Symmetry problem and anti-symmetry of the Fourier coefficients

let's say I have a dataset

EchoH1SAT = {{- 0.010000000000005116}, 0.`}, {0.01000000000000556}, -0.0016729627707055}}, {0.03000000001099998, do-it-yourself articles, do-it-yourselfers, do-it-yourselfers, and office supplies. 0.09000000000000341`, -0.031181593750000076`}, {0.10999999999999943`, -0.040817236130445744`}, {0.12999999999999545`, -0.05189135257766407`}, {0.15000000000000568`, -0.05912155849700829`}, {0.1700000000000017`, -0.0637636364488487`}, {0.18999999999999773`, -0 06536318750000025 `}, {0.20999999999999375`, -0.06136318750000025`}, {0.23000000000000398`, -0.055.53631875000004`}, 0.30, of -0.30, of -30%, of -0.0999999999999399`,`, 0.039363187500}}, {0.4300000000000068` , -0. 05:30 0.5899999999080 `}, {0.6700000000000017`, -0.03136318750000022`}, {0.6899999999999977`, -0.0363631875000007`}, 0.709999999999, and 0, 0-0, 0, 0, 0, 0, 0, 0` .0.018363.100.23.28.28.28.28.28.28.28.28.28.28.28.28.28.28.28.28.28.28.28.28.100.100.1008 And In Stock}, {0.8900000000000006`, - 0.0423631800Follow 1.080000000000051`, -0.066363180014}, {0} `}, {1.12999999999955`, -0 The most recent is the most recent, the most recent being the most recent. }, {1.34999999999999994 -0.06736318750025}, {1.37000000004545`, `}, {1.46999999999999`, -0.03136318180000022`, -0.02963618180000022}, {0}; -0.02636318750010`}, {1.59000000630, framing, .0, 1.5 1.5, 1.50000, 00000000,}, {1.6899999999999977`, -0.022363187500000103`, {0.09999399999937`, -0.02436318750000105}, - 1. 0805303187500000105}, -0.0303631800000103}, -0.0303631800000103,, -0.023363187500000215`}, {1.8100000000 Keep the same thing}, {1.9300000000000068`, -0.0033631870001976`}, {1.95000000002828`, 0.0.006363000000311`}, {1.9699999999999999999999999999999999999999 of the day of the month from January 2017.]Most of our article is the closest to it: -, 0.017363187500021`, (2.04999999999999`, -0.0990000003400), {2.0699999999993`, 0.56999999993, so that our products comply with Article 6 we need to use them (2.1500990000000057`, 0.0076368124999991), {2.1799999709}, {2.1899999999999801, 6, 7, 6909999999999, as well as for DIY products, products from the products section, 2.6999999996 are also required. . `}, {2.3900000000000006`, -0.00`, -0.00636318750000011}}, {2.51000000000005, {2.530000000000001}, -0.03036000000000, and -0.03036000000001, and -0.03036000000000, and -0.030360000001, {0} {0}} {{}} }} {{}} {{}}} {} * * * * * * * * = * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Article 3 The following products are in good condition, {2.8299999999999983`, -0.003363180000001976`}, (2.0099699000001976), {2.8900000000000001976`}, (2.08000099000001976`), -0.006. ยป 0.018636812499999822` } {3.069999999999993, 0.01868998998990000, {3.09998000063434, 8/198996998808, and more, it is being reoriented, and it is also being redirected, it is necessary to keep 0099099999999, and 3.30999999994, {3.20999999994, {3.9999999904, { 3.20999999998, and / or / and / or); , 0.044636812499999845`}, {3.3100000000000023`, 0.04663681249999985`}, {3.3299999999999983`, 0.04663681249999985`}, {3.3499999999999943`, 0.04663681249999985`}, {3.3700000000000045`, 0.04263681249999984`}, {3.3900000000000006`, 0.03763681249999973`}, {3.4099999999999966`,, 032636812499999834 `3.6000000007`, rank, {3.549999999999997`, 0.0016368124999996958`}, {3.569999999999993`, 0.0016368124999996958}, {3.5900000000000034 & # 39 ;, 3026998999999993`, 0.006368124999996958} and 3.599999999999999), 306,999,999,999,999, and 3.59999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999900, 18016998999999993, 0.0016368124999996958} {3.59990000000034, 303,699,999,999,993, 0056368124999601930}, {3.6069906806806, 155008068078078071980780780719780780780780.jpg most brands}, {3.769999999999996`, 0.018636812499999822`}, {3.7900000000000063`, 0.022636812499999825`}, {3.8100000000000023`, 0.02763681249999972`}, {3.8299999999999983`, 0 .02963681249999972`}, {3.8499999999999943`, 0.03163681249999972`}, {3.8700000000000045`, 0.03163681249999972`}, {3.8900000000000006`, 0.03163681249999972`}, {3.9099999999999966`, 0.032636812499999834`}, {3.930000000000007`, 0.033636812499999724`}, {3.950000000000003`, 0.03663681249999984 `}, {3.969999999999999`, 0.04163681249999973`}, {3.989999999999995`, 0.045636812499999735`}, {4.010000000000005, 109999999999999`, 0.06163681249999975`}, {4.1299999999999955`, 0.06163681249999975`}, {4.150000000000006`, 0.06163681249999975`}, {4.170000000000002` , 0.06163681249999975`}, {4.189999999999998`, 0.05963681249999975`}, {4.209999999999994`, 0.056636812499999856`}, {4.230000000000004`, 0.054636812499999854`}, {4.25`, 0.05063681249999985`}, {4.269999999999996`, 0.04863681249999985`}, {4.290000000000006`, 0.04863681249999985 `}, {4.310000000000002`, 0.04863681249999985`}, {4.329999999999998`, 0.04863681249999985`}, {4.349999999999994`,, 05163681249999974`}, {}}, {4.489999999999995`, 0.077636812 4999997`}, {4.510000000000005`, 0.0776368129999996`}, {4.53000000000012,30,806,80999999996}, {4.53000009900990}, {4.53000009900990}, 00004; 590000000000003`, 0.06863681249999987`}, {4.609999999999999`, 0.06663681249999986`}, {4.6299999999999955`, 0.06363681249999975`}, {4.650000000000006`, 0.06163681249999975`}, {4.670000000000002`, 0.05963681249999975`}, {4.689999999999998`, 0.05963681249999975`}, {4.709999999999994` , 0.05963681249999975`}, {4.730000000000004`, 0.06063681249999986`}, {4.75`, 0.06163681249999975`}, {4.769999999999996`, 0.06363681249999975`}, {4.790000000000006`, 0.0620795088521887`}, {4.810000000000002`, 0.059369054634550764`}, {4.829999999999998`,, 053696366699785406 `}, {4.8499999999994`, 0.04688690248PP6257D86700CAD}, {4.9900C19999999999``, 7,80500006801, 2 rows, 2 rows, 2 rows of pieces, 2, {4.969999999999999`, 0.`}};

and I want to use NFourierTrigSeries to analyze this data and create a variance density spectrum

<< FourierSeries`
[CapitalDelta]x = 0.02;
Length1 = 5;
Echos1 = Interpolation[EchoHannatoS1];
Ground[{Echos1[x]}, {x, 0, Length1 - [CapitalDelta]X}]Profilecyclic1[x_] = By pieces[{{Echos1[x], 0 <x < Length1 - [CapitalDelta]x}}, 0];
Plot[Profilecyclic1[x], {x, 0 - 1, Length1 + 1}]
CyclicFunction1 = Interpolation[Table[{x, Profilecyclic1[x]}, {x, 0, Length1, 0.02}], PeriodicInterpolation -> True];
f0 = 1 / Length1; (* fundamental frequency *)
fc = 1 / [CapitalDelta]X; (* max frequency *)
M = IntegerPart[ fc/(2 f0)] ; (* order of the Fourier series *)
FourierFunction1[x_] =
NFourierTrigSeries[CyclicFunction1[x], x, M, FourierParameters -> {1, 2 [Pi] f 0}];
Ground[FourierFunction1[x], {x, 0, Length1}]A0 = FourierFunction1[x][[2, 1]]; (* Average value of the Fourier function *)
AllCoefficients = Reap[For[I=1i[For[I=1i[Pour[i=1i[For[i=1i< (M + 1), i++, Sow[i]; Sow[f0 i]; 
   Sow[Coefficient[FourierFunction1[x], Cos[ 2 [Pi] f0 i  x]]]; 
   Sow[Coefficient[FourierFunction1[x], Sin[ 2 [Pi] f0 i  x]]]; 
   Sow[[Sqrt]((Coefficient[FourierFunction1[x], 
         Sin[ 2 [Pi] f0 i  x]])^2 + (Coefficient[
         FourierFunction1[x], Cos[2 [Pi] f0 i  x]])^2)]];] [[2,1]]~Partition~5;
(* i'm creating a table with the values of i, f0i ,Ai, Bi,ai *)
MatrixForm[Prepend[AllCoefficients, {"i", "fi", "Ai", "Bi", "ai"}]];
Dimensions[AllCoefficients];
fi = AllCoefficients[[All, 2]];
Ai = AllCoefficients[[All, 3]];
Bi = AllCoefficients[[All, 4]] ;
ai = AllCoefficients[[All, 5]];
(* verify the Parseval's theorem --> the result should be approximately 0 *)
ParsevalValue = (f0 *
NIntegrate[(FourierFunction1[(FourierFunction1[(FourierFunction1[(FourierFunction1[x]) ^ 2, {x, 0, Length1}]) - (1 /
2 * (sum[ai[[i]]^ 2, {i, 1, M}]))
checkfunction1[x_] = Sum[Ai[[i]]* (Cos[2[2[2[2[Pi] f0 i x]), {i, 1, M}

+ Sum[Bi[[i]]*(Peach[2[2[2[2[Pi] f0 i x]), {i, 1, M}

;
Ground[checkfunction1[checkfunction1[checkfunction1[checkfunction1[x], {x, 0, Length1}]Spectrum = Transpose @ {fi, (ai ^ 2) / (2f0)};
ListLinePlot[{Spectrum}, PlotRange -> All]
S = Interpolation[Spectrum];
LogLogPlot[{S[f]}, {f, f0, M f0}, PlotLegends -> "Expressions", AxesLabel -> {"f", "S (f)"}, PlotRange -> {10 ^ (- 15), 10 ^ (- 3 )}]

The results seem to be correct. I have everything I need for my analysis. However, if I trace the Fourier coefficients, I get these results

ListPlot[Ai, PlotRange -> All, Filling -> Axis]
ListPlot[Bi, PlotRange -> All, Filling -> Axis]

and I do not understand why the coefficients Ai and Bi have lost their symmetrical and anti-symmetric properties.

Can any one point out the mistake I've made? maybe something about the Assumption of FourierTrigSeries or FourierParameters?

I ask these questions because I need the Fourier coefficients to have symmetrical properties in order to continue my analysis.