]: Kolmogoroff, A. "Une série de Fourier–Lebesgue divergente presque partout". *Fundamenta Mathematicae* 4 (1923), pp. 324–328. [^cn-dbr]: Citation needed: du Bois-Reymond's 1873 construction of a continuous function whose Fourier series diverges at a point; his note in the Göttingen *Nachrichten* of that year, with its page numbers, would settle the reference. [^wilbraham1848]: Wilbraham, H. "On a certain periodic function". *Cambridge and Dublin Mathematical Journal* 3 (1848), pp. 198–201. [^gibbs1899]: Gibbs, J. W. "Fourier's series". *Nature* 59 (December 29, 1898), p. 200, and *Nature* 59 (April 27, 1899), p. 606. [^bocher1906]: Bôcher, M. "Introduction to the theory of Fourier's series". *Annals of Mathematics* 7 (1906), pp. 81–152. ### Bibliography - Don Johnson. *Fundamentals of Electrical Engineering I*. 2014. CC BY. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 - John Dyer; Chad Davis. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 2020. CC BY-NC-SA. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods - Christian Tiberius; Max Mulder. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026. CC BY. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory - Michael Stiber; Bilin Stiber; Eric Larson. *Signal Computing: Digital Signals in the Software Domain*. 2020. CC BY-SA. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain ## External links - Fourier series, live p5.js sketch: https://editor.p5js.org/sciencenibber/full/tW8_OcN4Q - Fourier series, fork the sketch in the p5.js editor: https://editor.p5js.org/sciencenibber/sketches/tW8_OcN4Q <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->