Convolution - Wikiparency

# Convolution  ## Microsims — p5.js ### Convolution (p5.js) · `x * h` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/PEDdM5a-k" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Convolution — p5.js microsim"></iframe> </div> *Slide, multiply, and sum one signal against another — the single operation that computes any LTI system's output.* **Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/PEDdM5a-k) · movement *IX · Foundations & the rest of the toolbox* · library `p5js` ### Related microsims Live sims on neighbouring articles — 6 of them inside this article's own Wikipedia link tree: - [[Autocorrelation]] *(in tree)* - [[Digital_image_processing]] *(in tree)* - [[Digital_signal_processing]] *(in tree)* - [[Fast_Fourier_transform]] *(in tree)* - [[Functional_analysis]] *(in tree)* - [[Impulse_response]] *(in tree)* *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). 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library ![Animated: Convolution](https://upload.wikimedia.org/wikipedia/commons/6/6a/Convolution_of_box_signal_with_itself2.gif) *Animated: Convolution — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Information room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Convolution_of_box_signal_with_itself2.gif).* > In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( f {\displaystyle f} and g {\displaystyle g} ) that produces a third function ( f ∗ g {\displaystyle f*g} ). The term convolution refers to both the result function and to the process of computing it. ([Wikipedia](https://en.wikipedia.org/wiki/Convolution))  ## Media (PD/CC)  !Gif Library/Convolution/Convolution of box signal with itself2.gif *Convolution_of_box_signal_with_itself2.gif · Convolution_of_box_signal_with_itself.gif: Brian Amberg derivative work: Tinos (talk) · CC BY-SA 3.0 · [source](https://commons.wikimedia.org/wiki/File:Convolution_of_box_signal_with_itself2.gif)*  ## What it is Convolution is a mathematical operation on two functions that produces a third expressing how the shape of one is modified by the other — formed by sliding one function over the other, multiplying, and integrating (or summing) the overlap. ## How it works / why it matters For continuous signals it is defined as $(f * g)(t) = \int f(\tau)\,g(t-\tau)\,d\tau$, and for discrete signals as $(f * g)[n] = \sum_k f[k]\,g[n-k]$ — flip one signal, shift it, and accumulate the pointwise product. Its central role in signal processing comes from the fact that any **linear time-invariant** system computes its output as the convolution of the input with the system's **impulse response**. The **convolution theorem** makes it doubly powerful: convolution in the time domain equals simple multiplication of spectra in the frequency domain, so fast convolution is often done via the DFT. ## Signs & universals Instantiates: transformation · signal · frequency — a transformation combining signals that becomes multiplication in the frequency domain. ## Related Convolution is exactly how a [[Linear_time-invariant_system]] acts on its input — its output is the input convolved with the impulse response, the central theorem tying the two together. It is the mathematical engine of every Digital filter, and via the convolution theorem it is computed efficiently through the Discrete Fourier transform; in two dimensions it drives [[Digital_image_processing]] and in one dimension the effects of Audio signal processing.   *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Convolution) : [Wikitube](https://en.wikitube.io/wiki/Convolution) ## Previous hub tags Tree parent: [[Information_theory]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*