AaЪdGT▄yzzz{{}└HH $ d daFootnote TableFootnote*═Ю*═Ю. . / - п я\2 \ІHeading│АдT│Р┌Ь┐З└B┘E├┤┬┴ ┼▀ ▄█▌▐░▒▓⌠■z∙#√%≈'≤)≥+ /⌡M°C²G·I÷K═M║O╒QёSєU╔WіYїi╗]╘^╙`╚b╛eґgўk╞m╟o╠q╡sЁuЄw╣~І}Ї─╦{╧┌╨ *╩╪└Ґ├Ў┬©▄ю▌а█бNц║д )ефЕгOи7й9к;л=м?нAоCe 3пE яIAaрKdсMTтOуQyz ) 9 + )д{ *╨om█─m<$lastpagenum>n<$monthname> <$daynum>, <$year>o"<$monthnum>/<$daynum>/<$shortyear>p;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm>q"<$monthnum>/<$daynum>/<$shortyear>r<$monthname> <$daynum>, <$year>Іs"<$monthnum>/<$daynum>/<$shortyear>t <$fullfilename>u <$filename>v<$paratext[Title]>w<$paratext[Heading]>xyPagepage <$pagenum>zHeading & Page<$paratext> on page <$pagenum>{ <$curpagenum>⌡M| <$marker1>} <$marker2>M~ (Continued)+ (Sheet <$tblsheetnum> of <$tblsheetcount>)^5Уq┼5ъъgA6ЮЮAo7ААs8ББ~9ЦЦ─:ДД *;..└<У┴A▄=00█>11 )?22Е@339A44=B55CC66IAaD77TOE88F99╨G::█H;;anuI<<ohnJ== yeK>>ohnL??$orM@@;moN▐▐u, O░░:miP▒▒Q▓▓/daR⌠⌠aS■■ne>T∙∙e>ІU√√n>/V≈≈reaW≤≤ufiX≥≥fiY paZ⌡⌡[°°Hdi\²²Pa]··n>^══ ge_╒╒nag`ёё aєєM b╔╔cііMdїї)e╗╗bhef╘╘setg╙╙Уh╚╚ъgi╛╛ЮjґґАskўўБl╞╞Ц─m╟╟Дn╠╠.└o╡╡УpЁЁ0█·ДД1÷ФФ2Е═ГГ3║ХХ4=╒ИИ5ёЙЙ6IєКК7T╔ЛЛ8іММ9їНН:█╗ОО;a╘ПП<o╙ЯЯ= ╚РР>o╛СС?$ґТТ@;ўУУ▐u╞ЖЖ░:╟ВВ▒╠ЬЬ▓/╡ЫЫ⌠aЁЗЗ■nЄШШ∙e╣ЭЭ√nІЩЩ≈rЇЧЧ≤u╦ЪЪ≥╧ ╨⌡╩°H╪²Ґ·nЎ═ ©╒nюёаєMб ╔ц іMдї)е╗bф ╘sг╙х╚ъи╛ЮйґАкўБл╞Цм╟Дн╠.о╡УпЁ0яД1рФ2сГ3тХ4уИ5ж6вК7ьЛ8ы М9з!!Н:ш""О;э##П<щ$$Я=ч%%Р>ъ&&С?Ю''Т@А((У▐Б))Ж░Ц**В▒Д++Ь▓Е,,Ы⌠Ф--З■Г..Ш∙Х//Э√И00Щ≈Й11Ч≤К22ЪяЯ я╡╧р╡с╚⌡т╛у╚Hж╛╪вҐь╗·1 ы╣═з╔шўnэхющўчўєъўЮўА╨ц Table 1: БўїЦўДўbЕўф ФўГ╗╙2 Хў╚ИўЙЁЮКўйЛўМ╠ўНўОўЦПўмЯўРў╠С╠ТўУУ╠пЖўВўДЬўЫў2ЗўсШўЭўХЩўЧ╠5ЪўжюўК╠х8╧ы І Table 2: :гў; цэ#2.3 І$Я╗3 >І І?яЮ'@юю▐иБ)░╗6 ▒Ў╠▓╠Е,Ў╠-ЗЎЎ∙╠Х/╣ў0ЩўІ≤хК2 ўЯ!ц╧2.1 "Ё⌡#х$ўH%ў╪&╠'Ё·(ў)Ё*х+ў,ю-ў.ў/ў0х1Ё2ў: 3ў4Ё5Ё6о7ў8ў9їХў:ўИў;ўЙЁ<ўКў=ўЛў>╦М╠?ўНў@хОўAўПўB╘ЯўC╧РўDЎС╠E╠ТўFўУ╠GўЖўH╞ВўIюЬўJмЫўKЄЗўLмШўMюЭўNўЩўOўЧ╠PўQўюRўўSм╠TххU╘╧VўІWІ Ta Table 2: XЁўYЁ цZк[к І\к╗]к3 ^к_к?`кЮ'aкюbкюcкиdк░eкf╘Ўgк╠hЎ╠iхЎj╪╠kаЎlўЎmц╠2.2 nЎўo╧ўpхІqххrЎ ўsс!ц4 t╠u╠vІ Table 3: ўwІxІyІzЁ{Ё|Ё}Ё~с5 д─д│д┌Ё┐д: └д┘д├Ё┤Ё┬Ё┴Ё┼дХў▀дИў▄дЙЁ█ЁКў▌ЁЛў▐ЁМ╠░ЁНў▒ІОў▓ЁПў⌠ЁЯў■дРў∙дС╠√дТў≈ЁУ╠≤ЁЖў≥ЁВў╙©Ьў╚кЫў╛кЗўґ╘Шўў╘Эў╞кЩў╟кЧ╠╠к╡╘юЁ╘ўЄк╠╣кхІк╧Ї╘І╦╘ Ta╧кle ╨кў╩к ц╪╘Ґ╘ ІЎк╗©к3 юка╘?б╘Ю'цкюдкюениф╘░г╘хкЎик╠йк╠к╘Ўл╘╠мкЎнкЎок╠п╘.2 я╘ўркўскІткху╘ ўж╘!цвкькыкз╘ш╘: ўэкщкчкъ╘Ю╘АкБкЦкД╘дЕ╘─дФк│дГк┌ЁХк┐дЙЁ└дКЁ┘дщ]ЄAdщAЁДъyдdчAЁБЮzЁHHт┬ъBЦщЁHHт┬5╠`я ЁўHHт┬ЮBАчўHHт┬6к`р HМUXт АBБЮчHМUXт 7e к╙╙╙╙UUhс *Draft - ╨7/31/92╩ д2еH$т БBАч'H$т 8╘╙╙╙╙UU`т ЎHМUXт ЦBДъщHМUXт 9╠╘╙╙╙╙UU`у кH$т ДBЦщ╘H$т :╙╙╙╙UU`ж ╘ўdЕAФ{H;≥3K╨^╗ФBГЕH ЫшH ЫшдFootnoteдHE;≥?Э^╗ГBФХЕдHMЫшHMЫшASingle LineH'%ІЫшХBГЙЕИИFootnote ░ИFХЁ ≥HHlAЎDёf^╗ЙDХКЕHuHuDouble LineHHzЫшт%КBЙОЕЛМDouble LineтЛEМКтH1тМEЛКe 1т1ft h▌■+'НGР╣GJ╔╛equal[cong[over[times[char[r],char[m],char[s],id[times[char[e],char[r],char[r],char[o],char[r]]]],times[char[r],char[m],char[s],id[times[char[r],char[e],char[s],char[u],char[l],char[t]]]]],over[times[indexes[1,0,num[2,"2"],fract[id[plus[char[M],num[3,"3"]]],num[2,"2"]]],indexes[1,0,num[2,"2"],minus[char[B]]],id[num[0.3,"0.3"]]],times[char[r],char[m],char[s],id[times[char[I],char[n],char[i],char[t],char[i],char[a],char[l],char[A],char[r],char[r],char[a],char[y]]]]]],over[times[indexes[1,0,num[2,"2"],fract[id[plus[char[M],num[3,"3"]]],num[2,"2"]]],indexes[1,0,num[2,"2"],minus[char[B]]],id[num[0.3,"0.3"]]],fract[indexes[1,0,id[char[K]],fract[num[1,"1"],num[2,"2"]]],num[2,"2"]]]]^╗HT%т ОBКЕППSingle LineтуПEОМуdЯAСС│H²ZЮj▌■-'РHТb ННУ│HHт┬СBЯGHHт┬/║,h<ngtihT],n = *`VhaNumber of years = 552 D N[m_This represents the number of measurements of 1 minute repetitions for the systematic error to [2,R@N[cequal the shot noise error. ]]l С,"bSo the systematic component of the error becomes equal to the random error after 552 years, so it z@С,c@is negligible for the MDI with a maximum lifetime of six years. im°` [2 Other Bounds pІ У"3aAs a check, a few other bounds for the FFT truncation error in the literature are calculated and hдУ,"]found to be smaller, because of the exponent 2b which is double the exponent of that used by eр@УWelch [1]. ОЛ`ВFrom Weinstein [2] `Ь A hЫTotal FFT error ├ :`З T`ШНWhere (b+1) = 15 bits ┬n`ЭG0a = 2 (for complex multiplication) ti┬`Щ],N = 1024 = 210 ╒`Ъf 3Total FFT error @ 0.000163 (dimensionless) nu╪`tsFrom Tran-Thong [3] nsБhc &Total FFT error ┤ (dimensionless) oraIn [3] it is assumed that each multiplication performed in the FFT algorithm introduces an error rsodwhich is statistically independent of all other errors. Multiplication produces an error which can Hun/be modeled as a noise with a variance of ц. bo: uru_These upper bounds are all smaller compared to that calculated from Welchуs equation Number 1, cauH@u 2:and this leads to smaller possible systematic components. b`Ч].LWe still use Welchуs equation for a conservative estimate of the FFT error. |`El Q)/кЄ.■ЬHwfT(ҐЖЖh┌io` 7L'лГ╒LЫGBonF⌠ 5?(bSequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)3,"3"]],num[(*s9.00s*)0.3,"0.3"]]he 7ЪЪРJа'█'НTЗH7ifO-╧ВВУ┐whdШAТЩЩ|b.` 7L'лГ╒LЭGEonF⌠ 5?tySequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)4,"4"]],num[(*s9.00s*)0.8,"0.8"]]er HHт┬ЩBШthHHт┬└██У<di Щ`щFTOInitial array X0 contains the input data on which the FFT is performed "hшchwhereB = 15 bitsа <`Л i(N = number of input data points = 1024 VhК9WAK= average modulus squared of the initial (signal) array = ╪ rp`ч[1EM = number of stages in FFT = log2N = log21024 = 10 u┼`ъ1,Nwe find the rms error in a 15 bit fixed point FFT is 0.00083 (dimensionless). є ЮT(`This equation gives an approximate upper bound for the ratio of the rms error to the rms of the ar╡Ю(*cresult due to truncation. This bound increases as the ф or 1/2 bit per stage. In deriving the whюЮAdabove equation it is assumed that all errors are independent and, hence, the variance of the sum is deнЮ00hthe sum of the variances.It also assumes error-free multiplications by ╠1 and ╠ j, which are not э@Юaccounted for in the error. щЪUT UT╙╙`Гy Digitization Level and Scale aЪЧ`МT OTo Calculate the Scaling of the input data appropriate to carrying out the FFT (N3ЪЧ`Ц d2Input range of the Sunуs velocity = ╠ 4000 m/sec. MЪЧ`Дof&RMS velocity of the Sun = 2000 m/sec. gЪЧ`ЕM-Shot noise in the input data = S = 20 m/sec. g│ЪЧ` u ⌡ЪЧ`ФweaDigitization error when digitizing an input velocity signal to the least significant bit (m/sec) hаЪЧhэan = ┴ eГЪЧhheTotal error = ┼ or ЪЧ`!ar%To Calculate the Digitization Level: n#ЪЧ ld cAs D is a degree of freedom which is selected such that it does not add significantly to the total it 1ЪЧ@ll Jerror T, so let the digitization error be 1/10th of the shot noise error. #іЪЪ⌡і?оZ>2ЧG.I'Sї╛ e█cong[prompt[],over[times[char[alpha],indexes[1,0,num[2,"2"],minus[times[num[2,"2"],char[b]]]],indexes[1,0,char[N],num[2,"2"]]],num[12,"12"]]]ieZ>2ЪG@╡ґal╛equal[cong[prompt[],over[times[char[alpha],indexes[1,0,num[2,"2"],minus[times[num[2,"2"],char[b]]]],indexes[1,0,char[N],num[2,"2"]]],num[72,"72"]]],num[2.7e-05,"0.000027"]]ve⌠РюTXTNіыьH/ЯЪЧ2,ЧЧС├D n}xx'mjG?╪<e▄equal[indexes[1,1,char[N],char[j],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],plus[char[j],minus[num[1,"1"]]],num[1,"1"]],indexes[0,1,char[T],char[j]]]],num[2,"2"]],indexes[1,1,char[N],char[j],num[2,"2"]]]],over[num[1,"1"],num[2,"2"]]]]⌠РюСЗ│eZ >2H║Яfr2,ЪЪС┤d dAan~ЪЧ╗G/2 txx)mjHЬfthчh┬e HHт┬B#іHHт┬ Щ<coh#im▀ = 2 m/sec s[1*`$s[ eDh%b]/▄ = 2 ч LSB = ╣ = 6.93 m/sec ]]i^`ХG xh&FSo the number of bits required to represent the solar signal is █ ▓ (m[aTherefore 9+1= 10 bits are required, including the sign bit to represent the input data, and the T═@(error in the signal is then: nфh*G▌ Х`Б h+1,▐ m/sec. "1`-tr 16`.0,aSo, the error due to digitization is negligible in comparison with the shot noise in the signal. ]P`/de^10 bits are required to represent an input data that ranges from - 4000 m/sec to +4000 m/sec. vh0ЪSo, ░ m/sec. ≤`2,The conversion is: 1 bit scales to 8 m/sec. mj╡h3th1 bit ▒8 m/sec. e л`BhAs the data is stored and calculated as 15 bit integers, other scalings are also possible, see Table 1. ecHZЄHОЕ TableFootnote ґIВлG Wє┬aF °equal[char[D],over[times[string["Least'"],string["Significant'"],string["Bit"]],sqrt[num[12,"12"]]],over[times[char[L],char[S],char[B]],sqrt[num[12,"12"]]]]ngEmнtyGut#ІГu─ TZequal[char[T],sqrt[plus[indexes[1,0,char[D],num[2,"2"]],indexes[1,0,char[S],num[2,"2"]]]]]yЫ·╦╞IВл HЕШ√├Щ┴i t⌠r G .5░:aFermequal[char[D],times[over[num[1,"1"],num[10,"10"]],char[S]],cross[over[num[1,"1"],num[10,"10"]],num[20,"20"]]]0├)─ ┼~GmнtyH Ш iЧЫЩ┼fr2╘hэ&GTЄE═ЪFequal[char[D],over[times[char[L],char[S],char[B]],sqrt[num[12,"12"]]]]H?·╨k t⌠r Hth2, ▀e _"'⌠rGis0▒aFlcjequal[over[num[4000,"4000"],num[6.93,"6.93"]],cong[num[577.2,"577.2"],indexes[1,0,num[2,"2"],num[9,"9"]]]]H{╨`4╘hэ&H ~tn√├▄Gф_ВлGeq┌Ц/е═erUequal[char[T],sqrt[plus[indexes[1,0,char[D],num[2,"2"]],indexes[1,0,char[S],num[2,"2"]]]],sqrt[plus[times[over[num[1,"1"],num[100,"100"]],indexes[1,0,char[S],num[2,"2"]]],indexes[1,0,char[S],num[2,"2"]]]],sqrt[times[over[num[101,"101"],num[100,"100"]],indexes[1,0,char[S],num[2,"2"]]]],times[over[sqrt[num[101,"101"]],num[10,"10"]],char[S]]]m╞·╨a"'⌠rH~.2,█l[H5:`ў╦tВлH102,▐[oHЫ:`ф_ВлH0,2,▌ty╛╦tВлGЩW\:е═└equal[char[T],cross[over[sqrt[num[101,"101"]],num[10,"10"]],num[20,"20"]],times[num[2,"2"],sqrt[num[101,"101"]]],num[20.09,"20.09"]]H,√еЮGKaF"'Napprox[over[num[8000,"8000"],indexes[1,0,num[2,"2"],num[10,"10"]]],num[8,"8"]]dAde[9\ґ·╨.√еЮHd ░HHт┬BHHт┬C.>,<rtinh6nu⌡ ]`:[SgThis table gives the scaling factors. As the scale factor decreases, it makes the FFT error smaller. 1,06 Н]]jThe data are of 10-bits precision, and about 1/2 the least significant bit is lost at each stage; so, the DН"]irelative error for 10-bit data is about 0.1%; whereas, if the 10 bit data is represented as 15 bits, the 0R@Н relative error is about 0.003%. t`mComparison of Errors л▌ 7Щ^Since we want to see how the systematic component (if any?) will show up in the observations, °@7nucthe following calculation is performed and we claim that this is negligible, as can be seen below. [nuІ`;ex6Total FFT error = 0.00083 (dimensionless for 15 bits) пh8/Since 1 bit ▓8 m/sec in the scaling we use, Йh<d 6 ⌠total FFT error = 0.00083 x 8 = 0.00663 m/sec. `=^Random error from the table (Tim Brownуs table at the FFT stage - see Table 2) = 0.351 m/sec. p─К┘Gfa 8@ў, equiv[prompt[],prompt[]]alp─К┘Gda 8@ўitequiv[prompt[],prompt[]] t└РЬRp─ К┘HЦв▒Z─ 6Gor -@ =s therefore[prompt[]]if }ЬRp─ К┘HC15Цв▓VAx Gz0.@╚ aF▀equal[over[num[552,"552"],indexes[1,0,id[num[900,"900"]],fract[num[1,"1"],num[2,"2"]]]],over[num[552,"552"],num[30,"30"]],num[18.4,"18.4"]]ervQ(УцZ─ 6HfoЧЫ⌠ ph▄'2hп G├ n5Fягca⌠cong[indexes[1,0,matrix[1,1,over[num[0.351,"0.351"],over[num[0.000163,"0.000163"],cross[num[10,"10"],num[4,"4"]]]]],num[2,"2"]],num[14312,"14312"]]scaЪ0u\═T!G}d ;ґ─.u F╛equal[char[n],cong[indexes[1,0,matrix[1,1,over[num[0.351,"0.351"],num[0.0002075,"0.0002075"]]],num[2,"2"]],times[num[5.52,"5.52"],char[y],char[e],char[a],char[r],char[s]]]]G;Ж'#÷"G%eqШгroUindexes[1,0,matrix[1,1,over[num[0.351,"0.351"],num[0.000663,"0.000663"]]],num[2,"2"]]tHУГoIvЄ#╣<#H─┌*°╚┐┐,∙┐╟wл$G'orBьE═s }equal[over[char[n],cross[num[360,"360"],num[1440,"1440"]]],times[num[0.541,"0.541"],char[y],char[e],char[a],char[r],char[s]]]F^&8Ь=Ж'%÷%H┌'*],ь"",√,"∙Y┘42┴&G)]]K╛б6є,"╦cong[indexes[1,0,id[over[num[0.351,"0.351"],over[cross[num[0.00083,"0.00083"],num[4,"4"]],num[10,"10"]]]],num[2,"2"]],times[num[2.156,"2.156"],char[y],char[e],char[a],char[r],char[s]]][n²[╨`┘╟wл'H%)*3,2,$$,≈10p─К┘(G+nu 8@ў"]equiv[prompt[],prompt[]]Гс=⌡и\≈Y┘62┴)H'+*l[ШЕ&&,≤,0d*A],,,─75└ЬRp─ К┘+H)*.5Цв((,≥arHHт┬,B*Ж'HHт┬}+{{С<in1, >nubSince truncation does not lead to systematic error in this analysis, and we do not know about the >dsource content in the FFT error, let the systematic component of the error be arbitrarily 1/10th of 40$H>41the total FFT error.╦ h>`?[sUSystematic error = 1/10 x total FFT error = 1/10 x 0.00663" = 0.000663 m/sec. X A6єoThe contribution of the random error is reduced by the (number of measurements)"1/2 = n"1/2, the ]],fAnuWsystematic component of the error remains constant. So we can solve for the number of tA,\measurements n, for which the effect of the systematic error (SE) and random error (RE) are и\┌@Aequal. °`Ж,0#Set RE = SE after n measurements. бh@75∙ ╧ К┘ТhC.5n = √ FZNumber of years (for handling 10 bit input data stored as 15 bits) for one minute cadence $HFtiobservations = ≈ atJ`KnaENumber of years (for handling 10 bit input data stored as 11 bits) = tph9 l ≤ ▓`Ot t╛`GareIf we take l bit scales to 1/4 m/sec, corresponding to handling 10 bit input data stored as 15 bits, =фhPerH1 bit ≥ 1/4 m/sec for handling 10 bit input data stored as 15 bits. coЮ`Qra2Total FFT error = 0.00083 x 1/4 =0.0002075 m/sec. З`R"5Systematic error = 1/10 FFT error =0.00002075 m/sec. eM╘│÷-G/ S'тюlК f\indexes[1,0,id[(*i1i*)over[(*n*)num[0.351,"0.351"],num[2.075e-05,"0.00002075"]]],num[2,"2"]]icH`т.H1arH`т;⌡Ж`А ar╠m^=⌠O╘│÷/HЯ752,--С Э`т0HЭ`т=⌡(g ifWs b╠)░rЄ(1H.2░rЄ(>⌡K X┌#Numbers of bits of input data in a red@X16 bit sequence ╡ Dr░(2H13Dr░(?⌡/srr`Yin Scale factor dh5it1 bit °? m/sec╡r░ Є3H24ng░ Є@⌡`ZT 10 = Ё0D ░4H35D ░A⌡er.0`[8Ё│░╣Є5H46░╣ЄB⌡0..3`\,"11020Є]D╣░6H57HD╣░C⌡`]8/2 = 4АЄa░пЄ7H68░пЄD⌡H`^12т╣Dп░8H79 iDп░E⌡2`_4/2 = 2╣░КЄ9H8:░КЄF⌡a ed``13bitІqDК░:H9;3DК░G⌡`a2/2 = 1inІc░Є;H:<1 ░ЄH⌡4`b14ЇD░<H;=D░I⌡5`c 1/2 = 1/2Ї░!Є=H<>[░!ЄJ⌡`d15╦D!░>H=,"D!░K⌡`e1/2 в 1/2 = 1/4╦H┌ъ&╟?HХfaH┌ъ&╟Lhг`╧a33┌ъ&╟@Hfa33┌ъ&╟Mhг)f ╧p─К┘AGC4/ 8@ўequiv[prompt[],prompt[]]7ЪЪ:Jа0█'НTBH;^f╧ЫЫЪ└|°H┬ЬRp─ К┘CHЦвAA2°` 7L'лГ╒LDGGF⌠ 5?Sequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)5,"5"]],num[(*s9.00s*)0.5,"0.5"]]Є7ЪЪ^Jа0█'НTEH=`f╧ЭЭ┘` 7L'лГ╒LFGI=F⌠ 5?Sequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)6,"6"]],num[(*s9.00s*)7.9,"7.9"]]░7ЪЪ┌Jа0█'НTGH?bf╧DD ²` 7L'лГ╒LHGKF⌠ 5?Sequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)7,"7"]],num[(*s9.00s*)0.1,"0.1"]]7ЪЪіJа0█'НTIHAefв ╧FF· 7L,LГ╒LJGMF⌠ 5?Uequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)8,"8"]],num[(*s9.00s*)0.25,"0.25"]]7ЪЪйJа0█'НTKHCgf╧HH÷` 7L'лГ╒LLGOp─F⌠ 5?Sequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)9,"9"]],num[(*s9.00s*)0.1,"0.1"]]7ЪЪНJа6█'НTMHEkfp─╧JJ══ 7L/лГ╒LNGQлГ├⌠ 5?Wequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)10,"10"]],num[(*s9.00s*)0.03,"0.03"]]*s97ЪЪJа0█'НTOHImfНT╧LL║ +&Гў√PGS╒L⌠ 5?Eequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)11,"11"]],prompt[]])7ЪЪ6Jа; 'НTQHKof╧NN"╒b` 7L4LГ╒LRGU ├⌠ 5?╒LYequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)12,"12"]],num[(*s9.00s*)0.007,"0.007"]]77ЪЪZJа, 'НTSHMqfЪЪ╧PP'ё` 7L4LГ╒LTGW ├⌠ 5?GYequal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)13,"13"]],num[(*s9.00s*)0.005,"0.005"]],7ЪЪ~JаA 'НTUHOsfJа╧RR,є` ~ыAY ┤VGY7L'їa┬⌠equal[indexes[0,1,char[(*s9.00s*)N],num[(*s7.00s*)2,"2"]],cross[num[(*s9.00s*)3.2,"3.2"],indexes[1,0,num[(*s9.00s*)10,"10"],num[(*s7.00s*)2,"2"]]]]p─7ЪЪ╒JаA 'НTWHQufлГ╧TT1╔лГЗ─rА┴АXGi1,:jоТЛ)Nequal[indexes[1,1,char[N],num[1,"1"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[0,"0"],num[1,"1"]],indexes[0,1,char[T],num[1,"1"]]]],num[2,"2"]],indexes[1,1,char[N],num[1,"1"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[0,"0"]]7ЪЪxNN²YH9]fLГ╧VVЗіЗUV{А┴АZG]1,>ЙойB)Nequal[indexes[1,1,char[N],num[2,"2"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[1,"1"],num[1,"1"]],indexes[0,1,char[T],num[2,"2"]]]],num[2,"2"]],indexes[1,1,char[N],num[2,"2"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[320,"320"]]HЗ▀~!┴А[G^@ оYequal[indexes[1,1,char[N],num[3,"3"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[2,"2"],num[1,"1"]],indexes[0,1,char[T],num[3,"3"]]]],num[2,"2"]],indexes[1,1,char[N],num[3,"3"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[6.41,"6.41"]]З┌║┴А\G`s[BJо─m[equal[indexes[1,1,char[N],num[4,"4"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[3,"3"],num[1,"1"]],indexes[0,1,char[T],num[4,"4"]]]],num[2,"2"]],indexes[1,1,char[N],num[4,"4"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[17.96,"17.96"]]▌fe╣Ў}у·#А]HY;fА цZZШ╗Йо▌fe0──·$^HB=far[[╘1"З▀┌║┴А_GbexBJоs[equal[indexes[1,1,char[N],num[5,"5"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[4,"4"],num[1,"1"]],indexes[0,1,char[T],num[5,"5"]]]],num[2,"2"]],indexes[1,1,char[N],num[5,"5"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[11.87,"11.87"]]i▌feT└∙·$`HE?f[1─\\╙1,З┌║┴АaGeBJо─,nequal[indexes[1,1,char[N],num[6,"6"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[5,"5"],num[1,"1"]],indexes[0,1,char[T],num[6,"6"]]]],num[2,"2"]],indexes[1,1,char[N],num[6,"6"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[14.26,"14.26"]],▌few─└∙·$bHGAfex__ ╚4"З┌║┴АcGkN]BJоtЛm[equal[indexes[1,1,char[N],num[8,"8"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[7,"7"],num[1,"1"]],indexes[0,1,char[T],num[8,"8"]]]],num[2,"2"]],indexes[1,1,char[N],num[8,"8"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[12.98,"12.98"]]iЗ┌║┴АdGg[1BJоtЛdeequal[indexes[1,1,char[N],num[7,"7"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[6,"6"],num[1,"1"]],indexes[0,1,char[T],num[7,"7"]]]],num[2,"2"]],indexes[1,1,char[N],num[7,"7"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[14.26,"14.26"]]▌fe⌡└∙·$eHICf],─aa╛,idfA[1hh┌1,▌feд└∙·АgHKEfex lddґ6"HHт┬hBfN]HHт┬u?2Сx<,n2"h26.┬ l = 500 ; ┌ *h╙ Зг ┴▌feЕ▀tу·#aiHЗ9f lXXЖї1,З┌║┴АjGmexBJоtЛ1,equal[indexes[1,1,char[N],num[9,"9"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[8,"8"],num[1,"1"]],indexes[0,1,char[T],num[9,"9"]]]],num[2,"2"]],indexes[1,1,char[N],num[9,"9"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[12.98,"12.98"]],▌feХ└∙·АkHMIfex lccў[iВ⌡█!┴АlGoarG┬_tЛ,nequal[indexes[1,1,char[N],num[10,"10"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[9,"9"],num[1,"1"]],indexes[0,1,char[T],num[10,"10"]]]],num[2,"2"]],indexes[1,1,char[N],num[10,"10"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[11.68,"11.68"]]gH▌fe└∙·АmHOKf ljj╞BЖк░║┴АnGqIG▐tЛ!equal[indexes[1,1,char[N],num[11,"11"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[10,"10"],num[1,"1"]],indexes[0,1,char[T],num[11,"11"]]]],num[2,"2"]],indexes[1,1,char[N],num[11,"11"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[0.351,"0.351"]]i▌fe0▐ЎАoHQMfde lll#╟"8Жк░║┴АpGs,nIG▐tЛum!equal[indexes[1,1,char[N],num[12,"12"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[11,"11"],num[1,"1"]],indexes[0,1,char[T],num[12,"12"]]]],num[2,"2"]],indexes[1,1,char[N],num[12,"12"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[0.014,"0.014"]],▌feT▓▐АqHSOf1, lnn(╠umЖк░║┴АrGu0,IG▐tЛ,"!equal[indexes[1,1,char[N],num[13,"13"],num[1,"1"]],indexes[1,0,matrix[1,1,plus[indexes[1,0,id[times[indexes[1,1,char[N],num[12,"12"],num[1,"1"]],indexes[0,1,char[T],num[13,"13"]]]],num[2,"2"]],indexes[1,1,char[N],num[13,"13"],num[2,"2"]]]],fract[num[1,"1"],num[2,"2"]]],num[0.015,"0.015"]],▌fex▓▐АsHUQf1, lpp-╡,n╙│ыtGw[1U@u─[2+id[indexes[1,1,char[N],char[j],num[1,"1"]]],"2▌fe°▓▐АuHWf], lrr2ЁfedvAxx┐ЖкиqJї ─╙│ыwHЬ7fG▐єYttЛЄdeHHт┬xBv,"HHт┬zz▐єh┴<,0me UT UT╙╙`[N3Propagation of FFT Error Through the Inner Product ,"1#ЪЧ I,ifIn reality we are doing a 2-D calculation, that is, we first do a 1-D FFT along the latitude and then 1ЪЧI_project it on the Legendre function along the longitude, so in effect we want to study the FFT es[?ЪЧI,"_error propagating through the dot product. The dot product adds noise which is independent of ],nMЪЧ@I[0-the input noise, and is not considered here. 1gЪЧ "1cThe final error after the dot product is proportional to the 1-D FFT error and the proportionality 1,uЪЧ@=factor is (Number of points in the dot product)"1/2. d▐ЪЧ`,ha+Number of points in the dot product = 900. f╘ЪЧ`r.For 1-D FFT, the time calculated = 552 years. цЪЧhM1Error after the dot product, time = ■ years. tВUR UT╙╙`s+Comparison of the Various FFT Error Bounds ЪЭ teTable 3 gives the time in years for handling 10-bit input data stored as 15 bits for the various FFT IЪЭte berror bounds in the literature, taking truncation error into account. From this table we see the -ЪЭHtit\assumption that the systematic error is 1/10th the total FFT error is quite reasonable.б +╙і╙╙UU #r [This table shows the error in an integer FFT for assumptions about Systematic Error in the MЪЧ7╙╔╙╙UU#th]input signal. It gives the number of years for handling 10 bit input data stored as 15 bits C╙є╙╙UU#ioWfor one minute cadence observations. It shows the number of observations required for berO╙ё╙╙UU#do\the Systematic Error to equal random error (mainly shot noise) in the signal at this stage ЪЧ[╙╒╙╙UU@#FoYin the calculation. It compares several estimates of the FFT error from the literature. cЄЄ.yG~╙╙ зZ/ar$times[num[2,"2"],sqrt[num[12,"12"]]]КШ·╦│VAx zHvs гx■haHpт_ЪЬ{A* 1Hpт_ЪЬ,╦(ter╙╙╙╙UU linIf systematic error is considered equal to the total FFT error rather than 1/10th of the FFT error, and 1 bit ╙╘╙╙UU@ rr?scales to 1/4 m/sec, for 10 bit input data stored as 15 bits, ╙╙╙╗╙╙UU` t e*╙ї╙╙UUh n І g6╙і╙╙UU` io aB╙╔╙╙UU`h ro#Time for the 1-D FFT = 5.52 years. ut N╙є╙╙UU`D th uZ╙ё╙╙UUhn ha0Error after the dot product, time = Їyears.╙╙I!9ь|G─ce%░─ Т pequal[over[num[552,"552"],sqrt[num[900,"900"]]],over[num[5.52,"5.52"],num[30,"30"]],over[num[1,"1"],num[6,"6"]]]roZ┬|:w[T}H─*l %═!!{І╙╙п╘hпФЄЄ.~H IЧЫyy╣im5∙[':╪G┌erйґ:Ґpequal[char[n],indexes[1,0,matrix[1,1,over[indexes[0,1,char[sigma],char[v]],times[char[S],char[E]]]],num[2,"2"]]]ФҐ║ЇK!9ь─H}#* 0Д||{ЇH╔P%2╛│G└%R╗ПЦўtimes[over[num[1,"1"],char[N]],sum[indexes[1,0,abs[times[indexes[0,1,char[X],num[0,"0"]],id[char[j]]]],num[2,"2"]],equal[char[j],num[0,"0"]],plus[char[N],minus[num[1,"1"]]]]]≈vЄЯеC7∙[):╪┌H#%*s ЪЪ,╧UUGvЄ!╣<┐G#$╩Z▒UUgequal[over[indexes[0,1,char[sigma],char[v]],sqrt[char[n]]],Rightarrow[times[char[S],char[E]],prompt[]]]UU├@┬J╔P'2╛└HЕШucAи││Щ╪╙╙ ⌠r┘G┬ceaF %over[num[14312,"14312"],num[30,"30"]]0тj.3j▄'4hп├H┬┤m[≈ ┴Ґ1"d┤AH┴┴└%═К╘·Є"⌠r┬H├▄┤H2,┘┘┴ЎЧЫHHт┬┴B┤GHHт┬├▌x<arde UT UT╙╙`~,oConclusion ,1,#ЪЧ ]]bFrom the above analysis we see that the FFT error is negligible in comparison to other sources of 1ЪЧGderror at that stage in the calculation. As the DFT is a linear operation, we can do the detrending ,c?ЪЧ@,ibefore or after the FFT. ucUR UT╙╙`References N],}ЪЭ ]]][1]P.D. Welch, рA fixed-point fast Fourier trnasform error analysis,с Trans. IEEE, Vol. AU-▀ЪЭ@eq17, 1969, pp. 151-157. [si╔ЪЭ [ca[2]C.J. Weinstein, рQuantization effects in digital filters,с MIT Lincoln Lab. Tech. Rep. 468, иЁЪЭ@*ASTIA Doc. DDC AD-706862, Nov. 21, 1969. мЪЭ 12^[3]B. Liu and Tran-Thong, рFixed-point fast Fourier transform error analysis,с Trans. IEEE, шЪЭ@!Vol. ASSP-24, 1976, pp. 563-573. ЄУЪЭ`G[4] L. Bacon, рFixed-Point Fast Fourier Transform,с SOI-TN-009, 1991. HUP UT╙╙`Appendix 1 ar?ЪЗhi╙╙+From Weinstein [2], number of years = Ґ bFrqЪЗhsi5Error after the dot product, time = Ў = 477 years ёЪЗhof ,From Tran-Thong [3], number of years = © heиЪЗhpth8Error after the dot product, time = ю = 17387 years. ЪЧОЪЗ`qbeHThe above is the time for handling 10 bit input data stored as 15 bits.n▄'2hп┼G▄nt8Fягtr■cong[indexes[1,0,matrix[1,1,over[num[0.351,"0.351"],over[num[2.7e-05,"0.000027"],cross[num[10,"10"],num[4,"4"]]]]],num[2,"2"]],num[521605,"521605"]]n &⌠r▀G▌LiaFh.'over[num[521605,"521605"],num[30,"30"]]Doc^н.3p▄'4hп▄H┬▌┤ЪЭ≈ ┼┼┴©B.H╛т#ЪЩ█AШfaH╛т#ЪЩЩа ns, ╙╙╙╙UU r jOne bit is used for overflow checking, and this is an extra overhead for the algorithm. If detrending is ╙╘╙╙UUr 1.ldone on the ground with the ramp present, then there is always overflow, so a division by 2 is performed to ЪЗ╙╗╙╙UU@r Eroremove the overflow, and there is no need for this extra overflow checking bit. Thereby the data is truncated.©К·Є(⌠r▌H▄┤te2,▀▀┴юe H┘ЪЭсЪЪ▐H╞vH┘ЪЭсЪЪNxб╙╙t `vas ╨sъ33┘ЪЭсЪЪ░Hvъ33┘ЪЭсЪЪOxб riovfw1"v╨n'33┘ЪЭсЪЪ▒Hv0,'33┘ЪЭсЪЪPxб 5,5"fx⌠rG╨o33┘ЪЭсЪЪ▓Hv[no33┘ЪЭсЪЪQxб .3hпfyЪЭ╨ zff╠ЪЭO33B⌠H╟╙vAzff╠ЪЭO33BRxбЩ$`z╙╙╙╩U≥≥╠ЪЭH■Hvfl≥≥╠ЪЭHSxб f af├eng╩ Y≥≥╠ЪЭ░∙H╙╚vgrY≥≥╠ЪЭ░Txбalve`┤onSE = total FFT errorЗ╩╗║≥≥╠ЪЭH√Hvlo║≥≥╠ЪЭHUxб rfecf┬tha╩izffСЪЭO33≈H╗╛vHzffСЪЭO33VxбЪЪ─`{Welch (years)╪≥≥СЪЭH≤H╛≥v≥≥СЪЭHWxб`ЪЭ18.4╪Y≥≥СЪЭH≥H≤ vY≥≥СЪЭHXxбH`─ЪЪ5.52╪║≥≥СЪЭH H≥⌡v║≥≥СЪЭHYxб`│1/6╪zffЪЭO33(⌡H ґvzffЪЭO33(ZxбA | Weinstein @|Щ(years)Ґz≥≥ЪЭH(°Hґ²v≥≥ЪЭH([xб─`┐477├ҐnY≥≥ЪЭH(²H°·vY≥≥ЪЭH(\xбxal─`└143onҐE║≥≥ЪЭH(·H²═v║≥≥ЪЭH(]xб─`┘4.77Ґ)Gкц÷G║ёЕf$Lfract[indexes[1,0,num[2,"2"],minus[times[num[2,"2"],char[b]]]],num[12,"12"]]Wezff6ЪЭO33(═H·ўvHzff6ЪЭO33(^xбx }Tran-Thong ЪЭ@}ЪЭ(years)HЎ [≥э+Gкц║HЯ ÷÷÷Сц≥≥6ЪЭH(╒Hўёv≥≥6ЪЭH(_xбЪЭ─`┼x17387ЎY≥≥6ЪЭH(ёH╒єvY≥≥6ЪЭH(`xбЪЭ─`▀x5216Ў║≥≥6ЪЭH(єHёvn ║≥≥6ЪЭH(axбЪЭ─`▄173.86HЎzffкЪЭO33(╔HvzffкЪЭO33(bxб Hf█≥≥Э©≥≥кЪЭH(іH╚їv≥≥кЪЭH(cxбЪЭ ▌Error after @▌xdot product©Y≥≥кЪЭH(їHі╗vY≥≥кЪЭH(dxбt[`▐"21-D FFTim©n║≥≥кЪЭH(╗Hї≈vWe║≥≥кЪЭH(exб ░xError after x@░dot productЪЭ©≈33┘ЪЭсЪЪ╘Hv≈33┘ЪЭсЪЪfxб ÷f▒H╨ёи≥≥╠ЪЭ░╙H⌠∙vи≥≥╠ЪЭ░gxб┼17`▓SE = 1/10 total FFT error╩Эи≥≥кЪЭH(╚H∙іvЪЭи≥≥кЪЭH(hxб≥≥`⌠1-D FFTЪЭ©и≥≥СЪЭH╛H≈≤vи≥≥СЪЭHixбff33`■552ff╪Эи≥≥ЪЭH(ґH⌡°vи≥≥ЪЭH(jxбЪЭ─`∙14312Ґи≥≥6ЪЭH(ўH═╒vи≥≥6ЪЭH(kxб─`√uc521605Y≥≥ЎЭzff≈ЪЭO33╞H▐╟vЪЭzff≈ЪЭO33lxбt[`┌FFmюnи≥≥≈ЪЭ ╟H╞⌠vWeи≥≥≈ЪЭ mxб`┴x1Assumption of magnitude of Systematic Error (SE)Эю≥≥≈ЪЭH╠Hv≥≥≈ЪЭHnxб ÷f≈HюёY≥≥≈ЪЭH╡HvY≥≥≈ЪЭHoxб ┼17f≤Eю1║≥≥≈ЪЭHЁHv║≥≥≈ЪЭHpxб f≥ю48ЦGЕ≥≥──5?H sqrt[char[N]]║ЪЪ┌ъ&╟ДHfx║ЪЪ┌ъ&╟·hг)ff≥≥f°╧S*√Мйа48ЕH└ ШЧЫЦЦЩф─1ЪЪ┌ъ&╟ФHfЪЭ1ЪЪ┌ъ&╟÷hг)f ╧┬fe┌ъ&╟ГHfЭ┬fe┌ъ&╟═hг)ЪЭfx╧[H■33-ХH?ИfnH■33-║hгЪЭ─`"xjйa33■@лл-ИHХЙfmpa33■@лл-╒hгЭ≥≥─`'Nameй║ЪЪ■░-ЙHИКfx║ЪЪ■░-ёhгёЪЭ─`)NTRANS (T!j) ─`ЙNoise Transfer Function≤й1ЪЪ■Vff-КHЙЛf1ЪЪ■Vff-єhгx─`1 Additive ─`К48Noise (N!j)5?й┬fe■·юK-ЛHКМf┬fe■·юK-╔hг─`4≥≥ Output Noise ─hЙйаЄ8йHHа33$МHЛНfHа33$іhгЪЪ&╟*╙╙╙UU`B#0кa33а@лл$НHМОfa33а@лл$їhгfe&╟*╙╙╙UU`H$-к║ЪЪа░$ОHНПf║ЪЪа░$╗hг33─`J-к1ЪЪаVff$ПHОЯf1ЪЪаVff$╘hг33лл─`L-к┬feа·юK$ЯHПРfNa┬feа·юK$╙hгx─`S-кhHЕ33$РHЯСfHЕ33$╚hгЙNo*╙╙╙UU`U#io1лa33Е@лл$СHРТfa33Е@лл$╛hгx*╙╙╙UU`f# Telescopeл║ЪЪЕ░$ТHСУfj║ЪЪЕ░$ґhгfe─hgилh1ЪЪЕVff$УHТЖf≥≥1ЪЪЕVff$ўhг8─h╚Н┐л┬feЕ·юK$ЖHУВfh┬feЕ·юK$╞hг─h╛їл3H 33$ВHЖЬfH 33$╟hгЪЪ*╙╙╙UU`ґ#2мЪa33 @лл$ЬHВЫfa33 @лл$╠hгЪЪff*╙╙╙UU`ў#CCD ExposeVffм║ЪЪ ░$ЫHЬЗfлл║ЪЪ ░$╡hгР─h╞йм1ЪЪ Vff$ЗHЫШf─1ЪЪ Vff$ЁhгH─h╟33ім┬fe ·юK$ШHЗЭf╙╙┬fe ·юK$ЄhгТ─h╠╗мH-33$ЭHШЩfUUH-33$╣hгH*╙╙╙UU`╡#3нa33-@лл$ЩHЭЧf─a33-@лл$ІhгЖ*╙╙╙UU`Ё# CCD Readн║ЪЪ-░$ЧHЩЪf║ЪЪ-░$ЇhгH─hЄюKкн1ЪЪ-Vff$ЪHЧf1ЪЪ-Vff$╦hгH─h╣33└н┬fe-·юK$HЪf╙╙┬fe-·юK$╧hгЫ─hІ╘нHQ33$HfUUHQ33$╨hг@З*╙╙╙UU`Ї#4оa33Q@лл$HfРa33Q@лл$╩hгffH*╙╙╙UU`╦#ЪЪ CCD Sumsо║ЪЪQ░$Hf─║ЪЪQ░$╪hгЭ─h╧ло1ЪЪQVff$Hf─1ЪЪQVff$ҐhгЩ─h╨┘о┬feQ·юK$Hf*╙┬feQ·юK$ЎhгH─h╩лл╙оHu33$Hf╙╙Hu33$©hгЪ*╙╙╙UU`╪#5пa33u@лл$Hfa33u@лл$юhгffH*╙╙╙UU`Ґ#ЪЪV Calc$п║ЪЪu░$H f─║ЪЪu░$аhг─hЎмп1ЪЪuVff$ H f─1ЪЪuVff$бhг─h©²п┬feu·юK$ H f*╙┬feu·юK$цhгH─hюлл╚пH≥33$H f╙╙H≥33$дhг*╙╙╙UU`а#6яa33≥@лл$H fЭa33≥@лл$еhгffH*╙╙╙UU`б#ЪЪFix bad pixя║ЪЪ≥░$ Hf║ЪЪ≥░$фhгH─hцюKня1ЪЪ≥Vff$H f1ЪЪ≥Vff$гhгH─hд33·я┬fe≥·юK$Hf╙╙┬fe≥·юK$хhг─hе╛яHҐ33$HfUUHҐ33$иhгH*╙╙╙UU`ф#7рa33Ґ@лл$Hf─a33Ґ@лл$йhг *╙╙╙UU`г#Sub V%oр║ЪЪҐ░$Hf║ЪЪҐ░$кhг@─hхор1ЪЪҐVff$Hf1ЪЪҐVff$лhг─hи÷р┬feҐ·юK$HfUU┬feҐ·юK$мhгЭ─hйґрHА33$HfбHА33$нhгH*╙╙╙UU`к#8сa33А@лл$Hf─a33А@лл$оhг*╙╙╙UU`л#Remapс║ЪЪА░$Hf─║ЪЪА░$пhг─hмпс1ЪЪАVff$Hf─1ЪЪАVff$яhг─hн═с┬feА·юK$Hf*╙┬feА·юK$рhгH─hоллўсH33$Hf╙╙H33$сhгH*╙╙╙UU`п#ЪЪ9тa33@лл$Hfa33@лл$тhг*╙╙╙UU`я$W# Subtractт║ЪЪ░$Hf║ЪЪ░$уhгH─hрюKят1ЪЪVff$Hf1ЪЪVff$жhгH─hс33║т┬fe·юK$Hf╙╙┬fe·юK$вhг─hт╞тH)33$H fUUH)33$ьhг*╙╙╙UU`у#10уa33)@лл$ H!fa33)@лл$ыhг*╙╙╙UU`ж#Apodizeу║ЪЪ)░$!H "fн║ЪЪ)░$зhг*╙─hвру1ЪЪ)Vff$"H!#fо1ЪЪ)Vff$шhг╙╙─hь╒у┬fe)·юK$#H"$fп┬fe)·юK$эhг@33─hы╟уhHM33$$H#%fHM33$щhгH*╙╙╙UU`з#11жa33M@лл$%H$&fa33M@лл$чhгH*╙╙╙UU`ш#ffFFTж║ЪЪM░$&H%'f║ЪЪM░$ъhгH─hэюKсж1ЪЪMVff$'H&(f1ЪЪMVff$ЮhгH─hщ33ёж┬feM·юK$(H')f╙╙┬feM·юK$АhгH─hчлл╠жHq33$)H(*f╙╙Hq33$Бhг"*╙╙╙UU`ъ#12$вa33q@лл$*H)+f─a33q@лл$Цhг#*╙╙╙UU`Ю# Dot Readв║ЪЪq░$+H*,f║ЪЪq░$ДhгH─hАюKтв1ЪЪqVff$,H+-f1ЪЪqVff$ЕhгH─hБ33єв┬feq·юK$-H,.f╙╙┬feq·юK$ФhгH─hЦлл╡вH∙33$.H-/f╙╙H∙33$ГhгH*╙╙╙UU`Д#13ьa33∙@лл$/H.0fa33∙@лл$ХhгH*╙╙╙UU`Е#ffData Comp.ь║ЪЪ∙░$0H/1fщ║ЪЪ∙░$Иhг╙╙─hФуь1ЪЪ∙Vff$1H02fч1ЪЪ∙Vff$Йhг╙╙─hГ╔ь┬fe∙·юK$2H1fъ┬fe∙·юK$Кhг──hХЁь _Хі²rк?x4G9ЮBе┤хfчequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)2,"2"]],sin[times[char[(*s9.00s*)c],matrix[(*n*)1,1,over[times[(*n*)char[(*s9.00s*)pi],char[(*f"MT"fs9.00s*)l]],num[(*s9.00s*)3000,"3000"]]]]],num[(*s9.00s*)0.955,"0.955"]] Хі²rAfx6G7B@⌡хfequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)1,"1"]],sin[times[indexes[1,0,char[(*s9.00s*)c],num[(*s7.00s*)2,"2"]],matrix[(*n*)1,1,over[times[(*n*)char[(*s9.00s*)pi],char[(*f"MT"fs9.00s*)l]],num[(*s9.00s*)1500,"1500"]]]]],num[(*s9.00s*)0.6899999999999999,"0.69"]]їЪЪЕЇ └│6&a╡7HwЗfUU≥L66ТиУx+Q√Qў√8G;-ю═ 5?cequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)3,"3"]],id[prompt[]],num[(*s9.00sn*)0.02,"0.02"]]їЪЪ Ї ┘▀&a╡9HiYf≥L44Ый─5x+MQў√:G=*ю═ 5?`equal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)4,"4"]],id[prompt[]],num[(*s9.00s*)2.8,"2.8"]]_ХїЪЪ:Jа[│AНT;H]Bfе┤╧88ЧкdeУx+Q√Qў√<G?.0-ю═ 5?n[bequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)5,"5"]],id[prompt[]],num[(*s9.00s*)0.66,"0.66"]]їЪЪ^JаU│AНT=H^Ef[(╧::л]]x+FVQў√>GA&@═ 5?\equal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)6,"6"]],id[prompt[]],num[(*s9.00s*)1,"1"]]s*їЪЪ┌Jа[│AНT?H`Gf*n╧<<мchx+FVQў√@GC0s&@═ 5?.0\equal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)7,"7"]],id[prompt[]],num[(*s9.00s*)1,"1"]]їЪЪіJаL│AНTAHbIf+╧>> н ъХі²vаfxBGEs[E@⌡хf00equal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)8,"8"]],sin[times[indexes[1,0,char[(*s9.00s*)c],num[(*s7.00s*)2,"2"]],matrix[(*n*)1,1,over[times[(*n*)char[(*s9.00s*)pi],char[(*f"MT"fs9.00s*)l]],num[(*s9.00s*)3000,"3000"]]]]],num[(*s9.00s*)0.955,"0.955"]]umїЪЪйJаL│AНTCHeKf│A╧@@ое┤їЪЪАЇ ┼│6&a╡EHgMf√Q≥LBBп.0x+FVQў√GGI1,&@═ 5?)T\equal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)9,"9"]],id[prompt[]],num[(*s9.00s*)1,"1"]]╣x+P√Qў√HGK+-═ 5?bequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)10,"10"]],id[prompt[]],num[(*s9.00s*)0.9,"0.9"]]їЪЪJаL│AНTIHkOf│A╧GGя*nux+UQў√JGMVQ0═ 5?eequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)11,"11"]],id[prompt[]],num[(*s9.00sn*)0.03,"0.03"]]sїЪЪ6JаZAНTKHmQfH╧HH!р╧5x+Y√Qў√LGOG3═ 5?s[fequal[indexes[0,1,char[(*s9.00s*)T],num[(*s7.00s*)12,"12"]],id[prompt[]],num[(*s9.00s*)0.035,"0.035"]]їЪЪZJа`AНTMHoSfma╧JJ&сes∙x+IжQў√NGQMT(─═ 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