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I have the given function:

points = {{395.4416644777464`, 
    207.63931734339303`}, {391.15890276860114`, 
    240.47382378017346`}, {382.59337935031067`, 
    219.06001523444706`}, {378.3106176411653`, 
    209.0669045797748`}, {369.74509422287485`, 
    177.65998537937617`}, {361.1795708045843`, 
    250.4669344348457`}, {355.46922185905726`, 
    204.78414287062958`}, {346.9036984407667`, 
    236.19106207102823`}, {341.19334949523966`, 
    184.79792156128497`}, {332.6278260769492`, 
    220.48760247082885`}, {326.9174771314222`, 
    214.77725352530183`}, {322.6347154222768`, 
    260.4600450895181`}, {316.9243664767498`, 
    219.06001523444706`}, {314.06919200398636`, 
    241.90141101655524`}, {304.076081349314`, 
    119.12890868772422`}, {295.5105579310235`, 
    327.5566451994606`}, {278.3795110944425`, 
    156.24617683364988`}, {269.8139876761519`, 
    251.8945216712275`}, {264.1036387306249`, 
    196.218619452339`}, {258.39328978509786`, 
    286.1566153443896`}, {249.8277663668073`, 
    233.33588759826466`}, {241.26224294851679`, 
    173.37722367023093`}, {231.26913229384448`, 
    184.79792156128497`}, {224.13119611193568`, 
    286.1566153443896`}, {215.56567269364515`, 
    160.52893854279512`}, {199.86221309344583`, 
    303.28766218097076`}, {184.1587534932465`, 
    72.01852988712619`}, {172.73805560219247`, 
    587.3775222209401`}, {162.74494494752017`, 
    83.43922777818022`}, {149.89665982008435`, 
    260.4600450895181`}, {145.61389811093912`, 
    254.74969614399106`}, {138.47596192903032`, 
    317.56353454478824`}, {131.33802574712158`, 
    183.37033432490318`}, {124.20008956521278`, 
    193.36344497957543`}, {119.91732785606749`, 
    160.52893854279512`}, {112.77939167415872`, 
    214.77725352530183`}, {108.49662996501345`, 
    199.07379392510256`}, {105.64145549224995`, 
    250.4669344348457`}, {101.35869378310466`, 
    324.70147072669704`}, {98.50351931034115`, 
    224.77036417997408`}, {91.36558312843238`, 
    220.48760247082885`}, {81.37247247376007`, 
    96.28751290561604`}, {75.66212352823305`, 
    200.50138116148423`}, {72.80694905546954`, 
    351.82562821795045`}, {54.24831498250671`, 
    154.8185895972681`}, {41.4000298550709`, 
    240.47382378017346`}, {35.68968090954388`, 
    236.19106207102823`}, {32.83450643678037`, 
    280.4462663988626`}, {12.848285127435787`, 
    131.97719381515992`}, {7.1379361819087705`, 227.62553865273765`}};

points2 = {#[[1]], #[[2]] - 72.01852988712619`} & /@ points;


Clear[t];
f[t_] = Piecewise[
   Partition[Sort[points2], 2, 
     1] /. {{a_?NumericQ, b_}, {c_, d_}} :> {b, a <= t < c}];

f1[t_] = f[(172.73805560219247` (t + Pi)/Pi)];

which for I generated the Fourier series:

FD[t_] = FourierSeries[f1[t], t, 20]

Then I'd like to plot the real parts of every Fouriercoefficients on the plot, from 1-20:

Re[FourierCoefficient[f1[t], t, 1]]
Re[FourierCoefficient[f1[t], t, 2]]
Re[FourierCoefficient[f1[t], t, 3]]
...

Re[FourierCoefficient[f1[t], t, 20]]

However, there are many points, and it is difficult to "harvest" them into a set that can be directly plotted with the Fourier series plot:

Four = Plot[{FD[t]}, {t, -3, 3}, PlotRange -> Full]

by

Show[Four, Epilog -> {PointSize -> Large, Point[{{0, 2}}]}]

Are there better ways?

Thanks

attached the image of the real parts of the fourier coefficients on the x axis of a fourier transform of a piecewise function (other than the functtion in this post)

enter image description here

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1 Answer 1

3
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You may extract the real part of the coefficients from FD[t]:

res = (FD[t] /. Plus -> List) /. x_ Exp[__] -> Re[x]

To plot this you use ListPlot (note the point at index 1 belongs to the DC coefficient, index 2 to the lowest frequency ...):

ListPlot[res]

enter image description here

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  • $\begingroup$ Thanks Daniel, but I couldn't get the plot of the coefficients to coincide with the Fourier plot. In fact they are completely unrelated. $\endgroup$ Apr 12, 2023 at 15:05
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    $\begingroup$ Look at the Fourier Series and check if it is in agreement with the plot. $\endgroup$ Apr 12, 2023 at 16:15
  • $\begingroup$ That is what I did. They have nothing in common. $\endgroup$ Apr 12, 2023 at 16:30
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    $\begingroup$ THis are the first few terms of the Fourier Series: 139.85 + (15.8404 + 4.76022 I) E^(-I t) + (15.8404 - 4.76022 I) E^( I t) + (4.64917 - 3.3024 I) E^(-2 I t) Note the plot does not show the DC component because it is too large. Therefore the real terms are: 15.8, 15.8, 4.64, .. and that is what the plot shows. $\endgroup$ Apr 12, 2023 at 17:31
  • 1
    $\begingroup$ The FourierTransform takes data into account from -Infinity to Infinity. The FourierSeries takes data from -P to Pi. You can change frequencies in the range -Pi..Pi by changing the original function above Pi. This does not change the FourierSeries but the FourierTransfrom! $\endgroup$ Apr 13, 2023 at 12:07

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