Editing Convolution (section)

== Visual explanation ==

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{{ordered list
| list_style = margin-left:1.6em;|Express each function in terms of a [[Free variables and bound variables|dummy variable]] <math>\tau.</math>|Reflect one of the functions: <math>g(\tau)</math> → <math>g(-\tau).</math>|Add an offset of the independent variable, <math>t</math>, which allows <math>g(-\tau)</math> to slide along the <math>\tau</math>-axis. If {{mvar|t}} is a positive value, then <math>g(t-\tau)</math> is equal to <math>g(-\tau)</math> that slides or is shifted along the <math>\tau</math>-axis toward the right (toward <math>+\infty</math>) by the amount of <math>t</math>. If <math>t</math> is a negative value, then <math>g(t-\tau)</math> is equal to <math>g(-\tau)</math> that slides or is shifted toward the left (toward <math>-\infty</math>) by the amount of <math>|t|</math>.|Start <math>t</math> at <math>-\infty</math> and slide it all the way to <math>+\infty</math>. Wherever the two functions intersect, find the integral of their product. In other words, at time <math>t</math>, compute the area under the function <math>f(\tau)</math> weighted by the weighting function <math>g(t - \tau).</math>
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The resulting [[waveform]] (not shown here) is the convolution of functions <math>f</math> and <math>g</math>.

If <math>f(t)</math> is a [[unit impulse]], the result of this process is simply <math>g(t)</math>. Formally:
: <math>\int_{-\infty}^\infty \delta(\tau) g(t - \tau)\, d\tau = g(t)</math>
| [[File:Convolution3.svg|center|452x452px|class=skin-invert-image]]
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| In this example, the red-colored "pulse", <math>\ g(\tau),</math> is an [[even function]] <math>(\ g(-\tau) = g(\tau)\ ),</math> so convolution is equivalent to correlation. A snapshot of this "movie" shows functions <math>g(t - \tau)</math> and <math>f(\tau)</math> (in blue) for some value of parameter <math>t,</math> which is arbitrarily defined as the distance along the <math>\tau</math> axis from the point <math>\tau = 0</math> to the center of the red pulse. The amount of yellow is the area of the product <math>f(\tau) \cdot g(t - \tau),</math> computed by the convolution/correlation integral. The movie is created by continuously changing <math>t</math> and recomputing the integral. The result (shown in black) is a function of <math>t,</math> but is plotted on the same axis as <math>\tau,</math> for convenience and comparison.
| [[File:Convolution of box signal with itself2.gif|475px|class=skin-invert-image]]
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| In this depiction, <math>f(\tau)</math> could represent the response of a [[resistor-capacitor circuit]] to a narrow pulse that occurs at <math>\tau = 0.</math> In other words, if <math>g(\tau) = \delta(\tau),</math> the result of convolution is just <math>f(t).</math> But when <math>g(\tau)</math> is the wider pulse (in red), the response is a "smeared" version of <math>f(t).</math> It begins at <math>t = -0.5,</math> because we defined <math>t</math> as the distance from the <math>\tau = 0</math> axis to the ''center'' of the wide pulse (instead of the leading edge).
| [[File:Convolution of spiky function with box2.gif|475px|class=skin-invert-image]]
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