{"id":2138,"date":"2023-04-07T21:01:25","date_gmt":"2023-04-07T21:01:25","guid":{"rendered":"https:\/\/vibromera.eu\/?p=2138"},"modified":"2026-07-02T19:45:53","modified_gmt":"2026-07-02T19:45:53","slug":"furje-transformacijas-pielietosana-vibracijas-signalu-analizei","status":"publish","type":"post","link":"https:\/\/vibromera.eu\/lv\/example\/application-of-the-fourier-transform-to-the-analysis-of-vibration-signals\/","title":{"rendered":"Furj\u0113 transform\u0101cijas izmanto\u0161ana vibr\u0101cijas sign\u0101lu anal\u012bzei."},"content":{"rendered":"

F\u016briera transform\u0101cijas pielietojums vibr\u0101cijas sign\u0101lu anal\u012bz\u0113<\/h1>\n

Andrejs \u0160elkovenko. Viens no Vibromera izstr\u0101d\u0101t\u0101jiem un dibin\u0101t\u0101js.
\nRaksta tulkojum\u0101 var b\u016bt neprecizit\u0101tes.<\/p>\n

Furj\u0113 transform\u0101cija un sign\u0101la spektrs<\/h2>\n

Daudzos gad\u012bjumos uzdevums ir ieg\u016bt (apr\u0113\u0137in\u0101t) spektrs<\/a> sign\u0101la ir \u0161\u0101ds. Ir analog\u0101-ciparu p\u0101rveidot\u0101js (ADC), kas, veicot paraugu \u0146em\u0161anu frekvence<\/a> Fd p\u0101rveido nep\u0101rtrauktu sign\u0101lu, kas laika posm\u0101 T non\u0101k t\u0101 ieej\u0101, ciparu paraugos \u2013 N vien\u012bb\u0101s. P\u0113c tam \u0161is paraugu mas\u012bvs tiek nodots k\u0101dai programmai (piem\u0113ram FourierScope<\/a><\/span>), kas izvada N\/2 da\u017eas skaitliskas v\u0113rt\u012bbas.<\/p>\n

Lai p\u0101rbaud\u012btu, vai programma darbojas pareizi, m\u0113s veidojam paraugu mas\u012bvu k\u0101 divu sin(10*2*pi*x)+0,5*sin(5*2*pi*x) summu un ievad\u0101m to programm\u0101. Programma uzz\u012bm\u0113ja sekojo\u0161o:
\n

\"Furj\u0113

1. att\u0113ls Sign\u0101la laika funkcijas grafiks<\/span><\/i><\/em><\/p><\/div><\/p>\n

 <\/p>\n

\"2.

2. att\u0113ls Sign\u0101la spektra grafiks<\/p><\/div>\n

 <\/p>\n

There are two harmonikas<\/a> spektra diagramm\u0101 \u2013 5 Hz ar amplit\u016bdu 0,5 V un 10 Hz ar amplit\u016bdu 1 V, viss atbilst s\u0101kotn\u0113j\u0101 sign\u0101la formulai. Viss ir k\u0101rt\u012bb\u0101, programma darbojas pareizi.<\/p>\n

Tas noz\u012bm\u0113, ka, ja ADC ieej\u0101 padodam re\u0101lu sign\u0101lu no divu sinuso\u012bdu mais\u012bjuma, m\u0113s sa\u0146emsim l\u012bdz\u012bgu spektru, kas sast\u0101v no div\u0101m harmonik\u0101m.<\/p>\n

T\u0101tad m\u016bsu \u012bsts <\/span><\/b><\/strong>izm\u0113r\u012btais sign\u0101ls 5 sek. ilgs<\/span><\/b><\/strong>, ko digitaliz\u0113 ar ADC, t. i., att\u0113lo ar diskr\u0113tu <\/span><\/b><\/strong>paraugus, ir diskr\u0113ti neperiodiski <\/span><\/b><\/strong>spektrs.
\nNo matem\u0101tisk\u0101 viedok\u013ca - cik daudz k\u013c\u016bdu ir \u0161aj\u0101 fr\u0101z\u0113? <\/span><\/i><\/em><\/p>\n

Tagad m\u0113\u0123in\u0101sim izm\u0113r\u012bt to pa\u0161u sign\u0101lu 0,5 sekun\u017eu laik\u0101.<\/p>\n

\"3.

3. att\u0113ls Funkcijas sin(10*2*pi*x)+0,5*sin(5*2*pi*x) grafiks 0,5 s m\u0113r\u012bjumu periodam.<\/p><\/div>\n

 <\/p>\n

\"4.

4. att\u0113ls Funkcijas spektrs<\/p><\/div>\n

 <\/p>\n

Kaut kas \u0161eit nav k\u0101rt\u012bb\u0101! Harmonika pie 10 Hz tiek z\u012bm\u0113ta norm\u0101li, bet harmonikas pie 5 Hz viet\u0101 ir da\u017eas neskaidras harmonikas.<\/p>\n

Internet\u0101 vi\u0146i saka, ka parauga beig\u0101s ir nepiecie\u0161ams pievienot nulles, un spektrs tiks sast\u0101d\u012bts norm\u0101li.<\/p>\n

\"5.

5. att\u0113ls Paraugam esam pievienoju\u0161i nulles l\u012bdz 5 sek.<\/p><\/div>\n

 <\/p>\n

\"6.

6. att\u0113ls. Ieg\u016btais spektrs.<\/p><\/div>\n

 <\/p>\n

Tas t\u0101 nemaz nav. Man b\u016bs j\u0101nodarbojas ar teoriju. Dosimies uz Vikip\u0113dija<\/b><\/strong><\/a><\/span>\u00a0- zin\u0101\u0161anu avots.<\/p>\n

Nep\u0101rtraukta funkcija un t\u0101s Furj\u0113 rindu atveidojums<\/h2>\n

Matem\u0101tiski m\u016bsu sign\u0101ls, kura ilgums ir T sekun\u017eu, ir k\u0101da funkcija f(x), kas dota interv\u0101l\u0101 {0, T} (X \u0161aj\u0101 gad\u012bjum\u0101 ir laiks). \u0160\u0101du funkciju vienm\u0113r var att\u0113lot k\u0101 harmonisko funkciju (sinusa vai kosinusa) summu:<\/p>\n

\"Nep\u0101rtraukta

\u00a0(1), kur:<\/p>\n

<\/p><\/div>\n

k ir trigonometrisk\u0101s funkcijas numurs (harmonisk\u0101s komponentes numurs, harmonikas numurs).
\nT - posms, kur\u0101 defin\u0113ta funkcija (sign\u0101la ilgums).
\nAk - k-t\u0101s harmonisk\u0101s komponentes amplit\u016bda,
\n\u03b8k- k-t\u0101s harmonisk\u0101s komponentes s\u0101kotn\u0113j\u0101 f\u0101ze
\nKo noz\u012bm\u0113 \"att\u0113lot funkciju k\u0101 rindu summu\"? Tas noz\u012bm\u0113, ka, saskaitot Furj\u0113 virknes harmonisko komponen\u0161u v\u0113rt\u012bbas katr\u0101 punkt\u0101, m\u0113s ieg\u016bstam m\u016bsu funkcijas v\u0113rt\u012bbu \u0161aj\u0101 punkt\u0101.
\n(Prec\u012bz\u0101k sakot, virknes vid\u0113j\u0101 kvadr\u0101tisk\u0101 novirze no funkcijas f(x) tiecas uz nulli, bet, neraugoties uz vid\u0113jo kvadr\u0101tisko konver\u0123enci, funkcijas Furj\u0113 virknei, visp\u0101r\u012bgi run\u0101jot, nav nepiecie\u0161ams, lai t\u0101 punkts p\u0113c punkta konver\u0123\u0113tu pie t\u0101s. )
\n\u0160o s\u0113riju var rakst\u012bt ar\u012b form\u0101:<\/p>\n

\"(2),\"

(2),<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

kur \"Furj\u0113 , k-t\u0101 kompleks\u0101 amplit\u016bda.<\/p>\n

 <\/p>\n

vai<\/p>\n

\"

(3)<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

Attiec\u012bbu starp koeficientiem (1) un (3) izsaka \u0161\u0101das formulas:<\/p>\n

\"Formula<\/p>\n

 <\/p>\n

 <\/p>\n

\"Furj\u0113<\/p>\n

 <\/p>\n

 <\/p>\n

\u0145emiet v\u0113r\u0101, ka visi \u0161ie tr\u012bs Furj\u0113 rindas att\u0113lojumi ir piln\u012bgi ekvivalenti. Da\u017ereiz, str\u0101d\u0101jot ar Furj\u0113 rind\u0101m, \u0113rt\u0101k ir izmantot imagin\u0101ra argumenta eksponentes sinusu un kosinusu viet\u0101, t. i., izmantot Furj\u0113 transform\u0101ciju kompleks\u0101 form\u0101. Ta\u010du mums ir \u0113rti lietot formulu (1), kur Furj\u0113 rinda ir att\u0113lota k\u0101 kosinusu summa ar atbilsto\u0161\u0101m amplit\u016bd\u0101m un f\u0101z\u0113m. Stingri run\u0101jot, re\u0101la sign\u0101la Furj\u0113 transform\u0101cija patie\u0161\u0101m dod kompleksus koeficientus (forma (3)): katrs koeficients satur gan savas harmonikas amplit\u016bdu, gan f\u0101zi. Re\u0101lam sign\u0101lam \u0161iem kompleksajiem koeficientiem piem\u012bt konjug\u0113t\u0101 (Herm\u012bta) simetrija \u2014 negat\u012bvo frekven\u010du puse vienk\u0101r\u0161i atspogu\u013co pozit\u012bvo frekven\u010du pusi un nesniedz papildu inform\u0101ciju. T\u0101p\u0113c no kompleksajiem koeficientiem m\u0113s vienm\u0113r varam p\u0101riet uz formulas (1) re\u0101laj\u0101m nenegat\u012bvaj\u0101m amplit\u016bd\u0101m Ak un f\u0101z\u0113m \u03b8k \u2014 un tie\u0161i \u0161o amplit\u016bdu spektru att\u0113lo anal\u012bzes programmas.<\/p>\n

 <\/p>\n

Apak\u0161\u0113j\u0101 l\u012bnija:<\/span><\/b><\/u><\/strong>
\n<\/span><\/b><\/strong>Sign\u0101lu spektr\u0101l\u0101s anal\u012bzes matem\u0101tiskais pamats ir Furj\u0113 transform\u0101cija.<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>Furj\u0113 transform\u0101cija \u013cauj att\u0113lot nep\u0101rtrauktu funkciju f(x) (sign\u0101lu), kas defin\u0113ta interv\u0101l\u0101 {0, T}, k\u0101 bezgal\u012bgi daudzu trigonometrisko funkciju (sinusa un\/vai kos\u012bna) summu (bezgal\u012bgu rindu) ar noteikt\u0101m amplit\u016bd\u0101m un f\u0101z\u0113m, kas ar\u012b tiek apskat\u012btas interv\u0101l\u0101 {0, T}. \u0160\u0101du rindu sauc par Furj\u0113 rindu.<\/span><\/b><\/strong><\/p>\n

\u0145emiet v\u0113r\u0101 v\u0113l da\u017eus punktus, kuru izpratne ir nepiecie\u0161ama, lai pareizi piem\u0113rotu Furj\u0113 transform\u0101ciju sign\u0101lu anal\u012bzei. Ja apl\u016bkosim Furj\u0113 rindu (sinuso\u012bdu summu) uz visas X ass, redz\u0113sim, ka \u0101rpus interv\u0101la {0, T} Furj\u0113 rindu funkcija periodiski atk\u0101rtos m\u016bsu funkciju.<\/p>\n

Piem\u0113ram, 7. att\u0113l\u0101 att\u0113lotaj\u0101 grafik\u0101 s\u0101kotn\u0113j\u0101 funkcija ir defin\u0113ta interv\u0101l\u0101 {-T\\2, +T\\2}, un Furj\u0113 rinda ir periodiska funkcija, kas defin\u0113ta uz visas x ass.<\/p>\n

Tas ir t\u0101p\u0113c, ka sinuso\u012bdas pa\u0161as par sevi ir periodiskas funkcijas, t\u0101p\u0113c ar\u012b to summa b\u016bs periodiska funkcija.<\/p>\n

\"7.

7. att\u0113ls Neperiodiskas avota funkcijas att\u0113lo\u0161ana ar Furj\u0113 rindu<\/p><\/div>\n

T\u0101d\u0113j\u0101di:<\/p>\n

M\u016bsu s\u0101kotn\u0113j\u0101 funkcija ir nep\u0101rtraukta, neperiodiska funkcija, kas defin\u0113ta uz k\u0101da T garuma posma.
\n\u0160\u012bs funkcijas spektrs ir diskr\u0113ts, t. i., to att\u0113lo k\u0101 bezgal\u012bgu harmonisko komponen\u0161u virkni - Furj\u0113 virkni.
\nPaties\u012bb\u0101 Furj\u0113 rinda defin\u0113 k\u0101du periodisku funkciju, kas sakr\u012bt ar m\u016bsu funkciju interv\u0101l\u0101 {0, T}, bet mums \u0161\u012b periodiskums nav b\u016btisks.<\/p>\n

N\u0101kamais.<\/p>\n

Harmonisko komponen\u0161u periodi ir interv\u0101la {0, T}, kur\u0101 defin\u0113ta s\u0101kotn\u0113j\u0101 funkcija f(x), reizin\u0101t\u0101ji. Citiem v\u0101rdiem sakot, harmonisko komponen\u0161u periodi ir sign\u0101la m\u0113r\u012bjuma ilguma reizin\u0101t\u0101ji. Piem\u0113ram, Furj\u0113 virknes pirm\u0101s harmonikas periods ir vien\u0101ds ar interv\u0101lu T, kur\u0101 defin\u0113ta funkcija f(x). Otr\u0101s harmonikas periods Furj\u0113 virkn\u0113 ir vien\u0101ds ar interv\u0101lu T\/2. Un t\u0101 t\u0101l\u0101k (sk. 8. att\u0113lu).<\/p>\n

\"8.

8. att\u0113ls Furj\u0113 virknes harmonisko komponen\u0161u periodi (frekvences) (\u0161eit T=2\u03c0)<\/p><\/div>\n

Attiec\u012bgi harmonisko komponen\u0161u frekvences ir 1\/T reizin\u0101t\u0101ji. Tas ir, harmonisko komponen\u0161u Fk frekvences ir Fk= k\\T, kur k ir v\u0113rt\u012bbas no 0 l\u012bdz \u221e, piem\u0113ram, k=0 F0=0; k=1 F1=1\\T; k=2 F2=2\\T;k=3 F3=3\\T;..... Fk= k\\T (pie nulles frekvences, konstanta komponente).<\/p>\n

Lai m\u016bsu s\u0101kotn\u0113j\u0101 funkcija ir sign\u0101ls, kas ierakst\u012bts laik\u0101 T=1 sek. Tad pirm\u0101s harmonikas periods b\u016bs vien\u0101ds ar m\u016bsu sign\u0101la ilgumu T1=T=1 sek, un harmonikas frekvence ir 1 Hz. Otr\u0101s harmonikas periods b\u016bs vien\u0101ds ar m\u016bsu sign\u0101la ilgumu, kas dal\u012bts ar 2 (T2=T\/2=0,5 s), un frekvence ir 2 Hz. Tre\u0161ajai harmoniskajai ir T3=T\/3 s, un frekvence ir 3 Hz. Un t\u0101 t\u0101l\u0101k.<\/p>\n

Soli starp harmoniskaj\u0101m \u0161aj\u0101 gad\u012bjum\u0101 ir 1 Hz.<\/p>\n

T\u0101d\u0113j\u0101di sign\u0101lu, kura ilgums ir 1 sekunde, var sadal\u012bt harmoniskaj\u0101s komponent\u0113s (lai ieg\u016btu spektru) ar frekvences iz\u0161\u0137irtsp\u0113ju 1 Hz.
\nLai palielin\u0101tu iz\u0161\u0137irtsp\u0113ju divas reizes l\u012bdz 0,5 Hz, m\u0113r\u012bjuma ilgums j\u0101palielina divas reizes l\u012bdz 2 sekund\u0113m. 10 sekun\u017eu garu sign\u0101lu var sadal\u012bt harmoniskaj\u0101s sast\u0101vda\u013c\u0101s (spektr\u0101) ar frekvences iz\u0161\u0137irtsp\u0113ju 0,1 Hz. Citu veidu, k\u0101 palielin\u0101t frekvences iz\u0161\u0137irtsp\u0113ju, nav. J\u016bs varat izp\u0113t\u012bt \u0161o sakar\u012bbu, izmantojot m\u016bsu FFT iz\u0161\u0137irtsp\u0113jas kalkulators<\/a>.<\/p>\n

Ir veids, k\u0101 m\u0101ksl\u012bgi palielin\u0101t sign\u0101la ilgumu, paraugu mas\u012bvam pievienojot nulles. Ta\u010du tas nepalielina re\u0101lo frekven\u010du iz\u0161\u0137irtsp\u0113ju.<\/p>\n

Diskr\u0113tie sign\u0101li un diskr\u0113t\u0101 F\u016briera transform\u0101cija<\/h2>\n

L\u012bdz ar digit\u0101lo tehnolo\u0123iju att\u012bst\u012bbu ir main\u012bju\u0161ies m\u0113r\u012bjumu datu (sign\u0101lu) glab\u0101\u0161anas veidi. Ja agr\u0101k sign\u0101lu var\u0113ja ierakst\u012bt magnetofona lent\u0113 un saglab\u0101t lent\u0113 analog\u0101 form\u0101, tad tagad sign\u0101lus digitaliz\u0113 un saglab\u0101 datn\u0113s datora atmi\u0146\u0101 k\u0101 skait\u013cu kopu (skait\u013cu).<\/p>\n

Parast\u0101 sign\u0101la m\u0113r\u012b\u0161anas un digitaliz\u0101cijas sh\u0113ma izskat\u0101s \u0161\u0101di.<\/p>\n

M\u0113r\u012b\u0161anas dev\u0113js -- Sign\u0101la normaliz\u0113t\u0101js -- ADC -- Dators
\n(9. att\u0113ls M\u0113r\u012b\u0161anas kan\u0101la sh\u0113ma)<\/p>\n

<\/span><\/i><\/em><\/p>\n

Sign\u0101ls no m\u0113rp\u0101rveidot\u0101ja non\u0101k ADC uz laika periodu T. Laika period\u0101 T sa\u0146emtie sign\u0101la r\u0101d\u012bjumi (paraugu \u0146em\u0161ana) tiek p\u0101rs\u016bt\u012bti uz datoru un saglab\u0101ti atmi\u0146\u0101.<\/p>\n

\"10.

10. att\u0113ls Digitaliz\u0113ts sign\u0101ls - N paraugi, kas sa\u0146emti laik\u0101 T<\/p><\/div>\n

K\u0101das ir pras\u012bbas sign\u0101lu digitaliz\u0101cijas parametriem? Ier\u012bci, kas p\u0101rveido ieejas analogo sign\u0101lu diskr\u0113t\u0101 kod\u0101 (ciparu sign\u0101l\u0101), sauc par analogo ciparu p\u0101rveidot\u0101ju (ADC) (\u00a9 Wiki).<\/p>\n

Viens no ADC pamatparametriem ir maksim\u0101lais paraugu \u0146em\u0161anas \u0101trums - nep\u0101rtraukta sign\u0101la paraugu \u0146em\u0161anas bie\u017eums laik\u0101. Paraugu \u0146em\u0161anas frekvenci m\u0113ra hercos. ((\u00a9 Wiki))<\/p>\n

Saska\u0146\u0101 ar Kote\u013c\u0146ikova teor\u0113mu, ja nep\u0101rtrauktam sign\u0101lam ir spektrs, kas ierobe\u017eots ar frekvenci Fmax, to var piln\u012bgi un viennoz\u012bm\u012bgi atjaunot no diskr\u0113tiem paraugiem, kas \u0146emti ar laika interv\u0101liem \u0394t \u2264 1\/(2*Fmax), t.i., ar diskretiz\u0101cijas frekvenci Fd \u2265 2*Fmax, kur Fd – diskretiz\u0101cijas frekvence; Fmax – sign\u0101la spektra maksim\u0101l\u0101 frekvence. Citiem v\u0101rdiem, sign\u0101la digitaliz\u0101cijas frekvencei (ADC diskretiz\u0101cijas frekvencei) j\u0101b\u016bt vismaz divreiz liel\u0101kai par maksim\u0101lo sign\u0101la frekvenci, kuru v\u0113lamies izm\u0113r\u012bt.<\/p>\n

Un kas notiks, ja m\u0113s \u0146emsim paraugus ar maz\u0101ku bie\u017eumu, nek\u0101 pieprasa Kotel\u0146ikova teor\u0113ma?<\/p>\n

\u0160aj\u0101 gad\u012bjum\u0101 ir \u201eizl\u012bdzin\u0101\u0161ana<\/a>\u0160aj\u0101 gad\u012bjum\u0101 rodas \"aliasing\" efekts (paz\u012bstams ar\u012b k\u0101 stroboskopiskais efekts, moir\u0113 efekts), kad augstfrekvences sign\u0101ls p\u0113c digitaliz\u0101cijas p\u0101rv\u0113r\u0161as par zemas frekvences sign\u0101lu, kas paties\u012bb\u0101 nepast\u0101v. att\u0113l\u0101 11. att\u0113l\u0101 sarkanais augstas frekvences sinusoid\u0101lais vilnis ir \u012bstais sign\u0101ls. Zem\u0101kas frekvences zilais sinusoid\u0101lais vilnis ir fikt\u012bvs sign\u0101ls, kas rodas t\u0101p\u0113c, ka paraugu \u0146em\u0161anas laik\u0101 ir pag\u0101jis vair\u0101k nek\u0101 pusperiods no augstfrekvences sign\u0101la.<\/p>\n

\"11.

11. att\u0113ls. Nepareiza zemas frekvences sign\u0101la par\u0101d\u012b\u0161an\u0101s pie nepietiekami augsta paraugu \u0146em\u0161anas \u0101truma<\/p><\/div>\n

 <\/p>\n

Lai izvair\u012btos no aliasinga efekta, tiek izmantots \u012bpa\u0161s anti-aliasinga filtrs (zemfrekvences filtrs<\/a>) ir novietots pirms ADC. Tas caurlaida frekvences, kas ir zem\u0101kas par pusi no ADC diskretiz\u0101cijas frekvences, un nogrie\u017e augst\u0101k\u0101s frekvences.<\/p>\n

Lai apr\u0113\u0137in\u0101tu sign\u0101la spektru, izmantojot t\u0101 diskr\u0113tos paraugus, diskr\u0113tais F\u016briera transform\u0101cija (DFT)<\/a> tiek izmantots. Atcerieties, ka diskr\u0113t\u0101 sign\u0101la spektrs \u201ep\u0113c defin\u012bcijas\u201c ir ierobe\u017eots l\u012bdz frekvencei Fmax, kas ir maz\u0101ka par pusi no diskretiz\u0101cijas frekvences Fd. T\u0101d\u0113j\u0101di diskr\u0113t\u0101 sign\u0101la spektru var att\u0113lot k\u0101 summu a gal\u012bgs <\/u>harmoniku skaits, at\u0161\u0137ir\u012bb\u0101 no bezgal\u012bg\u0101s summas nep\u0101rtraukta sign\u0101la Furj\u0113 virknei, kuras spektrs var b\u016bt neierobe\u017eots. Saska\u0146\u0101 ar Kotel\u0146ikova teor\u0113mu harmonikas maksim\u0101lajai frekvencei j\u0101b\u016bt t\u0101dai, lai t\u0101 atbilstu vismaz diviem paraugiem, t\u0101p\u0113c harmoniku skaits ir vien\u0101ds ar pusi no diskr\u0113t\u0101 sign\u0101la paraugu skaita. Tas ir, ja paraug\u0101 ir N paraugu, tad harmoniku skaits spektr\u0101 b\u016bs N\/2.<\/p>\n

Tagad apl\u016bkojiet diskr\u0113to Furj\u0113 transform\u0101ciju (DFT).<\/p>\n

\"Diskr\u0113t\u0101<\/p>\n

Sal\u012bdzinot to ar Furj\u0113 s\u0113riju<\/p>\n

 <\/p>\n

\"Diskr\u0113t\u0101s<\/p>\n

K\u0101 redzams, tie sakr\u012bt, iz\u0146emot to, ka laiks FFT ir diskr\u0113ts un harmoniku skaits ir ierobe\u017eots l\u012bdz N\/2, kas ir puse no paraugu skaita.<\/p>\n

DFT formulas tiek rakst\u012btas bezdimensiju veselos main\u012bgajos k, s, kur k ir sign\u0101la paraugu skaits, s ir spektr\u0101lo komponen\u0161u skaits.
\nV\u0113rt\u012bba s par\u0101da pilno harmonisko sv\u0101rst\u012bbu skaitu period\u0101 T (sign\u0101la m\u0113r\u012b\u0161anas ilgums). Diskr\u0113to Furj\u0113 transform\u0101ciju izmanto, lai harmoniku amplit\u016bdas un f\u0101zes atrastu skaitliski, t. i., \"dator\u0101\".<\/p>\n

K\u0101 jau min\u0113ts iepriek\u0161, sadalot neperiodisku funkciju (m\u016bsu sign\u0101lu) Furj\u0113 virkn\u0113, ieg\u016bt\u0101 Furj\u0113 virkne faktiski atbilst periodiskai funkcijai ar periodu T (12. att\u0113ls).<\/p>\n

 <\/p>\n

\"12.

12. att\u0113ls. Periodiska funkcija f(x) ar periodu T0, ar periodu T>T0<\/p><\/div>\n

 <\/p>\n

K\u0101 redzams 12. att\u0113l\u0101, funkcija f(x) ir periodiska ar periodu T0. Tom\u0113r, t\u0101 k\u0101 m\u0113r\u0101m\u0101 parauga garums T neatbilst funkcijas periodam T0, funkcija, kas ieg\u016bta k\u0101 F\u016briera rinda, punkt\u0101 T ir nep\u0101rtraukta. Rezult\u0101t\u0101 \u0161\u012bs funkcijas spektr\u0101 b\u016bs liels skaits augstfrekvences harmoniku. \u0160o par\u0101d\u012bbu sauc par spektr\u0101l\u0101 nopl\u016bde<\/a>, un praks\u0113 to samazina par logu veido\u0161ana<\/a> sign\u0101ls pirms transform\u0101cijas. Ja m\u0113r\u0101m\u0101 parauga T ilgums sakristu ar funkcijas T0 periodu, tad p\u0113c F\u016briera transform\u0101cijas ieg\u016btais spektrs satur\u0113tu tikai pirmo harmoniku (sinuso\u012bdu, kuras periods ir vien\u0101ds ar parauga ilgumu), jo funkcija f(x) ir sinuso\u012bda.<\/p>\n

Citiem v\u0101rdiem sakot, DFT programma \"nezina\", ka m\u016bsu sign\u0101ls ir \"sinuso\u012bd\u0101 vi\u013c\u0146a \u0161\u0137\u0113lums\", bet m\u0113\u0123ina k\u0101 virkni att\u0113lot periodisku funkciju, kurai ir p\u0101rtraukt\u012bba, ko rada atsevi\u0161\u0137u sinuso\u012bd\u0101 vi\u013c\u0146a da\u013cu p\u0101rtraukt\u012bba.<\/p>\n

Rezult\u0101t\u0101 spektr\u0101 par\u0101d\u0101s harmonikas, kur\u0101m kopum\u0101 b\u016btu j\u0101atspogu\u013co funkcijas forma, ieskaitot \u0161o p\u0101rr\u0101vumu.<\/p>\n

T\u0101d\u0113j\u0101di, lai ieg\u016btu \"pareizu\" spektru sign\u0101lam, kas ir vair\u0101ku sinuso\u012bdu ar da\u017e\u0101diem periodiem summa, ir nepiecie\u0161ams, lai vesels skaitlis periodu skaits <\/u>katrai sinuso\u012bdai j\u0101b\u016bt sign\u0101la m\u0113r\u012b\u0161anas period\u0101. Praks\u0113 \u0161o nosac\u012bjumu var izpild\u012bt, ja sign\u0101la m\u0113r\u012b\u0161anas ilgums ir pietiekami ilgs.<\/p>\n

 <\/p>\n

\"13.

13. att\u0113ls P\u0101rnesumk\u0101rbas kinem\u0101tisk\u0101s k\u013c\u016bdas sign\u0101la funkcijas un spektra piem\u0113rs<\/p><\/div>\n

 <\/p>\n

\u012as\u0101k\u0101 laik\u0101 att\u0113ls izskat\u0101s \"slikt\u0101ks\":<\/p>\n

 <\/p>\n

\"14.

14. att\u0113ls Rotora vibr\u0101ciju funkcijas un spektra piem\u0113rs<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

Praks\u0113 var b\u016bt gr\u016bti saprast, kur ir \"\u012bstie komponenti\" un kur \"artefakti\", ko izraisa komponentu periodu un sign\u0101la paraugu \u0146em\u0161anas ilguma neatbilst\u012bba vai \"l\u0113cieni un p\u0101rtraukumi\" vi\u013c\u0146u form\u0101. Protams, v\u0101rdi \"\u012bst\u0101s komponentes\" un \"artefakti\" ne velti ir doti p\u0113di\u0146\u0101s. Daudzu harmoniku kl\u0101tb\u016btne spektra grafik\u0101 nenoz\u012bm\u0113, ka m\u016bsu sign\u0101ls patie\u0161\u0101m sast\u0101v no t\u0101m. Tas ir t\u0101pat k\u0101 dom\u0101t, ka skaitlis 7 \"sast\u0101v\" no skait\u013ciem 3 un 4. Skaitli 7 var uzskat\u012bt par 3 un 4 summu - tas ir pareizi.<\/p>\n

T\u0101tad ar\u012b m\u016bsu sign\u0101lu... vai dr\u012bz\u0101k pat ne \"m\u016bsu sign\u0101lu\", bet periodisku funkciju, kas veidota, atk\u0101rtojot m\u016bsu sign\u0101lu (paraugu), var att\u0113lot k\u0101 harmoniku (sinuso\u012bdu) summu ar noteikt\u0101m amplit\u016bd\u0101m un f\u0101z\u0113m. Bet daudzos praks\u0113 svar\u012bgos gad\u012bjumos (sk. att\u0113lus iepriek\u0161) patie\u0161\u0101m ir iesp\u0113jams spektr\u0101 ieg\u016bt\u0101s harmonikas saist\u012bt ar\u012b ar re\u0101liem procesiem, kam ir ciklisks raksturs un kas b\u016btiski ietekm\u0113 sign\u0101la formu.<\/p>\n

Da\u017ei rezult\u0101ti<\/h2>\n

1. Re\u0101lam izm\u0113r\u012btam sign\u0101lam ar T sek. ilgumu, kas digitaliz\u0113ts ar ADC, t. i., att\u0113lots ar diskr\u0113tu paraugu kopu (N vien\u012bbu), ir diskr\u0113ts neperiodisks spektrs, ko att\u0113lo harmoniku kopa (N\/2 vien\u012bbas).<\/p>\n

2. Sign\u0101ls ir att\u0113lots ar re\u0101lu v\u0113rt\u012bbu kopu. T\u0101 DFT spektrs ir kompleksu koeficientu kopa ar konjug\u0113to simetriju; no tiem ieg\u016bst amplit\u016bdu spektru \u2014 re\u0101lu nenegat\u012bvu amplit\u016bdu (un f\u0101\u017eu) kopu pozit\u012bvaj\u0101s frekvenc\u0113s, un praks\u0113 tiek att\u0113lots tie\u0161i \u0161is vienpus\u0113jais amplit\u016bdu spektrs. Divpus\u0113j\u0101 kompleks\u0101 forma ar negat\u012bvaj\u0101m frekvenc\u0113m un vienpus\u0113j\u0101 amplit\u016bdas\/f\u0101zes forma ir ekvivalenti viena un t\u0101 pa\u0161a spektra att\u0113lojumi \u2014 sign\u0101lu anal\u012bz\u0113 parasti \u0113rt\u0101k ir str\u0101d\u0101t ar vienpus\u0113jo amplit\u016bdu spektru.<\/p>\n

3. Sign\u0101ls, kas izm\u0113r\u012bts laik\u0101 T, ir noteikts tikai laik\u0101 T. Kas notika pirms m\u0113s s\u0101k\u0101m m\u0113r\u012bt sign\u0101lu un kas notiks p\u0113c tam, zin\u0101tnei nav zin\u0101ms. Un m\u016bsu gad\u012bjum\u0101 tas nav interesanti. Laika ierobe\u017eot\u0101 sign\u0101la FFT dod t\u0101 \"re\u0101lo\" spektru t\u0101d\u0101 noz\u012bm\u0113, ka pie noteiktiem nosac\u012bjumiem \u013cauj apr\u0113\u0137in\u0101t t\u0101 komponen\u0161u amplit\u016bdu un frekvenci.<\/p>\n

 <\/p>","protected":false},"excerpt":{"rendered":"

Application of the Fourier Transform to the Analysis of Vibration Signals Andrei Shelkovenko. One of the developers and founder of Vibromera. The translation of the article may contain inaccuracies. Fourier transform and signal spectrum In many cases the task of obtaining (calculating) the spectrum of a signal is as follows. […]<\/p>\n","protected":false},"author":2,"featured_media":2161,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ai_generated_summary":"","footnotes":""},"categories":[4],"tags":[],"class_list":["post-2138","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-example"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/posts\/2138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/comments?post=2138"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/posts\/2138\/revisions"}],"predecessor-version":[{"id":102137,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/posts\/2138\/revisions\/102137"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/media\/2161"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/categories?post=2138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/lv\/wp-json\/wp\/v2\/tags?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}