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Let's say I have a function that represents how much light is produced within a particular volume:

gamma[t_] := Piecewise[{{0.2 Exp[-t/13] + 0.5 Exp[-t/35] + 0.3 Exp[-t/270],  t >= 0}}, 0]

And let's say that the light is collected at one end of this volume, and the arrival time is gaussian distributed for a particular source time (i.e. if we isolate all of the light emitted at t, then it will arrive in a gaussian distribution centered around t' with a particular standard deviation).

Thus far I have come up with this:

gaussian[t_, μ_, σ_] := 1/(σ Sqrt[2 Pi]) Exp[-(1/2) ((t - μ)/σ)^2]

SetAttributes[gaussianSmear, HoldAll]

gaussianSmear[t_, f_[a___, y_, b___], σ_] := NIntegrate[ 
  f[a, y, b] gaussian[t, y, σ], {y, t-6σ, t+6σ}]

This will work, but it is very slow. Is there a faster way to perform this? Or is there a built-in function of some kind? I'm aware of the function GaussianFilter, but it only seems to operate on discrete data, not an arbitrary function.

An example usage, original function:

gamma[t_] := Piecewise[{{0.2 Exp[-t/13] + 0.5 Exp[-t/35] + 0.3 Exp[-t/270],  t >= 0}}, 0]
neutron[t_] := Piecewise[{{0.2 Exp[-t/13] + 0.5 Exp[-t/59] + 0.3 Exp[-t/490], t >= 0}}, 0]
Plot[{gamma[t], neutron[t]}, {t, 0, 80}, 
 PlotLegends -> {"gamma", "neutron"}, PlotRange -> {0, 1}, 
 PlotLabel -> "Theoretical activity of crystal", 
 AxesLabel -> {"Time (ns)", "Output level"}]

enter image description here

An example usage, after smearing:

Plot[{gaussianSmear[t, gamma[y], 1], 
  gaussianSmear[t, neutron[y], 1]}, {t, -2, 80}, PlotRange -> {0, 1}, 
 PlotLegends -> {"gamma", "neutron"}, 
 PlotLabel -> "Light collected by SiPM", 
 AxesLabel -> {"Time (ns)", "Output level"}]

enter image description here

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  • $\begingroup$ x is not defined. An example of how to use gaussianSmear would be nice also. $\endgroup$ – C. E. Aug 26 '18 at 18:14
  • $\begingroup$ Sorry, I changed variables from x to t when translating this from my notebook in order to make my intentions more clear, and I missed an instance. $\endgroup$ – OmnipotentEntity Aug 26 '18 at 18:15
  • $\begingroup$ Examples added. $\endgroup$ – OmnipotentEntity Aug 26 '18 at 18:20
  • $\begingroup$ @OmnipotentEntity Thank you for the update. $\endgroup$ – C. E. Aug 26 '18 at 18:20
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Well, this is a bit embarrassing.

I just realized that what I was performing was essentially a convolution of my original wave form with a standard Gaussian with a mean of 0.

So in other words, I can simply write:

Convolve[gamma[t], gaussian[t, 0, 1], t, y]

To get a function that can be directly plotted and manipulated much faster than my NIntegrate monstrosity.

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