Using the Fourier Transform to Get Frequency and Phase

Question

Suppose I have samples of a periodic function, what is a good way to get frequency and phase information from it?

In particular, I would like to get a form like

a+b Cos[cx + d] 

Here is part of the sample

 {255,255,255,249,64,0,0,0,0,0,0,0,0,0,0,0,0,233,255,255,255,255,255,255,255,255,255,209,0,0,0,0,0,0,0,0,0,0,0,0,118,255,255,255,255,255,255,255,255,255,255,132,0,0,0,0,0,0,0,0,0,0,0,0,200,255,255,255,255,255,255,255,255,255,239,19,0,0,0,0,0,0,0,0,0,0,0,46,245,255,255,255,255,255,255,255,255,255,186,0} 

Answer 1

Using autocorrelation and FindFit []

 (*Your list*) ListPlot@l 

 (*trim the list*) l1 = Drop[l, ( First@Position [l, 0])[[1]] - 1]; l2 = Drop[l1, Length@l1 - ( Last@Position [l1, 0])[[1]] - 1]; (*autocorrelate*) ListLinePlot@ (ac = ListConvolve[l2, l2, {1, 1}]) 

 (*Find Period by taking means of maxs and mins spacings*) period = Mean@ Join[ Differences@ (maxs = Table[If[ac[[i - 1]] < ac[[i]] > ac[[i + 1]], i, Sequence @@ {}], {i, 2, Length@ac - 1}]), Differences@ (mins = Table[If[ac[[i - 1]] > ac[[i]] < ac[[i + 1]], i, Sequence @@ {}], {i, 2, Length@ac - 1}])]; (*Show it*) Show[ListLinePlot[(ac = ListConvolve[l2, l2, {1, 1}]), Epilog -> Inset[Framed[Style["Mean Period = " <> ToString@N @period, 20], Background -> LightYellow]]], Graphics[Join[{Arrowheads[{-.05, .05}]}, {Red}, Sequence @@@ Arrow[{{{#[[1]], Min@ac }, {#[[2]], Min@ac }}}] & /@ Partition[mins, 2, 1], {Blue}, Sequence @@@ Arrow[{{{#[[1]], Max@ac }, {#[[2]], Max@ac }}}] & /@ Partition[maxs, 2, 1]]]] 

 (*Now let fit the Cos[ ] to find the phase*) model = a + b Cos[x (2 Pi)/period + phase]; ff = FindFit[l, model, {a, b, phase}, x, Method -> NMinimize, MaxIterations -> 100]; (*Show results*) Show[ListPlot[l, PlotRange -> All, Epilog -> Inset[Framed[Style["Phase = " <> ToString@N @(phase /. ff), 20], Background -> LightYellow]]], Plot[model /. ff, {x, 1, 100}]] 

Answer 2

See the FourierDCT ref page .

Your data is very similar to the SquareWave feature. On a manual inspection, it appears that your data matches SquareWave[{0, 255}, (x + 5)/23 ] .