Fourier Transforms in Maple - Maple Programming Help

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Fourier Transforms in Maple

 Fourier transforms in Maple can be categorized as either transforms on expressions or transforms on signal data.
 To compute the Fourier transform of an expression, use the inttrans[fourier] command. For more details on this command, see the inttrans[fourier] help page.
 To compute the Fourier transform of signal data, the following commands are available:
 • SignalProcessing[DFT] : This command computes the discrete Fourier transform of an Array of signal data points. The SignalProcessing[DFT] command works for any size Array. For more information, see SignalProcessing[DFT].
 • SignalProcessing[FFT] : Similar to the SignalProcessing[DFT] command, SignalProcessing[FFT] computes the discrete Fourier transform of an Array of signal data points. The difference between the two commands is that the SignalProcessing[FFT] command uses the fast Fourier transform algorithm. Note: SignalProcessing[FFT] requires that the size of the Array must be a power of 2, greater than 2. If the Array passed to SignalProcessing[FFT] does not meet this requirement, a warning message is displayed and the SignalProcessing[DFT] command is used instead. For more information, see SignalProcessing[FFT].
 • DiscreteTransforms[FourierTransform] : The DiscreteTransforms[FourierTransform] provides similar functionality to that of SignalProcessing[DFT]. There are some options available in DiscreteTransforms[FourierTransform], such as padding, that are not available in SignalProcessing[DFT]. For more information, see DiscreteTransforms[FourierTransform].
 Note: Typically, SignalProcessing[DFT] and SignalProcessing[FFT] are slightly more efficient than DiscreteTransforms[FourierTransform].
 The table below provides a summarized comparison of the discrete Fourier transform commands mentioned above.

 Feature SignalProcessing[FFT] SignalProcessing[DFT] DiscreteTransforms[Fourier Transform] input single rtable yes yes yes input two rtables (Re/Im) yes yes yes higher-dimensional transforms yes yes yes specify single dim for higher-dimensional transforms no no yes output single rtable yes yes yes output two rtables (Re/Im) yes yes yes padding no no yes apply transform only to initial segment no no yes in place yes yes yes specify output rtable yes yes no specify working storage no no yes size of Array: power of 2 yes yes yes size of Array: other yes (warning and then dispatch to DFT) yes yes

Examples

 > $\mathrm{signal}≔\mathrm{sin}\left(t\right){ⅇ}^{-\frac{{t}^{2}}{100}}$
 ${\mathrm{signal}}{≔}{\mathrm{sin}}{}\left({t}\right){}{{ⅇ}}^{{-}\frac{{{t}}^{{2}}}{{100}}}$ (1)
 > $\mathrm{plot}\left(\mathrm{signal},t=-30..30\right)$ > $\mathrm{transform}≔{\mathrm{inttrans}}_{\mathrm{fourier}}\left(\mathrm{signal},t,s\right)$
 ${\mathrm{transform}}{≔}{-}{10}{}{I}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{sinh}}{}\left({50}{}{s}\right){}{{ⅇ}}^{{-}{25}{}{{s}}^{{2}}{-}{25}}$ (2)

The transform is purely imaginary:

 > $\mathrm{evalc}\left(\mathrm{ℜ}\left(\mathrm{transform}\right)\right)$
 ${0}$ (3)

This is what the imaginary part looks like:

 > $\mathrm{plot}\left(\mathrm{ℑ}\left(\mathrm{transform}\right),s=-3..3\right)$ > ${\mathrm{inttrans}}_{\mathrm{invfourier}}\left(\mathrm{transform},s,t\right)$
 ${\mathrm{sin}}{}\left({t}\right){}{{ⅇ}}^{{-}\frac{{{t}}^{{2}}}{{100}}}$ (4)

Turn the original signal into data by sampling:

 > $\mathrm{data}≔\mathrm{Array}\left(1..80,i→\mathrm{evalf}\left(\genfrac{}{}{0}{}{\mathrm{signal}}{\phantom{t=\frac{i}{4}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{signal}}}{t=\frac{i}{4}}\right)\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}0.2472493801& 0.4782284717& 0.6778153055& 0.8330982086& 0.9342719766& 0.9753019572& 0.9543081382& 0.8736433648& 0.7396636850& 0.5622125499& 0.3538622664& 0.1289739763& -0.09734987013& -0.3103399929& -0.4965810793& -0.6449045459& -0.7470908727& -0.7983356335& -0.7974529355& -0.7468109761& -0.6520150601& -0.5213720678& -0.3651855797& -0.1949415782& -0.02245018091& 0.1409909856& 0.2853511479& 0.4024873310& 0.4865936932& 0.5344562763& 0.5455063352& 0.5216811081& 0.4671149321& 0.3876949535& 0.2905236192& 0.1833342324& 0.07390606951& -0.03047787547& -0.1234939721& -0.2001341823& -0.2569380786& -0.2920941320& -0.3054141956& -0.2981943590& -0.2729827805& -0.2332802508& -0.1832019433& -0.1271290188& -0.06937673750& -0.01390182491& 0.03593343608& 0.07752902163& 0.1091418791& 0.1299085975& 0.1398024388& 0.1395353879& 0.1304188346& 0.1141980569& 0.09287587668& 0.06853983412& 0.04320521664& 0.01868354009& -0.003517056003& -0.02225629117& -0.03679098198& -0.04677160933& -0.05221092953& -0.05343083141& -0.05099444295& -0.04563063549& -0.03815763932& -0.02941158576& -0.02018456742& -0.01117540149& -0.002954832805& 0.004054455326& 0.009583845211& 0.01351257059& 0.01585170961& 0.01672117554\end{array}\right]$ (5)
 > $\mathrm{SignalProcessing}:-\mathrm{SignalPlot}\left(\mathrm{data}\right)$ > $\mathrm{tdata}≔\mathrm{SignalProcessing}:-\mathrm{DFT}\left(\mathrm{data}\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}0.4497659902145549+0.0{}I& 0.5052107976574163+0.03906636874940997{}I& 0.8711587723414446+0.07448652662727094{}I& 1.1182760567766632-1.6032057898841618{}I& -0.8339749012551638-0.6640820060944201{}I& -0.3159645805618081-0.12387884553360848{}I& -0.16663098558056866-0.07884710782613985{}I& -0.10535201213814539-0.059562131612666616{}I& -0.07240929585556202-0.048279430331170076{}I& -0.05222881581212239-0.040706219532542554{}I& -0.03882429156840338-0.035192046627913816{}I& -0.029408211717446228-0.030953615257113244{}I& -0.022514831311322547-0.027566764942940535{}I& -0.01730473009676248-0.024779825343899695{}I& -0.013265506494983152-0.022433066673257066{}I& -0.010068426817571105-0.020419723054675615{}I& -0.007493983240128059-0.018665435725976523{}I& -0.005390693235827055-0.01711665405828176{}I& -0.0036510898568611646-0.015733725210829757{}I& -0.002197117052166677-0.014486588000811401{}I& -0.0009709198499760314-0.013351987698461087{}I& 0.00007114657962727471-0.012311614766041167{}I& 0.0009625191227188982-0.011350826945342068{}I& 0.0017291645449145677-0.010457750572218883{}I& 0.0023914911868130355-0.009622634659130522{}I& 0.0029657088319229798-0.00883737750161765{}I& 0.003464811917516336-0.008095174947573733{}I& 0.0038993009931257852-0.007390253615141137{}I& 0.004277719195538448-0.006717666840270438{}I& 0.004607055731170729-0.0060731353106488785{}I& 0.004893052577030947-0.005452922509981582{}I& 0.005140439896249555-0.004853734699977014{}I& 0.005353117624996999-0.004272640914574546{}I& 0.005534296909190233-0.003707006932207723{}I& 0.005686610182988066-0.0031544417156110626{}I& 0.005812197276695991-0.0026127514510301993{}I& 0.0059127721776296585-0.002079901359869103{}I& 0.0059896751052553494-0.0015539818621112334{}I& 0.006043910803150312-0.0010331800204289058{}I& 0.006076178185685466-0.0005157520706090525{}I& 0.006086888632727769-7.757919228897728{}{10}^{-18}{}I& 0.00607617818568548+0.0005157520706090301{}I& 0.006043910803150335+0.0010331800204288989{}I& 0.0059896751052553685+0.0015539818621112117{}I& 0.005912772177629706+0.0020799013598689917{}I& 0.0058121972766957825+0.002612751451030189{}I& 0.0056866101829880335+0.0031544417156110895{}I& 0.005534296909190161+0.0037070069322077104{}I& 0.005353117624996999+0.004272640914574546{}I& 0.005140439896249608+0.004853734699976993{}I& 0.004893052577031012+0.00545292250998158{}I& 0.004607055731170938+0.006073135310648669{}I& 0.004277719195538379+0.006717666840270417{}I& 0.0038993009931257636+0.007390253615141087{}I& 0.0034648119175163176+0.008095174947573721{}I& 0.0029657088319229863+0.008837377501617644{}I& 0.002391491186813019+0.009622634659130522{}I& 0.0017291645449145827+0.010457750572218876{}I& 0.0009625191227188866+0.01135082694534206{}I& 0.00007114657962730806+0.01231161476604118{}I& -0.0009709198499760225+0.013351987698461087{}I& -0.0021971170521666713+0.014486588000811377{}I& -0.003651089856861188+0.015733725210829764{}I& -0.005390693235826993+0.017116654058281736{}I& -0.007493983240128059+0.018665435725976523{}I& -0.010068426817571098+0.020419723054675668{}I& -0.01326550649498307+0.02243306667325704{}I& -0.017304730096762463+0.024779825343899664{}I& -0.022514831311322574+0.027566764942940525{}I& -0.02940821171744621+0.03095361525711325{}I& -0.03882429156840338+0.03519204662791382{}I& -0.052228815812122374+0.04070621953254256{}I& -0.07240929585556202+0.048279430331170076{}I& -0.10535201213814538+0.05956213161266663{}I& -0.16663098558056866+0.07884710782613985{}I& -0.3159645805618081+0.12387884553360849{}I& -0.8339749012551637+0.6640820060944201{}I& 1.1182760567766632+1.6032057898841616{}I& 0.8711587723414448-0.07448652662727094{}I& 0.5052107976574162-0.039066368749409955{}I\end{array}\right]$ (6)

The following gives a warning: FFT not applicable, using DFT instead.

 > $\mathrm{tdata2}≔\mathrm{SignalProcessing}:-\mathrm{FFT}\left(\mathrm{data}\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}0.4497659902145549+0.0{}I& 0.5052107976574163+0.03906636874940997{}I& 0.8711587723414446+0.07448652662727094{}I& 1.1182760567766632-1.6032057898841618{}I& -0.8339749012551638-0.6640820060944201{}I& -0.3159645805618081-0.12387884553360848{}I& -0.16663098558056866-0.07884710782613985{}I& -0.10535201213814539-0.059562131612666616{}I& -0.07240929585556202-0.048279430331170076{}I& -0.05222881581212239-0.040706219532542554{}I& -0.03882429156840338-0.035192046627913816{}I& -0.029408211717446228-0.030953615257113244{}I& -0.022514831311322547-0.027566764942940535{}I& -0.01730473009676248-0.024779825343899695{}I& -0.013265506494983152-0.022433066673257066{}I& -0.010068426817571105-0.020419723054675615{}I& -0.007493983240128059-0.018665435725976523{}I& -0.005390693235827055-0.01711665405828176{}I& -0.0036510898568611646-0.015733725210829757{}I& -0.002197117052166677-0.014486588000811401{}I& -0.0009709198499760314-0.013351987698461087{}I& 0.00007114657962727471-0.012311614766041167{}I& 0.0009625191227188982-0.011350826945342068{}I& 0.0017291645449145677-0.010457750572218883{}I& 0.0023914911868130355-0.009622634659130522{}I& 0.0029657088319229798-0.00883737750161765{}I& 0.003464811917516336-0.008095174947573733{}I& 0.0038993009931257852-0.007390253615141137{}I& 0.004277719195538448-0.006717666840270438{}I& 0.004607055731170729-0.0060731353106488785{}I& 0.004893052577030947-0.005452922509981582{}I& 0.005140439896249555-0.004853734699977014{}I& 0.005353117624996999-0.004272640914574546{}I& 0.005534296909190233-0.003707006932207723{}I& 0.005686610182988066-0.0031544417156110626{}I& 0.005812197276695991-0.0026127514510301993{}I& 0.0059127721776296585-0.002079901359869103{}I& 0.0059896751052553494-0.0015539818621112334{}I& 0.006043910803150312-0.0010331800204289058{}I& 0.006076178185685466-0.0005157520706090525{}I& 0.006086888632727769-7.757919228897728{}{10}^{-18}{}I& 0.00607617818568548+0.0005157520706090301{}I& 0.006043910803150335+0.0010331800204288989{}I& 0.0059896751052553685+0.0015539818621112117{}I& 0.005912772177629706+0.0020799013598689917{}I& 0.0058121972766957825+0.002612751451030189{}I& 0.0056866101829880335+0.0031544417156110895{}I& 0.005534296909190161+0.0037070069322077104{}I& 0.005353117624996999+0.004272640914574546{}I& 0.005140439896249608+0.004853734699976993{}I& 0.004893052577031012+0.00545292250998158{}I& 0.004607055731170938+0.006073135310648669{}I& 0.004277719195538379+0.006717666840270417{}I& 0.0038993009931257636+0.007390253615141087{}I& 0.0034648119175163176+0.008095174947573721{}I& 0.0029657088319229863+0.008837377501617644{}I& 0.002391491186813019+0.009622634659130522{}I& 0.0017291645449145827+0.010457750572218876{}I& 0.0009625191227188866+0.01135082694534206{}I& 0.00007114657962730806+0.01231161476604118{}I& -0.0009709198499760225+0.013351987698461087{}I& -0.0021971170521666713+0.014486588000811377{}I& -0.003651089856861188+0.015733725210829764{}I& -0.005390693235826993+0.017116654058281736{}I& -0.007493983240128059+0.018665435725976523{}I& -0.010068426817571098+0.020419723054675668{}I& -0.01326550649498307+0.02243306667325704{}I& -0.017304730096762463+0.024779825343899664{}I& -0.022514831311322574+0.027566764942940525{}I& -0.02940821171744621+0.03095361525711325{}I& -0.03882429156840338+0.03519204662791382{}I& -0.052228815812122374+0.04070621953254256{}I& -0.07240929585556202+0.048279430331170076{}I& -0.10535201213814538+0.05956213161266663{}I& -0.16663098558056866+0.07884710782613985{}I& -0.3159645805618081+0.12387884553360849{}I& -0.8339749012551637+0.6640820060944201{}I& 1.1182760567766632+1.6032057898841616{}I& 0.8711587723414448-0.07448652662727094{}I& 0.5052107976574162-0.039066368749409955{}I\end{array}\right]$ (7)

tdata and tdata2 are the same, up to tiny float inaccuracies:

 > $\mathrm{verify}\left(\mathrm{tdata},\mathrm{tdata2},'\mathrm{Array}'\left('\mathrm{float}'\left(10,\mathrm{test}=2\right)\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{tdata3}≔\mathrm{DiscreteTransforms}:-\mathrm{FourierTransform}\left(\mathrm{data}\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}0.4497659902145549+0.0{}I& 0.5052107976574163+0.039066368749409955{}I& 0.8711587723414446+0.07448652662727094{}I& 1.1182760567766632-1.603205789884162{}I& -0.8339749012551638-0.6640820060944201{}I& -0.3159645805618081-0.12387884553360849{}I& -0.16663098558056866-0.07884710782613988{}I& -0.1053520121381454-0.05956213161266657{}I& -0.07240929585556202-0.04827943033117009{}I& -0.05222881581212236-0.040706219532542574{}I& -0.038824291568403356-0.03519204662791384{}I& -0.029408211717446116-0.03095361525711323{}I& -0.022514831311322495-0.027566764942940535{}I& -0.01730473009676253-0.024779825343899796{}I& -0.013265506494983152-0.022433066673257066{}I& -0.010068426817571093-0.02041972305467564{}I& -0.007493983240128061-0.018665435725976523{}I& -0.005390693235827055-0.01711665405828174{}I& -0.0036510898568611677-0.015733725210829785{}I& -0.002197117052166725-0.014486588000811432{}I& -0.0009709198499759817-0.013351987698461092{}I& 0.00007114657962733987-0.012311614766041179{}I& 0.0009625191227189075-0.011350826945342072{}I& 0.0017291645449145458-0.010457750572218868{}I& 0.0023914911868130415-0.009622634659130522{}I& 0.0029657088319229924-0.008837377501617671{}I& 0.0034648119175163237-0.008095174947573733{}I& 0.00389930099312586-0.0073902536151410625{}I& 0.004277719195538548-0.0067176668402704875{}I& 0.004607055731170927-0.0060731353106488785{}I& 0.004893052577030996-0.005452922509981607{}I& 0.005140439896249604-0.004853734699977054{}I& 0.005353117624997005-0.004272640914574546{}I& 0.005534296909190279-0.0037070069322076974{}I& 0.005686610182988094-0.0031544417156110804{}I& 0.005812197276695914-0.0026127514510301112{}I& 0.005912772177629639-0.002079901359869095{}I& 0.005989675105255347-0.0015539818621112295{}I& 0.0060439108031502976-0.001033180020428902{}I& 0.006076178185685449-0.0005157520706089898{}I& 0.0060868886327277725-7.757919228897728{}{10}^{-18}{}I& 0.006076178185685475+0.0005157520706090549{}I& 0.00604391080315032+0.0010331800204289082{}I& 0.005989675105255343+0.00155398186211128{}I& 0.005912772177629716+0.002079901359869018{}I& 0.005812197276695808+0.0026127514510301112{}I& 0.005686610182988059+0.00315444171561106{}I& 0.005534296909190154+0.0037070069322076944{}I& 0.005353117624997005+0.004272640914574546{}I& 0.005140439896249597+0.004853734699976988{}I& 0.004893052577031059+0.0054529225099815915{}I& 0.0046070557311709195+0.006073135310648729{}I& 0.004277719195538376+0.006717666840270428{}I& 0.0038993009931257133+0.007390253615141106{}I& 0.0034648119175163003+0.008095174947573737{}I& 0.0029657088319229767+0.00883737750161764{}I& 0.0023914911868130207+0.009622634659130528{}I& 0.0017291645449145692+0.01045775057221889{}I& 0.0009625191227188874+0.011350826945342079{}I& 0.00007114657962720333+0.012311614766041179{}I& -0.0009709198499760143+0.013351987698461097{}I& -0.0021971170521666618+0.01448658800081144{}I& -0.003651089856861185+0.015733725210829736{}I& -0.005390693235826972+0.017116654058281704{}I& -0.007493983240128061+0.018665435725976523{}I& -0.010068426817571101+0.020419723054675647{}I& -0.01326550649498307+0.022433066673257063{}I& -0.01730473009676242+0.02477982534389959{}I& -0.022514831311322602+0.027566764942940518{}I& -0.029408211717446266+0.03095361525711326{}I& -0.03882429156840338+0.03519204662791383{}I& -0.052228815812122444+0.040706219532542554{}I& -0.07240929585556204+0.048279430331170076{}I& -0.1053520121381454+0.05956213161266663{}I& -0.16663098558056866+0.07884710782613985{}I& -0.3159645805618082+0.12387884553360848{}I& -0.8339749012551638+0.6640820060944201{}I& 1.1182760567766632+1.6032057898841618{}I& 0.8711587723414446-0.07448652662727087{}I& 0.5052107976574163-0.03906636874940996{}I\end{array}\right]$ (9)

Moreover, tdata and tdata3 are the same, up to tiny float inaccuracies:

 > $\mathrm{verify}\left(\mathrm{tdata},\mathrm{tdata3},'\mathrm{Array}'\left('\mathrm{float}'\left(10,\mathrm{test}=2\right)\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{SignalProcessing}:-\mathrm{SignalPlot}\left({⟨{\mathrm{~}}_{\mathrm{ℜ}}\left(\mathrm{tdata}\right),{\mathrm{~}}_{\mathrm{ℑ}}\left(\mathrm{tdata}\right)⟩}^{\mathrm{%T}},'\mathrm{compactplot}'\right)$  > $\mathrm{original}≔\mathrm{SignalProcessing}:-\mathrm{InverseFFT}\left(\mathrm{tdata}\right)$
 $\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}0.24724938010000005-5.788806807038671{}{10}^{-17}{}I& 0.4782284717-4.6306843959555885{}{10}^{-18}{}I& 0.6778153055-8.813190380294797{}{10}^{-17}{}I& 0.8330982086000001-6.514854332728859{}{10}^{-17}{}I& 0.9342719765999999-3.849326999814238{}{10}^{-17}{}I& 0.9753019572-4.372165814963856{}{10}^{-17}{}I& 0.9543081381999999-1.0713724607539688{}{10}^{-16}{}I& 0.8736433648+1.1462631311827254{}{10}^{-17}{}I& 0.7396636849999999+7.27845345235421{}{10}^{-18}{}I& 0.5622125499-4.186869669466468{}{10}^{-17}{}I& 0.35386226639999996+8.639718032697117{}{10}^{-17}{}I& 0.12897397629999988-3.1013698720138586{}{10}^{-17}{}I& -0.09734987012999996+3.5982754599275804{}{10}^{-17}{}I& -0.3103399928999999+1.0523039104519782{}{10}^{-16}{}I& -0.4965810792999999-3.1136855829179936{}{10}^{-17}{}I& -0.6449045459000001+4.2494308227418165{}{10}^{-17}{}I& -0.7470908727+3.35757149293384{}{10}^{-17}{}I& -0.7983356335+1.3239115740348083{}{10}^{-18}{}I& -0.7974529354999998+5.249906607569585{}{10}^{-17}{}I& -0.7468109761+1.8871850934219758{}{10}^{-17}{}I& -0.6520150600999999-4.1077204126344286{}{10}^{-17}{}I& -0.5213720677999999+5.063547722021758{}{10}^{-17}{}I& -0.36518557969999993-2.6385105329169736{}{10}^{-17}{}I& -0.19494157819999997-5.034994399150584{}{10}^{-17}{}I& -0.02245018091000007+4.520793273779095{}{10}^{-17}{}I& 0.14099098560000015-3.2724974232940717{}{10}^{-17}{}I& 0.2853511479+3.907489070614328{}{10}^{-17}{}I& 0.402487331+8.469533485416189{}{10}^{-18}{}I& 0.48659369319999984+9.698082920104402{}{10}^{-18}{}I& 0.5344562762999999-8.932425093008308{}{10}^{-17}{}I& 0.5455063352-2.1762828718436853{}{10}^{-19}{}I& 0.5216811080999999+2.6482383785783187{}{10}^{-17}{}I& 0.4671149320999999+7.212827541007565{}{10}^{-17}{}I& 0.3876949535+8.14606194735691{}{10}^{-17}{}I& 0.2905236192-3.3022244070640787{}{10}^{-17}{}I& 0.18333423239999988+1.3039786540038024{}{10}^{-16}{}I& 0.07390606950999985-2.5043265028372553{}{10}^{-17}{}I& -0.030477875470000006+1.1462895400597365{}{10}^{-16}{}I& -0.12349397209999992+3.961109875421382{}{10}^{-17}{}I& -0.2001341822999999+5.8889948737155596{}{10}^{-18}{}I& -0.2569380785999999+2.3740175388370566{}{10}^{-17}{}I& -0.29209413199999995+2.457843447149258{}{10}^{-17}{}I& -0.3054141955999998-1.8190162000157572{}{10}^{-17}{}I& -0.2981943589999999-5.7721481393800585{}{10}^{-18}{}I& -0.27298278049999997+2.5840450106396485{}{10}^{-18}{}I& -0.2332802507999998+3.2618811210719874{}{10}^{-18}{}I& -0.1832019433+4.9607848690802386{}{10}^{-17}{}I& -0.12712901879999977-1.2903000941256319{}{10}^{-16}{}I& -0.06937673750000004-1.886823916508005{}{10}^{-17}{}I& -0.013901824909999944-4.5812523542074915{}{10}^{-17}{}I& 0.03593343608000013-4.031822888859644{}{10}^{-17}{}I& 0.0775290216300001-1.0226046617704087{}{10}^{-16}{}I& 0.10914187909999994-9.030053709139358{}{10}^{-17}{}I& 0.12990859750000008-1.0343740793553614{}{10}^{-16}{}I& 0.13980243880000012-5.483011258062978{}{10}^{-17}{}I& 0.13953538790000003+1.7036267749938947{}{10}^{-17}{}I& 0.1304188345999999-8.184830882269641{}{10}^{-17}{}I& 0.11419805689999987-4.6252474029640126{}{10}^{-17}{}I& 0.09287587667999987+5.420140833553328{}{10}^{-18}{}I& 0.06853983411999992-1.1345784577491752{}{10}^{-17}{}I& 0.04320521663999989+2.4471999464506593{}{10}^{-17}{}I& 0.01868354008999981-4.7606716955327633{}{10}^{-17}{}I& -0.003517056003000017-4.908741794575067{}{10}^{-18}{}I& -0.022256291169999994+2.37790804659256{}{10}^{-17}{}I& -0.03679098197999992-8.071254299971714{}{10}^{-18}{}I& -0.04677160933000001+6.568608741830469{}{10}^{-18}{}I& -0.05221092953000003+4.06939146588404{}{10}^{-17}{}I& -0.053430831409999796+6.477669952037744{}{10}^{-17}{}I& -0.05099444294999982+6.45112800847227{}{10}^{-17}{}I& -0.04563063549000001+1.2015062864008614{}{10}^{-16}{}I& -0.038157639320000006+3.1982773214475455{}{10}^{-17}{}I& -0.029411585760000056+7.327520661516035{}{10}^{-17}{}I& -0.02018456741999993-2.004933932523007{}{10}^{-17}{}I& -0.011175401490000025+1.1050934674793172{}{10}^{-17}{}I& -0.0029548328049999457+2.448292107023198{}{10}^{-17}{}I& 0.00405445532599999-5.334304181791691{}{10}^{-17}{}I& 0.009583845211000034+4.511353717067075{}{10}^{-17}{}I& 0.01351257058999993-6.604288736668821{}{10}^{-17}{}I& 0.015851709609999994+1.7161159567782773{}{10}^{-17}{}I& 0.016721175539999883+6.255571657698161{}{10}^{-17}{}I\end{array}\right]$ (11)
 > $\mathrm{verify}\left(\mathrm{data},\mathrm{original},'\mathrm{Array}'\left('\mathrm{float}'\left(10,\mathrm{test}=2\right)\right)\right)$
 ${\mathrm{true}}$ (12)
 >