1
$\begingroup$

I am trying to find the resonant frequency of a particle by Fourier analysis of the trajectory. Here is the data file.

 trajectory=   {{0.`, 0.3297799999999995`}, {0.03817`, 
   0.38034000000000034`}, {0.07254`, 0.4272500000000008`}, {0.10458`, 
   0.47228000000000137`}, {0.13548`, 0.5169700000000006`}, {0.16608`, 
   0.5624800000000008`}, {0.19706`, 0.6098700000000008`}, {0.22919`, 
   0.6603600000000007`}, {0.26336`, 0.7155099999999983`}, {0.30094`, 
   0.7777199999999986`}, {0.3443`, 0.8512099999999982`}, {0.37148`, 
   0.8979500000000016`}, {0.39865`, 0.9450399999999988`}, {0.41224`, 
   0.9686599999999999`}, {0.41564`, 0.9745600000000003`}, {0.41733`, 
   0.9775099999999988`}, {0.41903`, 0.9804700000000004`}, {0.42073`, 
   0.9834199999999989`}, {0.42413`, 0.9893299999999989`}, {0.42752`, 
   0.9952299999999994`}, {0.43432`, 1.00704`}, {0.44111`, 
   1.0188300000000012`}, {0.4547`, 1.042390000000001`}, {0.48187`, 
   1.0892300000000006`}, {0.50904`, 1.1354900000000008`}, {0.56339`, 
   1.2252200000000002`}, {0.60962`, 1.297229999999999`}, {0.63444`, 
   1.3337399999999988`}, {0.65523`, 1.362960000000001`}, {0.676`, 
   1.390789999999999`}, {0.69685`, 1.4172599999999989`}, {0.71785`, 
   1.4422999999999995`}, {0.73902`, 1.4658300000000004`}, {0.7604`, 
   1.4877399999999987`}, {0.78201`, 1.5079199999999986`}, {0.80384`, 
   1.5262400000000014`}, {0.82587`, 1.5425700000000013`}, {0.84808`, 
   1.5567799999999998`}, {0.87041`, 1.56879`}, {0.89277`, 
   1.578520000000001`}, {0.91509`, 1.5859699999999997`}, {0.93729`, 
   1.5911700000000017`}, {0.95931`, 1.59422`}, {0.98111`, 
   1.5952400000000004`}, {0.98145`, 1.5952400000000004`}, {0.98161`, 
   1.5952400000000004`}, {0.9817`, 1.5952400000000004`}, {0.98174`, 
   1.5952400000000004`}, {0.98174`, 1.5952400000000004`}, {0.98174`, 
   1.5952400000000004`}, {0.98174`, 1.5952400000000004`}, {0.98174`, 
   1.5952400000000004`}, {0.98174`, 1.5952400000000004`}, {0.98174`, 
   1.5952400000000004`}, {0.98174`, 1.5952400000000004`}, {0.98174`, 
   1.5952400000000004`}, {0.98174`, 1.5952400000000004`}, {0.98175`, 
   1.5952400000000004`}, {0.98175`, 1.5952400000000004`}, {0.98176`, 
   1.5952400000000004`}, {0.98178`, 1.5952400000000004`}, {0.98182`, 
   1.5952400000000004`}, {0.98189`, 1.5952400000000004`}, {0.98204`, 
   1.5952400000000004`}, {0.98234`, 1.5952400000000004`}, {0.98293`, 
   1.5952400000000004`}, {0.98412`, 1.5952300000000008`}, {0.9865`, 
   1.5951999999999984`}, {0.99126`, 1.5950600000000001`}, {1.00078`, 
   1.594529999999999`}, {1.01982`, 1.5924399999999999`}, {1.04099`, 
   1.5885999999999996`}, {1.06193`, 1.5833399999999997`}, {1.08273`, 
   1.576789999999999`}, {1.10344`, 1.5690999999999988`}, {1.12413`, 
   1.5603799999999985`}, {1.14487`, 1.5507400000000011`}, {1.16576`, 
   1.54026`}, {1.19142`, 1.5265699999999995`}, {1.22475`, 
   1.5078700000000005`}, {1.23734`, 1.5006599999999999`}, {1.24363`, 
   1.4970399999999984`}, {1.24678`, 1.4952299999999994`}, {1.24835`, 
   1.4943300000000015`}, {1.24992`, 1.4934200000000004`}, {1.2515`, 
   1.492519999999999`}, {1.25307`, 1.4916100000000014`}, {1.25622`, 
   1.489799999999999`}, {1.25936`, 1.48799`}, {1.26565`, 
   1.4843800000000016`}, {1.27824`, 1.4771900000000002`}, {1.29082`, 
   1.4700799999999994`}, {1.31599`, 1.4562099999999987`}, {1.36633`, 
   1.4307599999999994`}, {1.40436`, 1.414480000000001`}, {1.4264`, 
   1.4065399999999997`}, {1.44798`, 1.3999699999999997`}, {1.46969`, 
   1.3946900000000007`}, {1.49165`, 1.3908300000000011`}, {1.51396`, 
   1.3885100000000001`}, {1.52534`, 1.3879899999999985`}, {1.53103`, 
   1.3878999999999984`}, {1.53245`, 1.3878999999999984`}, {1.5325`, 
   1.3878999999999984`}, {1.53252`, 1.3878999999999984`}, {1.53252`, 
   1.3878999999999984`}, {1.53253`, 1.3878999999999984`}, {1.53253`, 
   1.3878999999999984`}, {1.53253`, 1.3878999999999984`}, {1.53253`, 
   1.3878999999999984`}, {1.53253`, 1.3878999999999984`}, {1.53253`, 
   1.3878999999999984`}, {1.53253`, 1.3878999999999984`}, {1.53253`, 
   1.3878999999999984`}, {1.53253`, 1.3878999999999984`}, {1.53253`, 
   1.3878999999999984`}, {1.53254`, 1.3878999999999984`}, {1.53255`, 
   1.3878999999999984`}, {1.53257`, 1.3878999999999984`}, {1.53261`, 
   1.3878999999999984`}, {1.53268`, 1.3878999999999984`}, {1.53284`, 
   1.3878999999999984`}, {1.53315`, 1.3878999999999984`}, {1.53378`, 
   1.3878999999999984`}, {1.53503`, 1.3879100000000015`}, {1.53752`, 
   1.3879400000000004`}, {1.54252`, 1.3880799999999986`}, {1.55251`, 
   1.3886299999999991`}, {1.57249`, 1.39086`}, {1.59663`, 
   1.395620000000001`}, {1.62135`, 1.4029200000000017`}, {1.64644`, 
   1.4129000000000005`}, {1.67156`, 1.4255299999999984`}, {1.69634`, 
   1.4406099999999995`}, {1.72044`, 1.4577799999999996`}, {1.72334`, 
   1.46002`}, {1.72407`, 1.4605800000000002`}, {1.72416`, 
   1.4606600000000007`}, {1.72417`, 1.4606600000000007`}, {1.72417`, 
   1.4606700000000004`}, {1.72417`, 1.4606700000000004`}, {1.72417`, 
   1.4606700000000004`}, {1.72417`, 1.4606700000000004`}, {1.72417`, 
   1.4606700000000004`}, {1.72417`, 1.4606700000000004`}, {1.72417`, 
   1.4606700000000004`}, {1.72417`, 1.4606700000000004`}, {1.72417`, 
   1.4606700000000004`}, {1.72418`, 1.4606700000000004`}, {1.72418`, 
   1.4606700000000004`}, {1.72418`, 1.46068`}, {1.72419`, 
   1.46068`}, {1.72421`, 1.4606999999999992`}, {1.72425`, 
   1.4607300000000016`}, {1.72433`, 1.4607899999999994`}, {1.72449`, 
   1.4609200000000016`}, {1.72481`, 1.4611599999999996`}, {1.72544`, 
   1.4616599999999984`}, {1.72671`, 1.4626599999999996`}, {1.72924`, 
   1.4646700000000017`}, {1.73431`, 1.4687899999999985`}, {1.74445`, 
   1.4773500000000013`}, {1.76473`, 1.495750000000001`}, {1.78627`, 
   1.5171700000000001`}, {1.80702`, 1.5395900000000005`}, {1.8271`, 
   1.5628899999999994`}, {1.84659`, 1.5869900000000001`}, {1.86559`, 
   1.6118200000000016`}, {1.88417`, 1.6373399999999982`}, {1.90243`, 
   1.6635099999999987`}, {1.92042`, 1.6903299999999994`}, {1.93822`, 
   1.7177800000000012`}, {1.95589`, 1.7458600000000004`}, {1.9735`, 
   1.7745999999999995`}, {1.99372`, 1.8084099999999985`}, {2.01811`, 
   1.8501399999999997`}, {2.04925`, 1.9045300000000012`}, {2.07159`, 
   1.944040000000001`}, {2.07718`, 1.953949999999999`}, {2.07997`, 
   1.9588999999999999`}, {2.08276`, 1.9638600000000004`}, {2.08346`, 
   1.9650999999999996`}, {2.08416`, 1.9663399999999989`}, {2.08486`, 
   1.9675800000000017`}, {2.08625`, 1.9700600000000001`}, {2.08765`, 
   1.9725399999999986`}, {2.09044`, 1.9774899999999995`}, {2.09603`, 
   1.987400000000001`}, {2.10161`, 1.9972999999999992`}, {2.11279`, 
   2.0170600000000007`}, {2.13513`, 2.0563199999999995`}, {2.15747`, 
   2.0949899999999992`}, {2.20216`, 2.16957`}, {2.23528`, 
   2.2214300000000016`}, {2.25387`, 2.2488899999999994`}, {2.27223`, 
   2.27468`}, {2.2908`, 2.29927`}, {2.30967`, 
   2.3225799999999985`}, {2.32896`, 2.3445099999999996`}, {2.34877`, 
   2.3649099999999983`}, {2.36924`, 2.3835799999999985`}, {2.39053`, 
   2.4002599999999994`}, {2.41282`, 2.4145900000000005`}, {2.43633`, 
   2.4260899999999985`}, {2.46125`, 2.434100000000001`}, {2.48763`, 
   2.4377899999999997`}, {2.49104`, 2.4379099999999987`}, {2.49275`, 
   2.437940000000001`}, {2.4936`, 2.437940000000001`}, {2.49403`, 
   2.437940000000001`}, {2.49404`, 2.437940000000001`}, {2.49405`, 
   2.437940000000001`}, {2.49405`, 2.437940000000001`}, {2.49405`, 
   2.437940000000001`}, {2.49405`, 2.437940000000001`}, {2.49405`, 
   2.437940000000001`}, {2.49405`, 2.437940000000001`}, {2.49405`, 
   2.437940000000001`}, {2.49405`, 2.437940000000001`}, {2.49405`, 
   2.437940000000001`}, {2.49405`, 2.437940000000001`}, {2.49405`, 
   2.437940000000001`}, {2.49405`, 2.437940000000001`}, {2.49406`, 
   2.437940000000001`}, {2.49406`, 2.437940000000001`}, {2.49406`, 
   2.437940000000001`}, {2.49408`, 2.437940000000001`}, {2.4941`, 
   2.437940000000001`}, {2.49415`, 2.437940000000001`}, {2.49424`, 
   2.437940000000001`}, {2.49443`, 2.437940000000001`}, {2.4948`, 
   2.437940000000001`}, {2.49554`, 2.4379300000000015`}, {2.49703`, 
   2.4379099999999987`}, {2.50001`, 2.437809999999999`}, {2.50075`, 
   2.43778`}, {2.50094`, 2.4377700000000004`}, {2.50103`, 
   2.4377700000000004`}, {2.50104`, 2.4377700000000004`}, {2.50105`, 
   2.4377700000000004`}, {2.50105`, 2.4377700000000004`}, {2.50105`, 
   2.4377700000000004`}, {2.50105`, 2.4377700000000004`}, {2.50105`, 
   2.4377700000000004`}, {2.50105`, 2.4377700000000004`}, {2.50105`, 
   2.4377700000000004`}, {2.50105`, 2.4377700000000004`}, {2.50106`, 
   2.4377700000000004`}, {2.50106`, 2.4377700000000004`}, {2.50106`, 
   2.4377700000000004`}, {2.50106`, 2.4377700000000004`}, {2.50106`, 
   2.4377700000000004`}, {2.50106`, 2.4377700000000004`}, {2.50107`, 
   2.4377700000000004`}, {2.50108`, 2.4377700000000004`}, {2.5011`, 
   2.437760000000001`}, {2.50115`, 2.437760000000001`}, {2.50124`, 
   2.437760000000001`}, {2.50143`, 2.437750000000001`}, {2.5018`, 
   2.4377299999999984`}, {2.50254`, 2.4376800000000003`}, {2.50403`, 
   2.43759`}, {2.507`, 2.437339999999999`}, {2.51295`, 
   2.43666`}, {2.52484`, 2.4345500000000015`}, {2.54863`, 
   2.427299999999999`}, {2.57353`, 2.4154799999999987`}, {2.5968`, 
   2.4006100000000004`}, {2.6187`, 2.38335`}, {2.63949`, 
   2.3641599999999983`}, {2.6594`, 2.3433600000000006`}, {2.67859`, 
   2.3211799999999982`}, {2.6972`, 2.2977999999999987`}, {2.71534`, 
   2.2733500000000006`}, {2.73313`, 2.2479200000000006`}, {2.75063`, 
   2.221589999999999`}, {2.76794`, 2.194420000000001`}, {2.78511`, 
   2.1664400000000015`}, {2.80223`, 2.1376799999999996`}, {2.82105`, 
   2.105170000000001`}, {2.84387`, 2.064730000000001`}, {2.87281`, 
   2.012270000000001`}, {2.89312`, 1.9749499999999998`}, {2.90327`, 
   1.9562099999999987`}, {2.90835`, 1.9468199999999989`}, {2.91342`, 
   1.937429999999999`}, {2.91469`, 1.9350899999999989`}, {2.91596`, 
   1.932739999999999`}, {2.91659`, 1.931560000000001`}, {2.91723`, 
   1.9303899999999992`}, {2.91786`, 1.9292200000000008`}, {2.91913`, 
   1.926870000000001`}, {2.92167`, 1.9221700000000013`}, {2.92421`, 
   1.9174800000000012`}, {2.92929`, 1.908100000000001`}, {2.93944`, 
   1.88936`}, {2.94959`, 1.8706700000000005`}, {2.9699`, 
   1.83352`}, {3.01051`, 1.7607500000000016`}, {3.06707`, 
   1.6646099999999997`}, {3.09491`, 1.6202499999999986`}, {3.11322`, 
   1.5923300000000005`}, {3.13155`, 1.5654499999999985`}, {3.15014`, 
   1.5393599999999985`}, {3.16905`, 1.514050000000001`}, {3.18836`, 
   1.4895700000000005`}, {3.20814`, 1.465959999999999`}, {3.22845`, 
   1.4432800000000015`}, {3.24936`, 1.4216499999999996`}, {3.27092`, 
   1.40118`}, {3.27371`, 1.3986799999999988`}, {3.2744`, 
   1.398060000000001`}, {3.27444`, 1.398019999999999`}, {3.27447`, 
   1.3979999999999997`}, {3.27447`, 1.39799`}, {3.27447`, 
   1.39799`}, {3.27447`, 1.39799`}, {3.27447`, 1.39799`}, {3.27447`, 
   1.39799`}, {3.27447`, 1.39799`}, {3.27447`, 1.39799`}, {3.27447`, 
   1.39799`}, {3.27447`, 1.39799`}, {3.27447`, 1.39799`}, {3.27448`, 
   1.39799`}, {3.27449`, 1.3979800000000004`}, {3.27451`, 
   1.3979600000000012`}, {3.27454`, 1.3979299999999988`}, {3.27462`, 
   1.3978600000000014`}, {3.27477`, 1.3977199999999996`}, {3.27508`, 
   1.3974499999999992`}, {3.27568`, 1.3969200000000015`}, {3.2769`, 
   1.3958399999999997`}, {3.27932`, 1.3937099999999987`}, {3.28418`, 
   1.3895300000000006`}, {3.29389`, 1.381450000000001`}, {3.31331`, 
   1.3664400000000008`}, {3.33674`, 1.3503700000000016`}, {3.36062`, 
   1.3362599999999993`}, {3.3847`, 1.3243100000000005`}, {3.40871`, 
   1.314630000000001`}, {3.43242`, 1.3072000000000017`}, {3.4557`, 
   1.3018900000000002`}, {3.4785`, 1.2985100000000003`}, {3.50082`, 
   1.2968599999999988`}, {3.51177`, 1.2966300000000004`}, {3.51314`, 
   1.2966200000000008`}, {3.51323`, 1.2966200000000008`}, {3.51327`, 
   1.2966200000000008`}, {3.51329`, 1.2966200000000008`}, {3.5133`, 
   1.2966200000000008`}, {3.5133`, 1.2966200000000008`}, {3.5133`, 
   1.2966200000000008`}, {3.5133`, 1.2966200000000008`}, {3.5133`, 
   1.2966200000000008`}, {3.5133`, 1.2966200000000008`}, {3.51331`, 
   1.2966200000000008`}, {3.51331`, 1.2966200000000008`}, {3.51331`, 
   1.2966200000000008`}, {3.51331`, 1.2966200000000008`}, {3.51331`, 
   1.2966200000000008`}, {3.51331`, 1.2966200000000008`}, {3.51331`, 
   1.2966200000000008`}, {3.51332`, 1.2966200000000008`}, {3.51334`, 
   1.2966200000000008`}, {3.51338`, 1.2966200000000008`}, {3.51346`, 
   1.2966200000000008`}, {3.51361`, 1.2966200000000008`}, {3.51391`, 
   1.2966300000000004`}, {3.51451`, 1.2966300000000004`}, {3.51571`, 
   1.2966300000000004`}, {3.51811`, 1.2966599999999993`}, {3.52292`, 
   1.296759999999999`}, {3.53254`, 1.2971700000000013`}, {3.55177`, 
   1.2987600000000015`}, {3.57301`, 1.301639999999999`}, {3.59404`, 
   1.3055699999999995`}, {3.61501`, 1.3104300000000002`}, {3.63602`, 
   1.3161400000000008`}, {3.65716`, 1.322610000000001`}, {3.67936`, 
   1.3300599999999996`}, {3.70822`, 1.3404999999999987`}, {3.72847`, 
   1.34816`}, {3.73859`, 1.3520399999999988`}, {3.74365`, 
   1.3539999999999992`}, {3.74871`, 1.35595`}, {3.74997`, 
   1.3564399999999992`}, {3.75124`, 1.3569299999999984`}, {3.7525`, 
   1.3574200000000012`}, {3.75377`, 1.3579100000000004`}, {3.7563`, 
   1.3588799999999992`}, {3.76136`, 1.3608399999999996`}, {3.76642`, 
   1.3627900000000004`}, {3.77654`, 1.3666699999999992`}, {3.79678`, 
   1.3742899999999985`}, {3.83726`, 1.388539999999999`}, {3.87774`, 
   1.4006399999999992`}, {3.90948`, 1.4079900000000016`}, {3.93189`, 
   1.4117899999999999`}, {3.95397`, 1.4142299999999999`}, {3.97605`, 
   1.41526`}, {3.97881`, 1.4152799999999992`}, {3.97951`, 
   1.4152899999999988`}, {3.97985`, 1.4152899999999988`}, {3.98002`, 
   1.4152899999999988`}, {3.98007`, 1.4152899999999988`}, {3.98009`, 
   1.4152899999999988`}, {3.9801`, 1.4152899999999988`}, {3.98011`, 
   1.4152899999999988`}, {3.98011`, 1.4152899999999988`}, {3.98011`, 
   1.4152899999999988`}, {3.98011`, 1.4152899999999988`}, {3.98011`, 
   1.4152899999999988`}, {3.98011`, 1.4152899999999988`}, {3.98011`, 
   1.4152899999999988`}, {3.98011`, 1.4152899999999988`}, {3.98011`, 
   1.4152899999999988`}, {3.98011`, 1.4152899999999988`}, {3.98011`, 
   1.4152899999999988`}, {3.98012`, 1.4152899999999988`}, {3.98013`, 
   1.4152899999999988`}, {3.98014`, 1.4152899999999988`}, {3.98018`, 
   1.4152899999999988`}, {3.98026`, 1.4152899999999988`}, {3.9804`, 
   1.4152899999999988`}, {3.9807`, 1.4152899999999988`}, {3.9813`, 
   1.4152799999999992`}, {3.98249`, 1.4152799999999992`}, {3.98487`, 
   1.4152500000000003`}, {3.98963`, 1.415140000000001`}, {3.99915`, 
   1.4146999999999998`}, {4.0182`, 1.4128900000000009`}, {4.04049`, 
   1.409130000000001`}, {4.06293`, 1.403500000000001`}, {4.08547`, 
   1.395900000000001`}, {4.10807`, 1.3862699999999997`}, {4.13068`, 
   1.374600000000001`}, {4.15324`, 1.3609000000000009`}, {4.1757`, 
   1.3452300000000008`}, {4.19801`, 1.3276900000000005`}, {4.22013`, 
   1.30837`}, {4.24205`, 1.2874100000000013`}, {4.26375`, 
   1.26492`}, {4.28526`, 1.2410099999999993`}, {4.30659`, 
   1.2158000000000015`}, {4.32776`, 1.1893600000000006`}, {4.34883`, 
   1.1617800000000003`}, {4.36982`, 1.1331200000000017`}, {4.3908`, 
   1.1034299999999995`}, {4.4118`, 1.0727499999999992`}, {4.43431`, 
   1.0389400000000002`}, {4.46102`, 0.9977499999999999`}, {4.49389`, 
   0.9457999999999984`}, {4.53741`, 0.875589999999999`}, {4.55403`, 
   0.8484900000000017`}, {4.57065`, 0.8213299999999997`}, {4.57897`, 
   0.8077299999999994`}, {4.58104`, 0.8043300000000002`}, {4.58208`, 
   0.8026300000000006`}, {4.58312`, 0.800930000000001`}, {4.58416`, 
   0.7992300000000014`}, {4.5852`, 0.7975299999999983`}, {4.58728`, 
   0.7941299999999991`}, {4.58936`, 0.7907299999999999`}, {4.59351`, 
   0.7839300000000016`}, {4.60182`, 0.7703400000000009`}, {4.61014`, 
   0.7567699999999995`}, {4.62676`, 0.7296599999999991`}, {4.66001`, 
   0.6757899999999992`}, {4.7265`, 0.5701399999999985`}, {4.83406`, 
   0.4075799999999994`}, {4.90305`, 0.30923999999999907`}, {4.95196`, 
   0.24202999999999975`}, {5.00008`, 0.1775500000000001`}, {5.05362`, 
   0.10722000000000165`}, {5.11854`, 
   0.02308000000000021`}, {5.20627`, -0.09035000000000082`}, \
{5.26804`, -0.171009999999999`}, {5.32982`, -0.2526699999999984`}, \
{5.36071`, -0.2938699999999983`}, {5.3916`, -0.33523999999999887`}, \
{5.40705`, -0.3559599999999996`}, {5.41091`, -0.3611499999999985`}, \
{5.41477`, -0.3663300000000014`}, {5.4167`, -0.36891999999999925`}, \
{5.41863`, -0.3715100000000007`}, {5.42056`, -0.3741100000000017`}, \
{5.42442`, -0.379290000000001`}, {5.42829`, -0.3844700000000003`}, \
{5.43601`, -0.3948399999999985`}, {5.44373`, -0.40520000000000067`}, \
{5.45918`, -0.425889999999999`}, {5.47462`, -0.4465399999999988`}, \
{5.50551`, -0.4876100000000001`}, {5.5364`, -0.5282300000000006`}, \
{5.59818`, -0.6072999999999986`}, {5.65996`, -0.682030000000001`}, \
{5.70066`, -0.7280200000000008`}, {5.72839`, -0.7575900000000004`}, \
{5.75161`, -0.7811200000000014`}, {5.77474`, -0.8033900000000003`}, \
{5.79786`, -0.8244000000000007`}, {5.82097`, -0.8441100000000006`}, \
{5.84409`, -0.8624900000000011`}, {5.86724`, -0.879529999999999`}, \
{5.8904`, -0.8951899999999995`}, {5.91358`, -0.9094699999999989`}, \
{5.93677`, -0.9223700000000008`}, {5.95997`, -0.9339200000000005`}, \
{5.98317`, -0.9441500000000005`}, {6.00638`, -0.9531099999999988`}, \
{6.02959`, -0.9608600000000003`}, {6.05283`, -0.9674900000000015`}, \
{6.07613`, -0.9730899999999991`}, {6.09952`, -0.9777500000000003`}, \
{6.12305`, -0.9815900000000006`}, {6.14768`, -0.9848199999999991`}, \
{6.17702`, -0.9878199999999993`}, {6.21375`, -0.9906499999999987`}, \
{6.23992`, -0.9923199999999994`}, {6.24647`, -0.9927199999999985`}, \
{6.24974`, -0.9929099999999984`}, {6.25137`, -0.9930100000000017`}, \
{6.25301`, -0.9931100000000015`}, {6.25628`, -0.993310000000001`}, \
{6.25955`, -0.9935100000000006`}, {6.26609`, -0.9939199999999992`}, \
{6.27918`, -0.9947499999999998`}, {6.30535`, -0.9966299999999997`}, \
{6.33152`, -0.9989300000000014`}, {6.38386`, -1.0055900000000015`}, \
{6.42101`, -1.0126600000000003`}, {6.44521`, -1.0185899999999997`}, \
{6.46882`, -1.0255299999999998`}, {6.49229`, -1.0336700000000008`}, \
{6.51568`, -1.0431199999999983`}, {6.53904`, -1.0539999999999985`}, \
{6.56238`, -1.0663899999999984`}, {6.58571`, -1.0803699999999985`}, \
{6.60898`, -1.0959900000000005`}, {6.63212`, -1.1132400000000011`}, \
{6.65507`, -1.1320799999999984`}, {6.67773`, -1.152429999999999`}, \
{6.70003`, -1.1741799999999998`}, {6.72191`, -1.1971999999999987`}, \
{6.74332`, -1.221350000000001`}, {6.76425`, -1.2465100000000007`}, \
{6.78472`, -1.2725700000000018`}, {6.78722`, -1.2758600000000015`}, \
{6.78848`, -1.2775099999999995`}, {6.7891`, -1.27834`}, {6.78941`, \
-1.2787499999999987`}, {6.78945`, -1.2788000000000004`}, {6.78947`, \
-1.2788299999999992`}, {6.78948`, -1.2788399999999989`}, {6.78948`, \
-1.2788499999999985`}, {6.78948`, -1.2788499999999985`}, {6.78948`, \
-1.2788499999999985`}, {6.78948`, -1.2788499999999985`}, {6.78948`, \
-1.2788499999999985`}, {6.78948`, -1.2788499999999985`}, {6.78948`, \
-1.2788499999999985`}, {6.78948`, -1.2788499999999985`}, {6.78949`, \
-1.2788499999999985`}, {6.78949`, -1.2788499999999985`}, {6.78949`, \
-1.2788600000000017`}, {6.7895`, -1.2788700000000013`}, {6.78952`, \
-1.2788900000000005`}, {6.78955`, -1.2789399999999986`}, {6.78962`, \
-1.2790299999999988`}, {6.78976`, -1.279209999999999`}, {6.79003`, \
-1.2795699999999997`}, {6.79058`, -1.2803000000000004`}, {6.79168`, \
-1.2817599999999985`}, {6.79387`, -1.2846900000000012`}, {6.79825`, \
-1.2906000000000013`}, {6.80702`, -1.3026100000000014`}, {6.82457`, \
-1.3274100000000004`}, {6.84381`, -1.3557699999999997`}, {6.8627`, \
-1.38476`}, {6.88131`, -1.4143700000000017`}, {6.89967`, \
-1.4446000000000012`}, {6.91785`, -1.4754199999999997`}, {6.93589`, \
-1.50685`}, {6.95384`, -1.5388899999999985`}, {6.97421`, \
-1.5760900000000007`}, {6.99802`, -1.6205399999999983`}, {7.02704`, \
-1.6758799999999994`}, {7.06523`, -1.75`}, {7.07269`, \
-1.7645599999999995`}, {7.08014`, -1.7791500000000013`}, {7.08201`, \
-1.7827899999999985`}, {7.08387`, -1.786439999999999`}, {7.08574`, \
-1.7900899999999993`}, {7.0876`, -1.79373`}, {7.08947`, \
-1.7973800000000004`}, {7.09319`, -1.8046700000000016`}, {7.10065`, \
-1.8192400000000006`}, {7.10811`, -1.8338`}, {7.12302`, \
-1.8628199999999993`}, {7.15286`, -1.9202499999999993`}, {7.18269`, \
-1.9763899999999985`}, {7.22407`, -2.0510100000000016`}, {7.24917`, \
-2.0938300000000005`}, {7.26772`, -2.1240500000000004`}, {7.28628`, \
-2.1528899999999993`}, {7.30505`, -2.1805200000000013`}, {7.32413`, \
-2.2068900000000014`}, {7.34361`, -2.23189`}, {7.36359`, \
-2.2553899999999985`}, {7.38421`, -2.2772000000000006`}, {7.39489`, \
-2.2875000000000014`}, {7.39756`, -2.2899600000000007`}, {7.39773`, \
-2.2901200000000017`}, {7.39773`, -2.2901200000000017`}, {7.39773`, \
-2.2901200000000017`}, {7.39773`, -2.2901200000000017`}, {7.39773`, \
-2.2901200000000017`}, {7.39773`, -2.2901200000000017`}, {7.39773`, \
-2.2901200000000017`}, {7.39773`, -2.2901200000000017`}, {7.39773`, \
-2.2901200000000017`}, {7.39774`, -2.2901200000000017`}, {7.39774`, \
-2.2901200000000017`}, {7.39774`, -2.2901200000000017`}, {7.39774`, \
-2.2901300000000013`}, {7.39775`, -2.290140000000001`}, {7.39777`, \
-2.2901500000000006`}, {7.39781`, -2.290189999999999`}, {7.39788`, \
-2.2902500000000003`}, {7.39803`, -2.2903899999999986`}, {7.39833`, \
-2.290659999999999`}, {7.39892`, -2.2912`}, {7.4001`, \
-2.2922699999999985`}, {7.40247`, -2.2943800000000003`}, {7.40721`, \
-2.298490000000001`}, {7.4167`, -2.3063000000000002`}, {7.43566`, \
-2.3201500000000017`}, {7.45929`, -2.33408`}, {7.48405`, -2.3446`}, \
{7.50968`, -2.351040000000001`}, {7.52262`, -2.35257`}, {7.52909`, \
-2.3529000000000018`}, {7.53232`, -2.35295`}, {7.53313`, \
-2.3529599999999995`}, {7.53323`, -2.3529599999999995`}, {7.53326`, \
-2.3529599999999995`}, {7.53326`, -2.3529599999999995`}, {7.53327`, \
-2.3529599999999995`}, {7.53327`, -2.3529599999999995`}};

This is the data corresponding to trajectory of ion inside the ion trap. To know the corresponding frequency, I did the fourier transform

I plotted the data

ListLinePlot[trajectory, AxesLabel -> {"t(\[Mu]s)", "x(t) (mm)"}]

Trajectory of ion

Next I plotted the Fourier transform of the data.

ListLinePlot[Abs[Fourier[trajectory[[All, 2]]]], PlotRange ->{{0, 1500}, {0, 0.5}}]

And got the following plot

enter image description here

I changed the plot range to see clear peaks

ListLinePlot[Abs[Fourier[trajectory[[All, 2]]]], 
 PlotRange -> {{0, 30}, {0, 50}}]

enter image description here

It seems that there is one peak at 2 Mhz. But there should be a major peak at 135 kHz.

and it should look like this.. In this picture there is major peak around 135 kHz

can anyone help me in solving this. I have gone through many questions asked about fourier transform but all of them assume a sine function and then create a table of data and sample the data accordingly, but as my data is generated after simulation, I can not sample it further.

Any suggestion will be helpful. Thanks.

Hi all, I followed the suggestions given by you guys and I am just about to reach the answer. I generated large number of data and plot of the data looks like this

enter image description here

time is in micro-sec and x(t) is in mm.

then I took x(t) and did its fourier transform. In the simulation, applied frequency of rf electric field is 2*pi*600 kHz, and resultant oscillation frequency of the ion is 2*pi*135 kHz. According to Nyquist-Shannon theorem, fs>= 2*f(max). So I took the sampling frequency fs=1200 kHz. With the following code

fft = Abs[Fourier[data[[All, 2]]]];    
fs = 1200000;
df = fs/Length[data];
f = Range[0, (Length[data] - 1)/Length[data]*fs, df] // N;
powerspectrum = Transpose[{f, fft}];
ListLinePlot[powerspectrum, PlotRange -> {{0, 1000}, {0, 700}}, 
AxesLabel -> {"f(Hz)", "Amplitude"}]

I got this spectrum

enter image description here

Major peak is around 100 Hz. My question is, (a) should I change the x(t) in meter and time in seconds? Because x axis of the power spectrum should be in kHz, (b) When I am taking sampling frequency greater than 2*fmax, say 10*fs, then my peaks are shifting, then how would I know what should be the proper sampling frequency?

Improve this question
$\endgroup$
7
  • 1
    $\begingroup$ Note that the spacing of your frequency samples is the reciprocal of the duration of your data (I assume that your samples are equally spaced). Your data set appears to have only 1.5 cycles at the lowest frequency. This not really enough to get an accurate frequency estimate - perhaps you can simulate more cycles. On the other hand, you seem to have a time step that is unnecessarily small (this won't give you the wrong answer, but may be inconvenient). $\endgroup$
    – mikado
    Dec 8, 2019 at 7:28
  • $\begingroup$ The data you posted, trajectory, corresponds to $t \in [0., 7.53327]$; your plot it seems is for $t \in [0,16.5]$. $\endgroup$
    – Anton Antonov
    Dec 8, 2019 at 14:26
  • 1
    $\begingroup$ I wrote some notes on Fourier here which might be useful. Note that you don't seem to have constructed a frequency axis. The axis you have looks like point numbers. As well as Anton Antonov's point you don't have equally spaced points on the time axis. This is needed. What are the units of your time axis? If they are seconds then approximately you have a sample rate of 86 samples per second which could give you a spectra with frequencies from 0 to 40 Hz. $\endgroup$
    – Hugh
    Dec 8, 2019 at 15:23
  • $\begingroup$ @Anton Antonov, because the whole data is too large, but as this is periodic, small data will contain the information. $\endgroup$
    – Kamal Kumar
    Dec 9, 2019 at 10:47
  • $\begingroup$ Dear @mikado, I have simulated more cycles, but due to a large array size, I posted only small size. And the step size is 0.001 microsec. $\endgroup$
    – Kamal Kumar
    Dec 9, 2019 at 10:57

1 Answer 1

Reset to default
0
$\begingroup$

I see this question as a duplicate of "Fourier Transform to help guess with NonLinearModelFit".

Following my answer in that discussion I get the following terms and plots.

enter image description here

enter image description here

From the comments

For the question:

The data you posted, trajectory, corresponds to t∈[0.,7.53327] ; your plot it seems is for t∈[0,16.5].

I got the response:

@Anton Antonov, because the whole data is too large, but as this is periodic, small data will contain the information.

As we can see in the plot above the data is not that periodic, hence the terms determined are likely not that representative.

Improve this answer
$\endgroup$
0

Not the answer you're looking for? Browse other questions tagged or ask your own question.