# Fourier analysis <!-- MICROSIMGEN:BEGIN v1.7 — generated by g08_place_microsims.py; three.js first (§15); do not hand-edit inside --> ## Microsims — p5.js ### Fourier analysis (p5.js) · `Σ sinusoids` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/Os2HKArEh" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Fourier analysis — p5.js microsim"></iframe> </div> *The whole idea that any signal is a sum of sinusoids, and the tools for finding them.* **Open in the editor:** [&#9654; fork this sketch](https://editor.p5js.org/sciencenibber/sketches/Os2HKArEh) · movement *II · Transforms & the frequency domain* · library `p5js` ### Related microsims Live sims on neighbouring articles — 6 of them inside this article's own Wikipedia link tree: - [[Calculus]] *(in tree)* - [[Complex_analysis]] *(in tree)* - [[Convolution]] *(in tree)* - [[Digital_image_processing]] *(in tree)* - [[Digital_signal_processing]] *(in tree)* - [[Discrete_cosine_transform]] *(in tree)* *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.* <!-- MICROSIMGEN:END --> ## Links (Wikipedia order) <!-- injected from _registry/childlinks/Fourier_analysis.json (2026-07-30T02:09:12Z) --> `*-algebra` · `2_Pallas` · `3_Juno` · `Abelian_group` · `Acoustics` · `Alexis_Clairaut` · `Algebraic_analysis` · `Aliasing` · `Almost_everywhere` · `Amplitude` · `Asteroid` · `Audrey_Terras` · `Babenko–Beckner_inequality` · `Babylonian_mathematics` · `Banach_space` · `Bessel's_inequality` · `Bispectrum` · `Bochner_space` · `Bounded_function` · `Brady_Haran` · `Brunn–Minkowski_theorem` · `C*-algebra` · [[Calculus]] · [[Calculus_of_variations]] · `Cambridge_University_Press` · `Carl_Friedrich_Gauss` · `Cauchy–Schwarz_inequality` · `Characteristic_function_(probability_theory)` · `Chebyshev's_inequality` · `Chebyshev_distance` · `Chirplet_transform` · `Circle` · `Clarkson's_inequalities` · `Combinatorics` · [[Complex_analysis]] · `Complex_number` · `Conjugate_Fourier_series` · `Continuous_function` · 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`Friedrich_Wilhelm_Bessel` · `Function_(mathematics)` · `Function_space` · [[Functional_analysis]] · `Fundamental_theorem_of_calculus` · `Gabor_transform` · `Generalized_Fourier_series` · [[Geometry]] · `Gian-Carlo_Rota` · `Group_representation` · `Hanner's_inequalities` · `Harmonic` · `Harmonic_analysis` · `Hausdorff–Young_inequality` · `Heat_equation` · [[Heat_transfer]] · `Hilbert_space` · `Hypercomplex_analysis` · `Hölder's_inequality` · `Infimum_and_supremum` · `Infinity` · `Infrared_spectroscopy` · `Integral` · [[Integral_transform]] · `Interlaced_video` · `Inverse_function` · `Isoperimetric_inequality` · `JPEG` · `Jaggies` · `Joseph_Fourier` · `Kepler's_equation` · `L-infinity` · [[Laplace_transform]] · [[Least-squares_spectral_analysis]] · `Least_squares` · `Lebesgue_integral` · `Lebesgue_measure` · `Leonhard_Euler` · `Limit_(mathematics)` · `Lis_Brack-Bernsen` · `List_of_Fourier-related_transforms` · `Lists_of_integrals` · `Locally_integrable_function` · `Lorentz_space` · `Lp_space` · `Magnitude_(mathematics)` · `Marcinkiewicz_interpolation_theorem` · `Markov's_inequality` · `Mathematical_analysis` · `Mathematics` · `Matrix_calculus` · `Measurable_function` · `Measurable_space` · `Measure_(mathematics)` · `Measure_space` · `Mellin_transform` · `Microlocal_analysis` · `Milman's_reverse_Brunn–Minkowski_inequality` · `Minkowski_distance` · `Minkowski_inequality` · `Minkowski–Steiner_formula` · `Multiplication_algorithm` · `Non-uniform_discrete_Fourier_transform` · [[Nuclear_magnetic_resonance]] · `Number_theory` · `Numerical_analysis` · [[Nyquist–Shannon_sampling_theorem]] · `Oceanography` · `Optics` · [[Ordinary_differential_equation]] · `Orthogonal_functions` · [[Oscillation]] · `Otto_E._Neugebauer` · `P-adic_analysis` · `P-adic_number` · `Parseval's_identity` · `Parseval's_theorem` · [[Partial_differential_equation]] · `Periodic_summation` · `Phase_(waves)` · [[Physics]] · `Plancherel_theorem` · `Poisson_summation_formula` · `Polarization_identity` · `Polynomial` · `Pontryagin_duality` · `Probability_theory` · `Protein` · `Prékopa–Leindler_inequality` · `Pythagorean_theorem` · `Quantum_Fourier_transform` · `Quasinorm` · `Quaternionic_analysis` · `Radio_scanner` · `Radio_wave` · `Real_analysis` · `Real_number` · `Representation_theory` · `Riemann–Lebesgue_lemma` · `Riesz–Fischer_theorem` · `Riesz–Thorin_theorem` · `Root_of_unity` · `Schwartz_space` · [[Science]] · `Seismic_wave` · `Sequence_space` · `Series_(mathematics)` · `Short-time_Fourier_transform` · [[Signal_processing]] · [[Sine_wave]] · `Sobolev_space` · `Sonar` · `Sound` · `Special_functions` · `Spectral_density` · `Spectral_density_estimation` · `Spectral_music` · `Spectrogram` · `Square-integrable_function` · `Statistics` · `Stochastic_differential_equation` · `Strip_aerial_photography` · `Summation` · `Symmetric_decreasing_rearrangement` · `Taxicab_geometry` · `The_Art_of_Computer_Programming` · [[Time_series]] · `Time–frequency_analysis` · `Topological_group` · `Trigonometric_functions` · `Trigonometric_interpolation` · `Trigonometric_series` · `Tropical_analysis` · `Two-sided_Laplace_transform` · [[Uncertainty_principle]] · `Uniform_norm` · `Unitary_operator` · `University_of_Nottingham` · `Vector_calculus` · `Vitale's_random_Brunn–Minkowski_inequality` · `Von_Neumann_algebra` · `Walsh_function` · `Wave_function` · `Wavelet` · `Whittaker–Shannon_interpolation_formula` · `Window_function` · `X-ray_crystallography` · `Young's_convolution_inequality` > Signal Processing concept · part of the Signal Processing Portal · movement II · !31 変換 henkan.svg <!-- RENDER-THUMB:START --> !480 *Rendered from the live microsim (▶ motion).* <!-- RENDER-THUMB:END --> <!-- GIFPLATE:BEGIN v1.0 g16 — Commons hotlink; do not hand-edit inside --> ## Images <figure class="wt-gifplate"> <img src="https://commons.wikimedia.org/wiki/Special:FilePath/Fourier_series_and_transform.gif" alt="Series & Transform" loading="lazy" decoding="async"> <figcaption><strong>Series & Transform</strong> — Illustrate complete decomposition of signals into frequency components.<br> <span class="wt-credit">Wikimedia Commons &middot; <strong>licence pending verification</strong> (run <code>g17_gif_verify.py</code> on a networked lane) &middot; <a href="https://commons.wikimedia.org/wiki/File:Fourier_series_and_transform.gif">Details</a></span></figcaption> </figure> *Still companion to the 1 live microsim above: the sim is the instrument, the plate is the glance. §15 keeps the player first; this sits in the image slot on [[Fourier_analysis]].* <!-- GIFPLATE:END --> ## See it next [](Fourier_transform) *→ Fourier transform* <!-- VISUAL-LINK:END --> --- Back to Signal Processing Portal · the room · Semiotic gateway ## What it is Fourier analysis is the general study of representing functions or signals as sums or integrals of sinusoids, and of the properties and applications of that decomposition. ## How it works / why it matters It encompasses the Fourier series (for periodic signals), the Fourier transform (for aperiodic signals), and their discrete counterparts, the DTFT and DFT, all resting on the idea that complex exponentials e^(j*omega*t) form an orthogonal basis. By moving between the time and frequency domains it turns convolution into multiplication and reveals the harmonic content of signals, underpinning nearly all of spectral analysis and filtering. ## Signs & universals Instantiates: frequency, spectrum, harmonic, oscillation. ## Related See Fourier transform, Fourier series, Discrete Fourier transform, [[Fast_Fourier_transform]], Discrete-time Fourier transform, and [[Frequency_domain]]. <!-- VISUAL-LINK:START --> <!-- CRAFT-LINK:START g12 --> *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].* <!-- CRAFT-LINK:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Fourier_analysis) : [Wikitube](https://en.wikitube.io/wiki/Fourier_analysis) ## Previous hub tags Tree parents: [[Decision_theory]] · [[Monte_Carlo_method]] · [[Reliability_engineering]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*