{"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":"application-of-the-fourier-transform-to-the-analysis-of-vibration-signals","status":"publish","type":"post","link":"https:\/\/vibromera.eu\/az\/example\/application-of-the-fourier-transform-to-the-analysis-of-vibration-signals\/","title":{"rendered":"Vibrasiya siqnallar\u0131n\u0131n t\u0259hlilind\u0259 Fourier \u00e7evrilm\u0259sinin t\u0259tbiqi."},"content":{"rendered":"

Vibrasiya Siqnallar\u0131n\u0131n Analizi \u00fc\u00e7\u00fcn Fur'e \u00c7evrilm\u0259sinin T\u0259tbiqi<\/h1>\n

Andrei \u015eelkovenko. Vibromera-n\u0131n inki\u015faf etdiricil\u0259rind\u0259n biri v\u0259 t\u0259sis\u00e7isi.
\nM\u0259qal\u0259nin t\u0259rc\u00fcm\u0259si qeyri-d\u0259qiqlikl\u0259r ehtiva ed\u0259 bil\u0259r.<\/p>\n

Fourier \u00e7evrilm\u0259si v\u0259 siqnal spektri<\/h2>\n

Bir \u00e7ox hallarda siqnal\u0131n spektr<\/a> al\u0131nmas\u0131 (hesablanmas\u0131) v\u0259zif\u0259si a\u015fa\u011f\u0131dak\u0131 kimidir. ADC var, bu da se\u00e7m\u0259 il\u0259 tezlik<\/a> Fd zaman T \u0259rzind\u0259 giri\u015fin\u0259 g\u0259l\u0259n davaml\u0131 siqnal\u0131 r\u0259q\u0259msal n\u00fcmun\u0259l\u0259r\u0259 \u2014 N \u0259d\u0259d\u0259 \u2014 \u00e7evirir. Sonra bu n\u00fcmun\u0259 qrupunun b\u0259zi proqrama (m\u0259s\u0259l\u0259n Fourier Spektroskopu<\/a><\/span>) hans\u0131 ki, N\/2 \u0259d\u0259d d\u0259y\u0259r \u00e7\u0131xar\u0131r.<\/p>\n

Proqram\u0131n d\u00fczg\u00fcn i\u015fl\u0259diyini yoxlamaq \u00fc\u00e7\u00fcn n\u00fcmun\u0259l\u0259r massivini iki sin(10*2*pi*x)+0.5*sin(5*2*pi*x) c\u0259mi kimi formala\u015fd\u0131r\u0131b proqram\u0131n i\u00e7ine daxil edirik. Proqram a\u015fa\u011f\u0131dak\u0131n\u0131 \u00e7\u0259kdi:
\n

\"Fourier

\u015e\u0259k.1 Siqnal\u0131n zaman funksiyas\u0131n\u0131n qrafiki<\/span><\/i><\/em><\/p><\/div><\/p>\n

 <\/p>\n

\"\u015e\u0259kil

\u015e\u0259kil 2. Siqnal spektrinin qrafiki<\/p><\/div>\n

 <\/p>\n

\u0130ki harmonikl\u0259r<\/a> spektr qrafikind\u0259 \u2014 0.5 V amplitudas\u0131 il\u0259 5 Hz v\u0259 1 V amplitudas\u0131 il\u0259 10 Hz, h\u0259r \u015fey orijinal siqnal d\u00fcsturunda oldu\u011fu kimidir. H\u0259r \u015fey yax\u015f\u0131d\u0131r, proqram d\u00fczg\u00fcn i\u015fl\u0259yir.<\/p>\n

Bu o dem\u0259kdir ki, \u0259g\u0259r ADC giri\u015fin\u0259 iki sinusoid qar\u0131\u015f\u0131\u011f\u0131ndan ibar\u0259t real siqnal ver\u0259riks\u0259, iki harmonikadan ibar\u0259t ox\u015far spektr \u0259ld\u0259 ed\u0259c\u0259yik.<\/p>\n

Bel\u0259likl\u0259, bizim h\u0259qiqi <\/span><\/b><\/strong>\u00f6l\u00e7\u00fclm\u00fc\u015f siqnal 5 saniy\u0259lik m\u00fcdd\u0259t<\/span><\/b><\/strong>, ADC t\u0259r\u0259find\u0259n r\u0259q\u0259msalla\u015fd\u0131r\u0131lm\u0131\u015f, y\u0259ni t\u0259msil olunmu\u015f ayr\u0131-ayr\u0131 <\/span><\/b><\/strong>n\u00fcmun\u0259l\u0259r, malikdir ayr\u0131-ayr\u0131 d\u00f6vr\u00fc olmayan <\/span><\/b><\/strong>Spektr.
\nFrom a mathematical point of view \u2013 how many errors in this phrase? <\/span><\/i><\/em><\/p>\n

Now let\u2019s try to measure the same signal for 0.5 sec.<\/p>\n

\"\u015e\u0259k.3.

\u015e\u0259k.3. \u00d6l\u00e7m\u0259 d\u00f6vr\u00fc 0,5 saniy\u0259 olan sin(10*2*pi*x)+0,5*sin(5*2*pi*x) funksiyas\u0131n\u0131n qrafiki<\/p><\/div>\n

 <\/p>\n

\"\u015e\u0259k.4

\u015e\u0259k.4 Funksiyan\u0131n spektri<\/p><\/div>\n

 <\/p>\n

Burada n\u0259s\u0259 s\u0259hvdir! 10 Hz-d\u0259ki harmonik normal \u015f\u0259kild\u0259 t\u0259svir olunub, 5 Hz-d\u0259ki harmonikin \u0259v\u0259zin\u0259 is\u0259 b\u0259zi qeyri-m\u00fc\u0259yy\u0259n harmonikl\u0259r var.<\/p>\n

\u0130nternetd\u0259 deyirl\u0259r ki, n\u00fcmun\u0259nin sonuna s\u0131f\u0131rlar \u0259lav\u0259 etm\u0259k laz\u0131md\u0131r v\u0259 spektr normal \u015f\u0259kild\u0259 \u00e7\u0259kil\u0259c\u0259k.<\/p>\n

\"\u015e\u0259kil

\u015e\u0259kil 5. N\u00fcmun\u0259y\u0259 5 saniy\u0259y\u0259 q\u0259d\u0259r s\u0131f\u0131rlar \u0259lav\u0259 etmi\u015fik.<\/p><\/div>\n

 <\/p>\n

\"\u015e\u0259k.6.

\u015e\u0259k.6. \u018fld\u0259 olunan spektr.<\/p><\/div>\n

 <\/p>\n

That\u2019s not it at all. I will have to deal with the theory. Let\u2019s go to vikipediya<\/b><\/strong><\/a><\/span>\u00a0\u2013 the source of knowledge.<\/p>\n

Davaml\u0131 funksiya v\u0259 onun Fourier silsil\u0259si t\u0259msil\u00e7iliyi<\/h2>\n

Riyazi bax\u0131mdan, T saniy\u0259 davaml\u0131 siqnal\u0131m\u0131z {0, T} intervallar\u0131nda verilmi\u015f f(x) funksiyas\u0131d\u0131r (bu halda X zaman dem\u0259kdir). Bel\u0259 bir funksiya h\u0259mi\u015f\u0259 a\u015fa\u011f\u0131dak\u0131 formadak\u0131 harmonik funksiyalar\u0131n (sinus v\u0259 ya kosinus) c\u0259mi kimi t\u0259qdim oluna bil\u0259r:<\/p>\n

\"Davaml\u0131

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

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

k trigonometrik funksiyan\u0131n n\u00f6mr\u0259sidir (harmonik komponentin n\u00f6mr\u0259si, harmonikin n\u00f6mr\u0259si)
\nT \u2013 segment where the function is defined (the duration of the signal)
\nAk - k-c\u0131 harmonik komponentin amplitudu,
\n\u03b8k - k-ci harmonik komponentin ilkin fazas\u0131
\nWhat does it mean to \u201crepresent the function as the sum of the series\u201d? It means that by adding the values of the harmonic components of the Fourier series at each point, we get the value of our function at that point.
\nDaha d\u0259qiq des\u0259k, s\u0131ra il\u0259 f(x) funksiyas\u0131 aras\u0131ndak\u0131 orta kvadrat x\u0259tan\u0131n qiym\u0259ti s\u0131f\u0131ra do\u011fru meyll\u0259n\u0259c\u0259k, lakin orta kvadrat yax\u0131nla\u015fmas\u0131na baxmayaraq, \u00fcmumilikd\u0259 bir funksiyan\u0131n Fourier s\u0131ras\u0131 n\u00f6qt\u0259-n\u00f6qt\u0259 ona yax\u0131nla\u015fmaya bil\u0259r.
\nBu s\u0131ra h\u0259m\u00e7inin a\u015fa\u011f\u0131dak\u0131 formada da yaz\u0131la bil\u0259r:<\/p>\n

\"(2),\"

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

 <\/p>\n

 <\/p>\n

 <\/p>\n

Burada \"Vibrasyon , k-c\u0131 kompleks amplituda.<\/p>\n

 <\/p>\n

or<\/p>\n

\"

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

 <\/p>\n

 <\/p>\n

 <\/p>\n

Koeffisientl\u0259r (1) v\u0259 (3) aras\u0131ndak\u0131 \u0259laq\u0259 a\u015fa\u011f\u0131dak\u0131 d\u00fcsturlarla ifad\u0259 olunur:<\/p>\n

\"Furye<\/p>\n

 <\/p>\n

 <\/p>\n

\"Furye<\/p>\n

 <\/p>\n

 <\/p>\n

Qeyd edin ki, Fourier s\u0131ras\u0131n\u0131n bu \u00fc\u00e7 t\u0259qdimat\u0131 tam ekvivalentdir. B\u0259z\u0259n Fourier s\u0131ralar\u0131 il\u0259 i\u015fl\u0259y\u0259rk\u0259n sinus v\u0259 kosinuslar \u0259v\u0259zin\u0259 x\u0259yali arqumentin \u00fcstl\u00fc funksiyalar\u0131ndan, y\u0259ni Fourier \u00e7evrilm\u0259sinin kompleks formas\u0131ndan istifad\u0259 etm\u0259k daha rahat olur. Amma bizim \u00fc\u00e7\u00fcn amplitudalar\u0131 v\u0259 fazalar\u0131 uy\u011fun olan kosinuslar\u0131n c\u0259mi \u015f\u0259klind\u0259 verilmi\u015f formula (1)-d\u0259n istifad\u0259 daha \u0259lveri\u015flidir. D\u0259qiq des\u0259k, real siqnal\u0131n Fourier \u00e7evrilm\u0259si h\u0259qiq\u0259t\u0259n kompleks \u0259msallar yarad\u0131r (forma (3)): h\u0259r \u0259msal \u00f6z harmonikas\u0131n\u0131n h\u0259m amplitudas\u0131n\u0131, h\u0259m d\u0259 fazas\u0131n\u0131 da\u015f\u0131y\u0131r. Real siqnal \u00fc\u00e7\u00fcn bu kompleks \u0259msallar qo\u015fma (Hermit) simmetriyaya malikdir \u2014 m\u0259nfi tezlik yar\u0131s\u0131 sad\u0259c\u0259 m\u00fcsb\u0259t tezlik yar\u0131s\u0131n\u0131 g\u00fczg\u00fc kimi t\u0259krarlay\u0131r v\u0259 \u0259lav\u0259 informasiya da\u015f\u0131m\u0131r. Buna g\u00f6r\u0259 d\u0259 kompleks \u0259msallardan h\u0259mi\u015f\u0259 formula (1)-d\u0259ki real, m\u0259nfi olmayan Ak amplitudalar\u0131na v\u0259 \u03b8k fazalar\u0131na ke\u00e7\u0259 bil\u0259rik \u2014 analiz proqramlar\u0131n\u0131n qrafik\u0259 \u00e7\u0131xard\u0131\u011f\u0131 da m\u0259hz bu amplituda spektridir.<\/p>\n

 <\/p>\n

N\u0259tic\u0259:<\/span><\/b><\/u><\/strong>
\n<\/span><\/b><\/strong>Siqnallar\u0131n spektral t\u0259hlili \u00fc\u00e7\u00fcn riyazi baza Fourier \u00e7evrilm\u0259sidir.<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>Fourier \u00e7evrilm\u0259si fasil\u0259siz funksiyan\u0131 f(x) (siqnal) {0, T} intervallar\u0131 \u00fcz\u0259rind\u0259 trigonometrik funksiyalar\u0131n (sinus v\u0259\/v\u0259 ya kosinus) m\u00fc\u0259yy\u0259n amplitudalar v\u0259 fazalar toplusu kimi ifad\u0259 etm\u0259y\u0259 imkan verir. Bel\u0259 s\u0131ra Fourier s\u0131ras\u0131 adlan\u0131r.<\/span><\/b><\/strong><\/p>\n

Bir ne\u00e7\u0259 \u0259lav\u0259 m\u0259qam\u0131 n\u0259z\u0259r\u0259 al\u0131n, onlar\u0131n anla\u015f\u0131lmas\u0131 Fourier \u00e7evrilm\u0259sinin siqnal analizind\u0259 d\u00fczg\u00fcn t\u0259tbiqi \u00fc\u00e7\u00fcn vacibdir. \u018fg\u0259r Fourier seriyas\u0131n\u0131 (sinusoidalar\u0131n c\u0259mini) b\u00fct\u00fcn X oxu boyunca n\u0259z\u0259rd\u0259n ke\u00e7irs\u0259k, {0, T} intervallar\u0131ndan k\u0259narda Fourier seriyas\u0131 funksiyas\u0131n\u0131n funksiyam\u0131z\u0131 periodik \u015f\u0259kild\u0259 t\u0259krarlad\u0131\u011f\u0131n\u0131 g\u00f6r\u0259rik.<\/p>\n

M\u0259s\u0259l\u0259n, 7-ci fiqurun qrafikind\u0259 ilkin funksiya {-T\/2, +T\/2} intervallar\u0131nda m\u00fc\u0259yy\u0259n edilir, Fourier silsil\u0259si is\u0259 b\u00fct\u00fcn x oxu boyunca m\u00fc\u0259yy\u0259n edilmi\u015f d\u00f6vri funksiyan\u0131 t\u0259msil edir.<\/p>\n

Bu, sinusoidl\u0259rin \u00f6zl\u0259ri periodik funksiyalar oldu\u011funa g\u00f6r\u0259, onlar\u0131n c\u0259mi d\u0259 periodik funksiya olacaq.<\/p>\n

\"\u015e\u0259kil

\u015e\u0259kil 7. D\u00f6vri olmayan m\u0259nb\u0259 funksiyas\u0131n\u0131n Fourier c\u0259min\u0259 g\u00f6r\u0259 t\u0259msil edilm\u0259si<\/p><\/div>\n

Bel\u0259likl\u0259:<\/p>\n

Orijinal funksiyam\u0131z T uzunlu\u011funda bir intervald\u0259 t\u0259yin olunmu\u015f fasil\u0259siz, d\u00f6vri olmayan funksiyad\u0131r.
\nThe spectrum of this function is discrete, i.e. it is represented as an infinite series of harmonic components \u2013 a Fourier series.
\n\u018fslind\u0259 Fourier silsil\u0259si m\u00fc\u0259yy\u0259n d\u00f6vri funksiyan\u0131 m\u00fc\u0259yy\u0259n edir, bu funksiya {0, T} intervallar\u0131nda bizim funksiyam\u0131zla \u00fcst-\u00fcst\u0259 d\u00fc\u015f\u00fcr, lakin bizim \u00fc\u00e7\u00fcn bu d\u00f6vrilik \u0259sas deyil.<\/p>\n

Sonrak\u0131<\/p>\n

Harmonik komponentl\u0259rin d\u00f6vrl\u0259ri, ilkin funksiya f(x)-in t\u0259yin olundu\u011fu {0, T} intervallar\u0131n\u0131n \u00e7oxluqlar\u0131d\u0131r. Ba\u015fqa s\u00f6zl\u0259, harmonikl\u0259rin d\u00f6vrl\u0259ri siqnal \u00f6l\u00e7m\u0259 m\u00fcdd\u0259tinin \u00e7oxluqlar\u0131d\u0131r. M\u0259s\u0259l\u0259n, Fourier seriyas\u0131nda birinci harmonikin d\u00f6vr\u00fc funksiya f(x)-in t\u0259yin olundu\u011fu intervala, y\u0259ni T-y\u0259 b\u0259rab\u0259rdir. Fourier seriyas\u0131nda ikinci harmonikin d\u00f6vr\u00fc is\u0259 T\/2 interval\u0131na b\u0259rab\u0259rdir. V\u0259 s. (Bax \u015e\u0259kil 8).<\/p>\n

\"\u015e\u0259k.

\u015e\u0259k. 8 Fourier seriyas\u0131n\u0131n harmonik komponentl\u0259rinin periodlar\u0131 (tezlikl\u0259ri) (burada T=2\u03c0)<\/p><\/div>\n

Accordingly, the frequencies of harmonic components are multiples of 1\/T. That is, frequencies of harmonic components Fk are Fk= k\\T, where k runs values from 0 to \u221e, for example, k=0 F0=0; k=1 F1=1\\T; k=2 F2=2\\T;k=3 F3=3\\T;\u2026. Fk= k\\T (at zero frequency, a constant component).<\/p>\n

Ba\u015flan\u011f\u0131c funksiyam\u0131z T=1 s m\u00fcdd\u0259tind\u0259 qeyd\u0259 al\u0131nm\u0131\u015f siqnald\u0131r. O zaman birinci harmonikin d\u00f6vr\u00fc siqnal\u0131m\u0131z\u0131n m\u00fcdd\u0259ti T1=T=1 s-\u0259 b\u0259rab\u0259r olacaq v\u0259 harmonikin tezliyi 1 Hz-dir. \u0130kinci harmonikin d\u00f6vr\u00fc siqnal\u0131m\u0131z\u0131n m\u00fcdd\u0259tinin 2-y\u0259 b\u00f6l\u00fcnm\u0259sin\u0259 b\u0259rab\u0259rdir (T2=T\/2=0,5 s) v\u0259 tezliyi 2 Hz-dir. \u00dc\u00e7\u00fcnc\u00fc harmonik \u00fc\u00e7\u00fcn T3=T\/3 s v\u0259 tezliyi 3 Hz-dir. V\u0259 s.<\/p>\n

Bu halda harmonikl\u0259r aras\u0131ndak\u0131 add\u0131m 1 Hz-dir.<\/p>\n

Bel\u0259likl\u0259, 1 saniy\u0259lik siqnal 1 Hz tezlik ay\u0131rdetm\u0259si il\u0259 harmonik komponentl\u0259r\u0259 ayr\u0131l\u0131b spektri \u0259ld\u0259 edil\u0259 bil\u0259r.
\nH\u0259lledici g\u00fcc\u00fc 2 d\u0259f\u0259 0.5 Hz-\u0259 q\u0259d\u0259r art\u0131rmaq \u00fc\u00e7\u00fcn, \u00f6l\u00e7m\u0259 m\u00fcdd\u0259tini 2 d\u0259f\u0259 2 saniy\u0259y\u0259 q\u0259d\u0259r art\u0131rmaq laz\u0131md\u0131r. 10 saniy\u0259lik siqnal harmonik komponentl\u0259r\u0259 (spektr\u0259) 0.1 Hz tezlik h\u0259llediciliyi il\u0259 par\u00e7alan\u0131r. Tezlik h\u0259llediciliyini art\u0131rmaq \u00fc\u00e7\u00fcn ba\u015fqa \u00fcsullar yoxdur. Bu m\u00fcnasib\u0259ti \u00f6yr\u0259n\u0259 bil\u0259rsiniz bizim FFT \u00c7\u00f6z\u00fcn\u00fcrl\u00fck Kalkulyatoru<\/a>.<\/p>\n

Sinyal m\u00fcdd\u0259tini s\u00fcni \u015f\u0259kild\u0259 art\u0131rmaq \u00fc\u00e7\u00fcn n\u00fcmun\u0259l\u0259r massivin\u0259 s\u0131f\u0131rlar \u0259lav\u0259 etm\u0259k m\u00fcmk\u00fcnd\u00fcr. Lakin bu, real tezlik ay\u0131rdetm\u0259sini art\u0131rm\u0131r.<\/p>\n

Diskret siqnallar v\u0259 diskret Fur'e \u00e7evrilm\u0259si<\/h2>\n

R\u0259q\u0259msal texnologiyan\u0131n inki\u015faf\u0131 il\u0259 \u00f6l\u00e7m\u0259 m\u0259lumatlar\u0131n\u0131n (sinyallar\u0131n) saxlanma \u00fcsullar\u0131 d\u0259yi\u015fib. \u018fvv\u0259ll\u0259r siqnal lentli maqnitofonla lent\u0259 analoq formada yaz\u0131la bilirdis\u0259, indi siqnallar r\u0259q\u0259msalla\u015fd\u0131r\u0131l\u0131r v\u0259 komp\u00fcter yadda\u015f\u0131ndak\u0131 fayllarda r\u0259q\u0259ml\u0259r (saylar) toplusu kimi saxlan\u0131l\u0131r.<\/p>\n

Siqnal\u0131n \u00f6l\u00e7\u00fclm\u0259si v\u0259 r\u0259q\u0259msalla\u015fd\u0131r\u0131lmas\u0131n\u0131n adi sxemi a\u015fa\u011f\u0131dak\u0131 kimidir.<\/p>\n

Measuring transducer \u2014- Signal normalizer \u2014- ADC \u2014\u2013 Computer
\n(\u015e\u0259k.9 \u00d6l\u00e7m\u0259 kanal\u0131n\u0131n sxemi)<\/p>\n

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

\u00d6l\u00e7\u00fc transd\u00fcserind\u0259n g\u0259l\u0259n siqnal T m\u00fcdd\u0259ti \u0259rzind\u0259 ADC-y\u0259 \u00f6t\u00fcr\u00fcl\u00fcr. T m\u00fcdd\u0259ti \u0259rzind\u0259 al\u0131nan siqnal oxunu\u015flar\u0131 (n\u00fcmun\u0259 g\u00f6t\u00fcrm\u0259) komp\u00fcter\u0259 g\u00f6nd\u0259rilir v\u0259 yadda\u015fda saxlan\u0131l\u0131r.<\/p>\n

\"\u015e\u0259k.10

Fig.10 Digitized signal \u2013 N samples received for time T<\/p><\/div>\n

Say\u0131salla\u015fd\u0131rma parametrl\u0259rini siqnalizasiya etm\u0259k \u00fc\u00e7\u00fcn t\u0259l\u0259bl\u0259r n\u0259dir? Giri\u015f analoq siqnal\u0131n\u0131 diskret koda (say\u0131sal siqnal) \u00e7evir\u0259n qur\u011fuya analoq-say\u0131sal \u00e7evirici (ADC) deyilir (\u00a9 Wiki).<\/p>\n

One of the basic parameters of ADC is the maximum sampling rate \u2013 the frequency of sampling of a signal which is continuous in time. Sample rate is measured in hertz. ((\u00a9 Wiki))<\/p>\n

Kotelnikov teoremin\u0259 \u0259sas\u0259n, \u0259g\u0259r davaml\u0131 siqnal\u0131n spektri Fmax tezliyi il\u0259 m\u0259hdudla\u015fd\u0131r\u0131l\u0131bsa, o, zaman intervallar\u0131 \u0394t \u2264 1\/(2*Fmax) il\u0259 g\u00f6t\u00fcr\u00fclm\u00fc\u015f diskret n\u00fcmun\u0259l\u0259rind\u0259n tam v\u0259 birm\u0259nal\u0131 \u015f\u0259kild\u0259 b\u0259rpa oluna bil\u0259r, y\u0259ni Fd \u2265 2*Fmax n\u00fcmun\u0259g\u00f6t\u00fcrm\u0259 tezliyi il\u0259, burada Fd – n\u00fcmun\u0259g\u00f6t\u00fcrm\u0259 tezliyi; Fmax – siqnal spektrinin maksimum tezliyidir. Ba\u015fqa s\u00f6zl\u0259, siqnal\u0131n r\u0259q\u0259msalla\u015fd\u0131rma tezliyi (ADC-nin n\u00fcmun\u0259g\u00f6t\u00fcrm\u0259 tezliyi) \u00f6l\u00e7m\u0259k ist\u0259diyimiz siqnal\u0131n maksimum tezliyind\u0259n \u0259n az\u0131 iki d\u0259f\u0259 y\u00fcks\u0259k olmal\u0131d\u0131r.<\/p>\n

Kotelnikov teoreminin teleb etdiyinden daha asagi tezlikle numuneler goturseydik, ne bas vererdi?<\/p>\n

Bu haldal\u0259q\u0259b<\/a>In this case there is an \u201caliasing\u201d effect (aka stroboscopic effect, moir\u00e9 effect), in which a high frequency signal after digitization turns into a low frequency signal, which in fact does not exist. In Fig. 11 the red sine wave of high frequency is the real signal. The blue sine wave of lower frequency is a fictitious signal, arising due to the fact that during the sampling time has time to pass more than half a period of the high-frequency signal.<\/p>\n

\"\u015e\u0259kil

\u015e\u0259kil 11. Yetersiz y\u00fcks\u0259k n\u00fcmun\u0259 g\u00f6t\u00fcrm\u0259 tezliyind\u0259 saxta a\u015fa\u011f\u0131 tezlikli siqnal\u0131n meydana \u00e7\u0131xmas\u0131<\/p><\/div>\n

 <\/p>\n

Aliasinq effektini qar\u015f\u0131lamaq \u00fc\u00e7\u00fcn, x\u00fcsusi anti-aliasinq filtri (a\u015fa\u011f\u0131 \u00f6t\u00fcrm\u0259 filtr<\/a>) ADC-d\u0259n \u0259vv\u0259l yerl\u0259\u015fdirilir. O, ADC se\u00e7m\u0259 tezliyinin yar\u0131s\u0131ndan a\u015fa\u011f\u0131 tezlikl\u0259ri ke\u00e7ir v\u0259 daha y\u00fcks\u0259k tezlikl\u0259ri k\u0259sir.<\/p>\n

Siqnal spektrini onun diskret n\u00fcmun\u0259l\u0259ri il\u0259 hesablamaq \u00fc\u00e7\u00fcn diskret Fur'e \u00e7evrilm\u0259si (DFT)<\/a> istifad\u0259 olunur. Yen\u0259 qeyd ets\u0259k ki, diskret siqnal\u0131n spektri \u00abt\u0259rif\u0259 g\u00f6r\u0259\u00bb se\u00e7m\u0259 tezliyinin Fd yar\u0131s\u0131ndan ki\u00e7ik Fmax tezliyin\u0259 m\u0259hdudla\u015f\u0131r. Bel\u0259likl\u0259, diskret siqnal\u0131n spektri a\u015fa\u011f\u0131dak\u0131lar\u0131n c\u0259mi il\u0259 t\u0259msil oluna bil\u0259r: m\u0259hdud <\/u>number of harmonics, in contrast to the infinite sum for the Fourier series of a continuous signal, whose spectrum can be unlimited. According to Kotelnikov\u2019s theorem, the maximum frequency of a harmonic must be such that it accounts for at least two samples, so the number of harmonics is equal to half the number of samples of a discrete signal. That is, if there are N samples in the sample, the number of harmonics in the spectrum will be N\/2.<\/p>\n

\u0130ndi diskret Fourier \u00e7evrilm\u0259sini (DFT) n\u0259z\u0259rd\u0259n ke\u00e7irin.<\/p>\n

\"Diskret<\/p>\n

Onu Fourier silsil\u0259si il\u0259 m\u00fcqayis\u0259 etm\u0259k<\/p>\n

 <\/p>\n

\"Diskret<\/p>\n

G\u00f6rd\u00fcy\u00fcm\u00fcz kimi, onlar \u00fcst-\u00fcst\u0259 d\u00fc\u015f\u00fcr, yegan\u0259 f\u0259rq odur ki, FFT-d\u0259 zaman diskretdir v\u0259 harmonikalar\u0131n say\u0131 n\u00fcmun\u0259l\u0259rin say\u0131n\u0131n yar\u0131s\u0131na b\u0259rab\u0259r olan N\/2 il\u0259 m\u0259hdudla\u015f\u0131r.<\/p>\n

DFT formullar\u0131 \u00f6l\u00e7\u00fcs\u00fcz b\u00fct\u00f6v d\u0259yi\u015f\u0259nl\u0259r k v\u0259 s-d\u0259 yaz\u0131l\u0131r, burada k siqnal n\u00fcmun\u0259l\u0259rinin say\u0131n\u0131, s is\u0259 spektral komponentl\u0259rin say\u0131n\u0131 g\u00f6st\u0259rir.
\nThe value s shows the number of full harmonic oscillations per period T (signal measurement duration). The discrete Fourier transform is used to find the amplitudes and phases of harmonics numerically, i.e. \u201con the computer\u201d.<\/p>\n

Yuxar\u0131da art\u0131q qeyd edildiyi kimi, d\u00f6vr\u00fc olmayan funksiyan\u0131 (sinyal\u0131m\u0131z\u0131) Fourier c\u0259rg\u0259l\u0259rin\u0259 ay\u0131rd\u0131qda, \u0259ld\u0259 olunan Fourier c\u0259rg\u0259si \u0259slind\u0259 T d\u00f6vr\u0259 malik d\u00f6vr\u00fc funksiyaya uy\u011fundur (\u015e\u0259k.12).<\/p>\n

 <\/p>\n

\"Fig.12.

\u015e\u0259k.12. D\u00f6vr\u00fc funksiya f(x) T0 d\u00f6vr\u00fc il\u0259, T>T0 d\u00f6vr\u00fc il\u0259<\/p><\/div>\n

 <\/p>\n

\u015e\u0259kil 12-d\u0259 g\u00f6r\u00fcn\u0259 bil\u0259c\u0259yi kimi, f(x) funksiyas\u0131 T0 periodu il\u0259 periodikdir. Bununla bel\u0259, \u00f6l\u00e7\u00fc n\u00fcmun\u0259sinin uzunlu\u011fu T, funksiyan\u0131n periodu T0-a b\u0259rab\u0259r olmad\u0131\u011f\u0131 \u00fc\u00e7\u00fcn, Fur'e seriyas\u0131 kimi al\u0131nan funksiya T n\u00f6qt\u0259sind\u0259 k\u0259silm\u0259lidir. N\u0259tic\u0259 etibaril\u0259, bu funksiyan\u0131n spektri \u00e7oxlu sayda y\u00fcks\u0259k tezlikli harmonikalar ehtiva ed\u0259c\u0259kdir. Bu fenomen kimi tan\u0131n\u0131r spektral s\u0131zma<\/a>, v\u0259 praktikada p\u0259nc\u0259r\u0259l\u0259m\u0259<\/a> siqnaldan \u0259vv\u0259l. \u018fg\u0259r \u00f6l\u00e7\u00fc n\u00fcmun\u0259sinin T m\u00fcdd\u0259ti funksiyan\u0131n T0 periodil\u0259 \u00fcst-\u00fcst\u0259 d\u00fc\u015f\u00fcrd\u00fcs\u0259, Fur'e \u00e7evrilm\u0259sind\u0259n sonra al\u0131nan spektr yaln\u0131z birinci harmonikas\u0131 (n\u00fcmun\u0259nin m\u00fcdd\u0259tin\u0259 b\u0259rab\u0259r period il\u0259 sinusoid) ehtiva ed\u0259rdi, \u00e7\u00fcnki f(x) funksiyas\u0131 sinusoiddir.<\/p>\n

In other words, the DFT program \u201cdoes not know\u201d that our signal is a \u201cslice of a sine wave\u201d, but tries to represent as a series a periodic function which has a discontinuity due to the discontinuity of separate pieces of the sine wave.<\/p>\n

N\u0259tic\u0259d\u0259 spektrd\u0259 harmonikalar peyda olur ki, onlar \u00fcmumilikd\u0259 bu diskontinuiteti d\u0259 daxil etm\u0259kl\u0259 funksiyan\u0131n formas\u0131n\u0131 t\u0259msil etm\u0259lidir.<\/p>\n

Thus, to get a \u201ccorrect\u201d spectrum of a signal which is a sum of several sinusoids with different periods, it is necessary that an d\u00f6vrl\u0259rin tam \u0259d\u0259di <\/u>H\u0259r bir sinusoid siqnal\u0131n \u00f6l\u00e7m\u0259 d\u00f6vr\u00fcnd\u0259 m\u00f6vcud olmal\u0131d\u0131r. Praktikada bu \u015f\u0259rt siqnal\u0131n kifay\u0259t q\u0259d\u0259r uzun m\u00fcdd\u0259t \u00f6l\u00e7\u00fclm\u0259si il\u0259 t\u0259min edil\u0259 bil\u0259r.<\/p>\n

 <\/p>\n

\"\u015e\u0259k.13

\u015e\u0259k.13 S\u00fcr\u0259t qutusunun kinematik s\u0259hv siqnal funksiyas\u0131 v\u0259 spektrinin n\u00fcmun\u0259si<\/p><\/div>\n

 <\/p>\n

At shorter duration the picture will look \u201cworse\u201d:<\/p>\n

 <\/p>\n

\"\u015e\u0259kil

\u015e\u0259kil 14. Rotorun vibrasiya funksiyas\u0131 v\u0259 spektrinin n\u00fcmun\u0259si<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

In practice, it can be difficult to understand where the \u201creal components\u201d and where the \u201cartifacts\u201d caused by inconsistency of component periods and signal sampling durations or \u201cjumps and breaks\u201d in the waveform. Of course, the words \u201creal components\u201d and \u201cartifacts\u201d are put in quotes for a reason. The presence of many harmonics on the spectrum graph does not mean that our signal actually consists of them. It is like thinking that number 7 \u201cconsists\u201d of numbers 3 and 4. The number 7 can be thought of as the sum of 3 and 4 \u2013 that is correct.<\/p>\n

So also our signal\u2026 or rather not even \u201cour signal\u201d, but a periodic function composed by repeating our signal (sample) can be represented as a sum of harmonics (sine waves) with certain amplitudes and phases. But in many cases important for practice (see figures above) it is indeed possible to relate the harmonics obtained in the spectrum also to real processes having cyclic character and contributing significantly to the form of the signal.<\/p>\n

B\u0259zi n\u0259tic\u0259l\u0259r<\/h2>\n

1. ADC t\u0259r\u0259find\u0259n T saniy\u0259 m\u00fcdd\u0259ti \u00fc\u00e7\u00fcn r\u0259q\u0259msalla\u015fd\u0131r\u0131lm\u0131\u015f real \u00f6l\u00e7\u00fclm\u00fc\u015f siqnal, y\u0259ni N \u0259d\u0259d diskret n\u00fcmun\u0259l\u0259r toplusu \u015f\u0259klind\u0259 t\u0259qdim olunan siqnal, diskret, d\u00f6vri olmayan spektr\u0259 malikdir v\u0259 bu spektr harmonikl\u0259r toplusu (N\/2 \u0259d\u0259d) il\u0259 t\u0259svir olunur.<\/p>\n

2. Siqnal real d\u0259y\u0259rl\u0259r toplusu il\u0259 t\u0259qdim olunur. Onun DFT spektri qo\u015fma simmetriyaya malik kompleks \u0259msallar toplusudur; onlardan amplituda spektri \u0259ld\u0259 olunur \u2014 m\u00fcsb\u0259t tezlikl\u0259rd\u0259 real, m\u0259nfi olmayan amplitudalar (v\u0259 fazalar) toplusu. Praktikada m\u0259hz bu birt\u0259r\u0259fli amplituda spektri g\u00f6st\u0259rilir. M\u0259nfi tezlikl\u0259ri olan iki t\u0259r\u0259fli kompleks forma il\u0259 birt\u0259r\u0259fli amplituda\/faza formas\u0131 eyni spektrin ekvivalent t\u0259qdimatlar\u0131d\u0131r \u2014 siqnal analizi \u00fc\u00e7\u00fcn ad\u0259t\u0259n birt\u0259r\u0259fli amplituda spektri il\u0259 i\u015fl\u0259m\u0259k daha rahatd\u0131r.<\/p>\n

3. The signal measured at time T is determined only at time T. What happened before we started measuring the signal and what will happen after that is unknown to science. And in our case it is not interesting. FFT of the time-limited signal gives its \u201creal\u201d spectrum, in the sense that under certain conditions it allows to calculate the amplitude and frequency of its components.<\/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\/az\/wp-json\/wp\/v2\/posts\/2138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/comments?post=2138"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/posts\/2138\/revisions"}],"predecessor-version":[{"id":102137,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/posts\/2138\/revisions\/102137"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/media\/2161"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/categories?post=2138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/az\/wp-json\/wp\/v2\/tags?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}