ld = {1/102960, -(8/45045), 2/1287, -(56/6435), 7/198, -(56/495), 14/
45, -(8/9), 0, 8/9, -(14/45), 56/495, -(7/198), 56/6435, -(2/1287),
8/45045, -(1/102960)};
g[x_] := x + 1/2 Sin[x] + RandomVariate[NormalDistribution[.1, .05]]
Plot[{g[x], 10 g'[x], 10 g''[x]}, {x, 0, 8 Pi}]
t = Table[{x, g[x]}, {x, 0, 8 Pi, 8 Pi/1500}];
ListPlot@ListCorrelate[ld, t[[All, 2]]]
ListPlot@ListCorrelate[ld, MovingAverage[t[[All, 2]], 30]]