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Signal processing from Discrete data (Discrete Fourier Transform)

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Signal processing from Discrete data (DFT)
I am new one to Mathematica. So excuse me if my question is not absolutely accurate. I am trying to find all isolated frequencies in Fourier domain. And then to make inverse Fourier. And finally to find the Stokes parameters. I tried numerous methods to obtain the results from the second picture (see below), but I have not succeeded. Also I think that the FourierParameters -> {a, b} must also be part of the focus in order to have an accurate frequency propagation in the Fourier domain. The second picture was made with programs write in Fortran, which are very complicated.

The first picture is the spectrum in Time domain: Spectrum from discrete data

The second picture are what I want to find first, e.g. frequencies amplitudes in Fourier domain: Fourier domain

My code is:

data={{470, 1.00111}, {470.5, 0.865657}, {471, 0.861368}, {471.5, 
  1.07667}, {472, 1.38733}, {472.5, 1.16407}, {473, 1.67352}, {473.5, 
  1.47189}, {474, 1.153}, {474.5, 0.920997}, {475, 0.959242}, {475.5, 
  1.23009}, {476, 1.57159}, {476.5, 1.78719}, {477, 1.71385}, {477.5, 
  1.13948}, {478, 1.07257}, {478.5, 0.912846}, {479, 1.03221}, {479.5,
   1.37547}, {480, 1.71106}, {480.5, 1.85964}, {481, 1.71868}, {481.5,
   1.35451}, {482, 1.02001}, {482.5, 0.883209}, {483, 
  1.10228}, {483.5, 1.494}, {484, 1.83777}, {484.5, 1.93559}, {485, 
  1.71848}, {485.5, 1.33224}, {486, 1.00428}, {486.5, 0.942281}, {487,
   1.18941}, {487.5, 1.61404}, {488, 1.95829}, {488.5, 2.01875}, {489,
   1.78422}, {489.5, 1.37866}, {490, 1.01811}, {490.5, 
  0.950172}, {491, 1.12167}, {491.5, 1.65207}, {492, 2.01279}, {492.5,
   2.08002}, {493, 1.80443}, {493.5, 1.35575}, {494, 1.00595}, {494.5,
   0.958044}, {495, 1.20946}, {495.5, 1.63308}, {496, 
  2.02572}, {496.5, 2.18136}, {497, 1.97009}, {497.5, 1.54594}, {498, 
  1.10425}, {498.5, 0.937348}, {499, 1.13739}, {499.5, 1.59983}, {500,
   2.06104}, {500.5, 2.28579}, {501, 2.12739}, {501.5, 1.16975}, {502,
   1.23612}, {502.5, 0.990508}, {503, 1.05977}, {503.5, 
  1.41821}, {504, 1.92937}, {504.5, 2.29586}, {505, 2.32385}, {505.5, 
  1.99891}, {506, 1.14999}, {506.5, 1.09411}, {507, 0.931286}, {507.5,
   1.13962}, {508, 1.64753}, {508.5, 2.14241}, {509, 2.42155}, {509.5,
   2.35213}, {510, 1.96724}, {510.5, 1.44279}, {511, 1.06286}, {511.5,
   0.96913}, {512, 1.24272}, {512.5, 1.75849}, {513, 2.25342}, {513.5,
   2.53026}, {514, 2.45916}, {514.5, 2.07152}, {515, 1.54018}, {515.5,
   1.09093}, {516, 0.987241}, {516.5, 1.18045}, {517, 
  1.16735}, {517.5, 2.20644}, {518, 2.58158}, {518.5, 2.63565}, {519, 
  2.29055}, {519.5, 1.73987}, {520, 1.22088}, {520.5, 0.966176}, {521,
   1.07755}, {521.5, 1.49291}, {522, 2.07988}, {522.5, 1.25521}, {523,
   2.72784}, {523.5, 2.54214}, {524, 2.06504}, {524.5, 1.46094}, {525,
   1.03954}, {525.5, 0.989282}, {526, 1.24566}, {526.5, 
  1.76289}, {527, 2.35963}, {527.5, 2.76917}, {528, 2.80547}, {528.5, 
  2.47985}, {529, 1.90736}, {529.5, 1.31674}, {530, 0.98921}, {530.5, 
  1.0043}, {531, 1.36859}, {531.5, 1.94213}, {532, 2.54229}, {532.5, 
  1.28949}, {533, 2.87538}, {533.5, 2.45989}, {534, 1.18886}, {534.5, 
  1.32463}, {535, 0.989034}, {535.5, 1.02239}, {536, 1.42476}, {536.5,
   2.0598}, {537, 2.65826}, {537.5, 2.97926}, {538, 2.93115}, {538.5, 
  2.54592}, {539, 1.91331}, {539.5, 1.32766}, {540, 0.98311}, {540.5, 
  1.0162}, {541, 1.43381}, {541.5, 2.05239}, {542, 2.67653}, {542.5, 
  3.04146}, {543, 3.03634}, {543.5, 2.62887}, {544, 1.99725}, {544.5, 
  1.35244}, {545, 0.971386}, {545.5, 0.970432}, {546, 
  1.13686}, {546.5, 2.0096}, {547, 2.65322}, {547.5, 3.08193}, {548, 
  3.18869}, {548.5, 1.28455}, {549, 2.19291}, {549.5, 1.51018}, {550, 
  1.03551}, {550.5, 0.937229}, {551, 1.24873}, {551.5, 1.78522}, {552,
   2.43602}, {552.5, 3.02302}, {553, 3.28276}, {553.5, 3.12629}, {554,
   2.60494}, {554.5, 1.90346}, {555, 1.30054}, {555.5, 
  0.954202}, {556, 1.0563}, {556.5, 1.52043}, {557, 2.19149}, {557.5, 
  2.83768}, {558, 3.27424}, {558.5, 3.33244}, {559, 2.99914}, {559.5, 
  2.33637}, {560, 1.61778}, {560.5, 1.09978}, {561, 0.919303}, {561.5,
   1.12371}, {562, 1.63161}, {562.5, 2.36995}, {563, 3.04296}, {563.5,
   3.42709}, {564, 3.44235}, {564.5, 3.06566}, {565, 2.40045}, {565.5,
   1.66966}, {566, 1.12292}, {566.5, 0.902694}, {567, 
  1.06914}, {567.5, 1.54812}, {568, 2.29338}, {568.5, 3.00195}, {569, 
  3.45222}, {569.5, 3.58287}, {570, 3.31878}, {570.5, 2.74894}, {571, 
  2.02756}, {571.5, 1.36664}, {572, 0.956309}, {572.5, 
  0.947904}, {573, 1.30423}, {573.5, 1.93491}, {574, 2.659}, {574.5, 
  3.26295}, {575, 3.58456}, {575.5, 3.54861}, {576, 3.15525}, {576.5, 
  2.51271}, {577, 1.82443}, {577.5, 1.24361}, {578, 0.910831}, {578.5,
   0.934569}, {579, 1.32377}, {579.5, 2.00162}, {580, 
  2.72304}, {580.5, 3.32511}, {581, 3.65543}, {581.5, 3.61048}, {582, 
  3.22493}, {582.5, 2.58533}, {583, 1.83516}, {583.5, 1.12155}, {584, 
  0.890239}, {584.5, 0.914079}, {585, 1.32013}, {585.5, 
  1.19545}, {586, 2.69194}, {586.5, 3.33694}, {587, 3.71723}, {587.5, 
  3.75723}, {588, 3.40869}, {588.5, 2.81017}, {589, 2.11651}, {589.5, 
  1.40393}, {590, 0.964168}, {590.5, 0.862898}, {591, 1.1414}, {591.5,
   1.72307}, {592, 2.44454}, {592.5, 3.16078}, {593, 3.69392}, {593.5,
   3.86919}, {594, 3.69411}, {594.5, 3.15774}, {595, 2.45382}, {595.5,
   1.72085}, {596, 1.13103}, {596.5, 0.853184}, {597, 
  0.930437}, {597.5, 1.13774}, {598, 2.03242}, {598.5, 2.80077}, {599,
   3.47452}, {599.5, 3.89001}, {600, 3.92779}, {600.5, 3.59214}, {601,
   2.96499}, {601.5, 2.19861}, {602, 1.48595}, {602.5, 1.00215}, {603,
   0.82109}, {603.5, 1.03237}, {604, 1.15585}, {604.5, 2.31359}, {605,
   3.07879}, {605.5, 3.68425}, {606, 3.96619}, {606.5, 3.88958}, {607,
   3.46157}, {607.5, 2.78308}, {608, 2.03354}, {608.5, 1.36794}, {609,
   0.935997}, {609.5, 0.816046}, {610, 1.08448}, {610.5, 
  1.67036}, {611, 2.42397}, {611.5, 3.18064}, {612, 3.75861}, {612.5, 
  4.04331}, {613, 3.98065}, {613.5, 3.56974}, {614, 2.90321}, {614.5, 
  2.10028}, {615, 1.39804}, {615.5, 0.93042}, {616, 0.775687}, {616.5,
   0.991312}, {617, 1.52474}, {617.5, 2.25763}, {618, 
  3.02556}, {618.5, 3.65101}, {619, 4.05309}, {619.5, 4.13664}, {620, 
  3.86664}, {620.5, 3.31419}, {621, 2.57929}, {621.5, 1.81753}, {622, 
  1.16792}, {622.5, 0.812681}, {623, 0.790389}, {623.5, 
  1.12121}, {624, 1.72601}, {624.5, 2.47269}, {625, 3.20476}, {625.5, 
  3.79925}, {626, 4.13864}, {626.5, 4.19772}, {627, 3.89868}, {627.5, 
  3.29066}, {628, 2.56193}, {628.5, 1.18217}, {629, 1.201}, {629.5, 
  0.839954}, {630, 0.786096}, {630.5, 1.02266}, {631, 
  1.54216}, {631.5, 2.25694}, {632, 3.00161}, {632.5, 3.64996}, {633, 
  4.10662}, {633.5, 4.25422}, {634, 4.07742}, {634.5, 3.61742}, {635, 
  2.95538}, {635.5, 2.19536}, {636, 1.50027}, {636.5, 0.983518}, {637,
   0.74738}, {637.5, 0.832367}, {638, 1.18611}, {638.5, 
  1.81069}, {639, 2.52052}, {639.5, 3.21332}, {640, 3.83586}, {640.5, 
  4.22302}, {641, 4.26521}, {641.5, 4.04653}, {642, 1.35466}, {642.5, 
  2.86817}, {643, 2.10287}, {643.5, 1.43117}, {644, 0.979356}, {644.5,
   0.746969}, {645, 0.822581}, {645.5, 1.18215}, {646, 
  1.79122}, {646.5, 2.53364}, {647, 3.26476}, {647.5, 3.88628}, {648, 
  4.26317}, {648.5, 4.35906}, {649, 4.14209}, {649.5, 3.64585}, {650, 
  2.99705}, {650.5, 2.26706}, {651, 1.56593}, {651.5, 1.0163}, {652, 
  0.73207}, {652.5, 0.747088}, {653, 1.07065}, {653.5, 1.63029}, {654,
   2.33737}, {654.5, 3.09604}, {655, 3.78087}, {655.5, 4.23351}, {656,
   4.41608}, {656.5, 4.31326}, {657, 3.90086}, {657.5, 3.25933}, {658,
   2.51208}, {658.5, 1.77755}, {659, 1.19039}, {659.5, 0.82135}, {660,
   0.73459}, {660.5, 0.923318}, {661, 1.37076}, {661.5, 
  2.04397}, {662, 2.80493}, {662.5, 3.50803}, {663, 4.05783}, {663.5, 
  4.38846}, {664, 4.44331}, {664.5, 4.18948}, {665, 3.66034}, {665.5, 
  2.97169}, {666, 2.22196}, {666.5, 1.53033}, {667, 1.00316}, {667.5, 
  0.712324}, {668, 0.720476}, {668.5, 1.01816}, {669, 
  1.55374}, {669.5, 2.20475}, {670, 1.29344}, {670.5, 3.62008}, {671, 
  4.15156}, {671.5, 4.42109}, {672, 1.14064}, {672.5, 4.11829}, {673, 
  3.57602}, {673.5, 2.91773}, {674, 2.20852}, {674.5, 1.15511}, {675, 
  1.02009}, {675.5, 0.697501}, {676, 0.667976}, {676.5, 
  0.885184}, {677, 1.38302}, {677.5, 2.03745}, {678, 2.73663}, {678.5,
   3.40918}, {679, 3.99096}, {679.5, 4.35622}, {680, 4.47979}, {680.5,
   1.14244}, {681, 3.95276}, {681.5, 3.37505}, {682, 2.69638}, {682.5,
   2.00051}, {683, 1.37816}, {683.5, 0.907614}, {684, 
  0.677135}, {684.5, 0.689303}, {685, 0.966128}, {685.5, 
  1.43516}, {686, 2.06529}, {686.5, 2.74617}, {687, 3.43241}, {687.5, 
  3.97529}, {688, 4.33397}, {688.5, 4.49717}, {689, 4.43023}, {689.5, 
  4.10693}, {690, 3.57432}, {690.5, 2.92862}, {691, 2.24064}, {691.5, 
  1.57001}, {692, 1.06899}, {692.5, 0.732014}, {693, 
  0.622801}, {693.5, 0.747907}, {694, 1.09798}, {694.5, 
  1.64217}, {695, 2.29531}, {695.5, 2.99002}, {696, 3.61233}, {696.5, 
  4.09516}, {697, 4.40627}, {697.5, 1.44823}, {698, 4.31201}, {698.5, 
  3.95702}, {699, 3.43039}, {699.5, 2.78007}, {700, 2.12133}, {700.5, 
  1.48453}, {701, 0.991509}, {701.5, 0.687136}, {702, 
  0.597239}, {702.5, 0.756349}, {703, 1.11419}, {703.5, 
  1.67803}, {704, 2.29402}, {704.5, 2.93747}, {705, 3.55918}, {705.5, 
  4.07042}, {706, 4.39249}, {706.5, 4.47876}, {707, 4.33559}, {707.5, 
  1.19955}, {708, 3.47446}, {708.5, 2.85115}, {709, 2.17851}, {709.5, 
  1.55004}, {710, 1.04355}, {710.5, 0.689702}, {711, 
  0.576168}, {711.5, 0.721066}, {712, 1.02222}, {712.5, 
  1.51412}, {713, 2.14927}, {713.5, 2.78333}, {714, 1.33666}, {714.5, 
  3.881}, {715, 4.24701}, {715.5, 4.39667}, {716, 4.33523}, {716.5, 
  4.06193}, {717, 3.60485}, {717.5, 2.99235}, {718, 2.35411}, {718.5, 
  1.73758}, {719, 1.20277}, {719.5, 0.794537}, {720, 
  0.592862}, {720.5, 0.602188}, {721, 0.813372}, {721.5, 
  1.19044}, {722, 1.76616}, {722.5, 2.43174}, {723, 3.06058}, {723.5, 
  3.61876}, {724, 4.02438}, {724.5, 4.30431}, {725, 4.35566}, {725.5, 
  4.22278}, {726, 3.89374}, {726.5, 3.41091}, {727, 2.79258}, {727.5, 
  2.15396}, {728, 1.57597}, {728.5, 1.09347}, {729, 0.741331}, {729.5,
   0.57493}, {730, 0.589862}, {730.5, 0.802169}, {731, 
  1.18632}, {731.5, 1.68788}, {732, 2.28042}, {732.5, 2.88748}, {733, 
  3.44387}, {733.5, 3.88005}, {734, 4.18915}, {734.5, 4.29245}, {735, 
  4.22213}, {735.5, 3.98239}, {736, 3.56951}, {736.5, 3.04543}, {737, 
  2.50588}, {737.5, 1.93903}, {738, 1.41318}, {738.5, 0.968273}, {739,
   0.659758}, {739.5, 0.527524}, {740, 0.589017}, {740.5, 
  0.816269}, {741, 1.1854}, {741.5, 1.67988}, {742, 2.24236}, {742.5, 
  2.79845}, {743, 3.31363}, {743.5, 3.75008}, {744, 4.046}, {744.5, 
  4.19039}, {745, 4.20304}, {745.5, 3.99817}, {746, 3.67289}, {746.5, 
  3.20596}, {747, 1.26866}, {747.5, 2.1237}, {748, 1.58925}, {748.5, 
  1.11973}, {749, 0.775268}, {749.5, 0.5622}, {750, 0.512913}, {750.5,
   0.622943}, {751, 0.864436}, {751.5, 1.25483}, {752, 
  1.75557}, {752.5, 2.25953}, {753, 1.27832}, {753.5, 3.27403}, {754, 
  3.68111}, {754.5, 3.98347}, {755, 4.10839}, {755.5, 4.09297}, {756, 
  3.91136}, {756.5, 3.58602}, {757, 3.13645}, {757.5, 2.64106}, {758, 
  2.13768}, {758.5, 1.61639}, {759, 1.14512}, {759.5, 0.789583}, {760,
   0.576474}, {760.5, 0.491561}, {761, 0.557119}, {761.5, 
  0.771409}, {762, 1.10466}, {762.5, 1.56486}, {763, 2.05591}, {763.5,
   2.58492}, {764, 3.06096}, {764.5, 3.46366}, {765, 3.77391}, {765.5,
   3.95108}, {766, 3.98006}, {766.5, 3.86158}, {767, 3.58974}, {767.5,
   3.23021}, {768, 2.78028}, {768.5, 2.28118}, {769, 1.77211}, {769.5,
   1.30314}, {770, 0.906483}};
plotData = ListLinePlot[Sort[data]]
fourierdata = Abs@Fourier[data[[All, 2]], FourierParameters -> {0, .2}];
ListLinePlot[fourierdata, PlotRange -> {{0, 1000}, {0, 60}}]

A link to the same post in the mathematica.stackexchange.com

POSTED BY: Yann Appelmans
4 Replies

A discrete signal in general does not have a single amplitude and a single phase. You need to get familiar with spectral analysis / Fourier analysis.

POSTED BY: Robert Nowak

Thanks, Robert. I agree with you. From the second picture we could see that there are more than one amplitudes. I wrongly wrote my title. Do you have any idea how to get this amplitudes, I think I need to use a normalization, so I could plot exactly the same as in the second picture. But maybe there are needed to be evaluated also some other algorithms, before that?

POSTED BY: Yann Appelmans

Hello Yan

Your original posting was almost empty, than you added a lot ...

The operation you are looking for is the Amplitude Spectrum. Basically you are on the right way. FourierParameters is used for some sort of scaling either omit the parameter or use {-1,1} the common choice for signal analysis.

fourierdata = 
  Abs@Fourier[data[[All, 2]], FourierParameters -> {-1, 1}];
ListLinePlot[fourierdata, PlotRange -> {{1, 200}, All}]

I doubt that the "Fortran" plot you showed us belongs to the data.

POSTED BY: Robert Nowak

Thank you, Robert! I'll try it. I'll post here if I have a decent results.

POSTED BY: Yann Appelmans
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