Convolution of interpolated data with function doesn't work

Question

I need to convolve one function with interpolated data:

function:

av1 = 1.3888*(12/20)*10^8;
bv1 = 2.5;
tr5[t_] = Exp[-(av1*t-bv1)^2];  

Data:

H4a={
{3.99932*10^-8, 2730.83720332692}, {4.04932*10^-8, 1630.73548351008},{4.09932*10^-8, 1277.35595196883}, {4.14932*10^-8, 1072.70360763529}, {4.19932*10^-8, 933.77632531279}, {4.24932*10^-8, 831.27749281985}, {4.29932*10^-8, 751.59481243293}, {4.34932*10^-8, 687.36617073509}, {4.39932*10^-8, 634.19609669597}, {4.44932*10^-8, 589.26940583597}, {4.49932*10^-8, 550.68537232488}, {4.54932*10^-8, 517.10668401615}, {4.59932*10^-8, 487.56071045906}, {4.64932*10^-8, 461.32048125234}, {4.69932*10^-8, 437.83006374543}, {4.74932*10^-8, 416.65596392318}, {4.79932*10^-8, 397.45444170215}, {4.84932*10^-8, 379.94891550006}, {4.89932*10^-8, 363.91396417635}, {4.94932*10^-8, 349.16376064501}, {4.99932*10^-8, 335.54355361763}, {5.04932*10^-8, 322.92328998620}, {5.09932*10^-8, 311.19276921945}, {5.14932*10^-8, 300.25791227827}, {5.19932*10^-8, 290.03785401278}, {5.24932*10^-8, 280.46265226146}, {5.29932*10^-8, 271.47146465654}, {5.34932*10^-8, 263.01108422334}, {5.39932*10^-8, 255.03475313929}, {5.44932*10^-8, 247.50119424772}, {5.49932*10^-8, 240.37381457968}, {5.54932*10^-8, 233.62004588761}, {5.59932*10^-8, 227.21079516706}, {5.64932*10^-8, 221.11998411525}, {5.69932*10^-8, 215.32416099476}, {5.74932*10^-8, 209.80217181913}, {5.79932*10^-8, 204.53488043315}, {5.84932*10^-8, 199.50492912035}, {5.89932*10^-8, 194.69653298106}, {5.94932*10^-8, 190.09530259197}, {5.99932*10^-8, 185.68809046291}}

Sd=Interpolation[H4a];

Convolution:

Ol3v22[y_?NumericQ] := NConvolve[Sd[t], tr5[t], t, y];
Plot[Ol3v22[y],{y, 40*10^-9, 60*10^-9},PlotRange -> All]

The result is just an empty plot!

Can you tell me where my mistake is?

thanks,

George

Answer 1

Notice that NConvolve is blue, this means it isn't one of the pre-defined functions.

You have to define the function yourself (or just grab Andrew Moylan's version here)

NConvolve[f_, g_, x_, y_?NumericQ] := 
 NIntegrate[f (g /. x -> y - x), {x, -Infinity, Infinity}]

Which makes the plot work just fine

Ol3v22[y_?NumericQ] := NConvolve[Sd[t], tr5[t], t, y];
Plot[Ol3v22[y], {y, 40*10^-9, 60*10^-9}, PlotRange -> All]

Edit: Can anyone tell me why it is letting me integrate an interpolating function from $\pm\infty$ when the function is only valid over a small range? If I were to ask for Sd[1.0] it would give me a warning that extrapolation was being used.