Wikimore

Description

English: An animation of the convolution of two boxcar functions, f(t) (red) and g(t) (blue)

f(t)=H\left(x+{\frac {1}{2}}\right)-H\left(x-{\frac {1}{2}}\right)

f(t)={\frac {1}{2}}\left[H\left(x+{\frac {1}{4}}\right)-H\left(x-{\frac {1}{4}}\right)\right]

,

where H(x) is the Heaviside step function.

Also shown is the time-reversed and shifted g(τ−t) in solid blue, the product f(t)g(τ−t) (green) and the convolution h(t)=f(t) ∗ g(t) (black, equal to the area of the product at the particular τ)

Date

5 March 2009

Source

Own work

Author

Inductiveload

Permission
(Reusing this file)

I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Category:Self-published work#Convolution%20Animation%20(Boxcar).gif Category:PD-self#Convolution%20Animation%20(Boxcar).gif

Mathematica Code

(*Function definitions*)
boxcar[x_, {a_, b_}] := UnitStep[x - a] - UnitStep[x - b]

f[t_] := boxcar[t, {-1/2, 1/2}];
g[t_] := boxcar[t, {-1/4, 1/4}]/2;
h[y_] := Module[
   {t},
   Convolve[f[t], g[t], t, y]
   ];

(*Parts of the drawing that only need to be computed once*)
constplots = Plot[
   {f[t], g[t], h[t]},
   {t, -2, 2},
   PlotStyle -> {Red, {Blue, Dashed}, Black},
   ClippingStyle -> False,
   Axes -> False,
   Exclusions -> False
   ];
anim =
  Table[Show[
    {
     (*The plots we generated earlier*)
     constplots,
     
     (*Plot the time reversed g[t], g[\[Tau]-t]*)
     Plot[g[\[Tau] - t],
      {t, -5, 5},
      PlotPoints -> 100,
      ClippingStyle -> False,
      PlotStyle -> Blue,
      Exclusions -> False,
      Axes -> False],
     
     (*Plot the product of the f[t] and g[\[Tau]-t]*)
     Plot[f[t]*g[\[Tau] - t],
      {t, -5, 5},
      PlotPoints -> 100,
      ClippingStyle -> False,
      Filling -> Axis,
      FillingStyle -> Directive[Green, Opacity[0.2]],
      PlotStyle -> Green,
      Exclusions -> False,
      Axes -> False],
     
     (*Plot the embellishments*)
     Graphics[
      {{Blue, Line[{{\[Tau], -0.05}, {\[Tau], 1.1}}]},
       {Red, Text[f, {.2, .96}, Background -> None]},
       {Blue, Text[g, {.2, .55}, Background -> None]},
       {Blue, 
        Text["g(\[Tau]-t)", {\[Tau] - 0.45, .4}, Background -> None]},
       {Green, 
        Text["f(t)g(\[Tau]-t)", {\[Tau] - 0.35, -0.04}, 
         Background -> None]},
       {Black, Text[Star[f, g], {0, .22}, Background -> None]},
       {Blue, Text["\[Tau]", {\[Tau], -0.1}, Background -> White]}
       }]
     },
    PlotRange -> {{-1.9, 1.9}, {-.2, 1}}],
   
   {\[Tau], -2, 2, .025}(*Table index is the time delay*)
   ];

ListAnimate[anim]
Export["anim2.gif", anim]

Category:Convolution Category:Animated GIF files

Wikimore

File:Convolution Animation (Boxcar).gif

Mathematica Code