Ανάλυση. σήματα και συστήματα

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1 Σήματα Συστήματα Ανάλυση ourier διακριτού χρόνου Κωνσταντίνος Κοτρόπουλος Τμήμα Πληροφορικής για σήματα και συστήματα Θεσσαλονίκη, Ιούνιος 23

2 Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό, όπως εικόνες, που υπόκειται σε άλλου τύπου άδειας χρήσης, η άδεια χρήσης αναφέρεται ρητώς. Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί στα πλαίσια του εκπαιδευτικού έργου του διδάσκοντα. Το έργο «Ανοικτά Ακαδημαϊκά Μαθήματα στο Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης»» έχει χρηματοδοτήσει μόνο τη αναδιαμόρφωση του εκπαιδευτικού υλικού. Το έργο υλοποιείται στο πλαίσιο του Επιχειρησιακού Προγράμματος «Εκπαίδευση και Δια Βίου Μάθηση»» και συγχρηματοδοτείται από την Ευρωπαϊκή Ένωση (Ευρωπαϊκό Κοινωνικό Ταμείο) και από εθνικούς πόρους. Θεσσαλονίκη, Ιούνιος 23

3 Kef laio 7 An lush ourier gia s mata kai sust mata diakritoô qrìnou H an lush ourier suneqoôc qrìnou (S.Q.) mac dðnei thn eukairða na katano soume tic i- diìthtec shm twn/susthm twn S.Q. AntikeÐmeno tou parìntoc kefalaðou eðnai h an lush ourier diakritoô qrìnou (D.Q.). H diapragm teush tou jèmatoc anaptôssetai parall lwc proc th melèth shm twn/susthm twn S.Q. Ta ergaleða pou ja melet soume èqoun tic dikèc touc diakritèc rðzec. Oi mèjodoi kai oi ènnoiec D.Q. eðnai jemeli deic sthn arijmhtik an - lush. 'Ontwc arijmhtikèc mèjodoi gia parembol, olokl rwsh kai diafìrish se akoloujðec arijm n rqisan na melet ntai apì ton eôtwna sta 6. H prìbleyh thc kðnhshc ouranðwn swm twn dosmènhc miac seir c parathr sewn kèntrise thn èreuna ton 8o kai 9o ai nec. MporoÔme na isquristoôme ìti Oi mèjodoi S.Q. apantoôntai sth fusik, sthn an lush hlektrik n kuklwm twn. Oi mèjodoi D.Q. apantoôntai sthn arijmhtik an lush, sthn an lush qronoseir n (p.q. oikonomik prìbleyh, an lush dhmografik n dedomènwn, prìbleyh exèlixhc fusik n fainomènwn). Ston 2o ai na, stic dekaetðec twn >4 kai >5 parathreðtai anagènnhsh twn teqnik n D.Q. kai qr sh thc an lushc ourier D.Q. Lìgoi pou sunèbalan sthn anagènnhsh aut eðnai: h qr sh yhfiak n upologist n gia ton upologismì metasqhmatism n ourier h sqedðash susthm twn D.Q. gia thn epexergasða arijmhtik n dedomènwn pou proèrqontai apì deigmatolhyða shm twn S.Q. ìpwc

4 2 K. Kotrìpouloc: S mata-sust mata yhfiakoð analutèc fwn c yhfiakoð analutèc f smatoc. Sta mèsa thc dekaetðac tou >6 <<anakalôptetai>> o Gr goroc Metasqhmatismìc ourier (ast ourier Transform) T pou eðnai kat llhloc gia apodotikèc yhfiakèc ulopoi seic el ttwse to qrìno upologismoô kat pollèc t xeic megèjouc apì O( 2 ) se O( log 2 ). Up rqoun pollèc omoiìthtec me thn an lush S.Q.: E n h eðsodoc kai èxodoc enìc G.Q.A. sust matoc D.Q. ekfrastoôn san grammikoð sunduasmoð migadik n ekjetik n, tìte oi suntelestèc thc anapar stashc thc exìdou mporoôn na ekfrastoôn se mia polô bolik morf sunart sei twn suntelest n thc anapar stashc thc eisìdou. Mia eureða kai qr simh om da shm twn mporeð na anaparastajeð san tètoioc grammikìc sunduasmìc. Up rqoun wstìso kai orismènec diaforèc: Antijètwc proc th seir apeðrwn ìrwn pou prokôptei sthn anapar stash me seir ourier periodik n shm twn S.Q., h anapar stash seir c ourier enìc periodikoô s matoc D.Q. eðnai peperasmènh. AxiopoÐhsh thc idiìthtac aut c gðnetai ston T. Ja orðsoume dôo metasqhmatismoôc ourier: to metasqhmatismì ourier D.Q. (Discrete-Time ourier Transform), T-DT kai to diakritì metasqhmatismì ourier (Discrete ourier Transform), DT. O metasqhmatismìc ourier D.Q. prokôptei apì th diakrit seir ourier apeirðzontac thn perðodo deigmatolhyðac, ìpwc akrib c proèkuye o metasqhmatismìc ourier S.Q. apì th seir ourier S.Q. All h diadikasða aut katal gei s> èna metasqhmatismì suneqoôc metablht c, pr gma bolo ìtan epexergazìmaste akoloujðec arijm n. Gia na jerapeôsoume aut th duskolða, kataskeu zoume to diakritì metasqhmatismì ourier pou apoteleð deigmatolhyða tou metasqhmatismoô ourier D.Q. se <<suqnìthtec>> pou antistoiqoôn se deðgmata <<suqnìthtac>> ta opoða prokôptoun apì omoiìmorfh

5 K. Kotrìpouloc: S mata-sust mata 3 deigmatolhyða tou diast matoc [, ). JumhjeÐte ìti oi <<suqnìthtec>> twn shm twn D.Q. eðnai gwnðec. An sumbolðsoume me Ω th <<suqnìthta>> D.Q., aut antistoiqeð sthn analogik suqnìthta ω, th gnwst mac kuklik suqnìthta (pou metriètai se rad/sec), sômfwna me th sqèsh Ω=ωT (7.) ìpou T eðnai h perðodoc deigmatolhyðac. 'Eqontac kat nou thn (7.) mporoôme na miloôme eurôtera gia suqnìthtec Ω qwrðc eisagwgik efex c. Ta omoiìmorfa deðgmata suqnìthtac den eðnai par oi timèc Ω k = k, k =,,...,. (7.2) QwrÐc kami aujairesða isqurizìmaste ìti o diakritìc metasqhmatismìc ourier, DT,den eðnai par mia nìja diakrit seir ourier. ApodotikoÐ algìrijmoi upologismoô tou DT eðnai oi algìrijmoi T. 7. Apìkrish G.Q.A. susthm twn diakritoô qrìnou se migadik ekjetik Gia na ex goume thn anapar stash seir c ourier sto D.Q. prèpei na anaptôxoume periodik s mata D.Q. san grammikoôc sunduasmoôc migadik n ekjetik n D.Q. Ja deðxoume ìti ta migadik ekjetik D.Q. eðnai idiosunart seic twn G.Q.A. susthm twn D.Q. Upojèste ìti èna G.Q.A. sôsthma D.Q. me kroustik apìkrish h[n] diegeðretai apì eðsodo x[n] =z n. (7.3) H èxodoc tou G.Q.A. sust matoc D.Q. ja dðnetai apì to jroisma thc sunèlixhc + [ + y[n] =(x h)[n] = h[k] x[n k] = h[k] z n k = z n ] h[k] z k k= k= k= }{{ } H(z) = H(z) }{{} z n (7.4) idiotim ìpou H(z) = + k= h[k] z k eðnai h idiotim pou antistoiqeð sthn idiosun rthsh z n. H exðswsh (7.4) mazð me thn idiìthta thc upèrjeshc upodhl noun ìti h anapar stash thc eisìdou +

6 4 K. Kotrìpouloc: S mata-sust mata wc grammikìc sunduasmìc migadik n ekjetik n odhgeð s> èna bolikì trìpo anapar stashc thc exìdou me th qr sh migadik n ekjetik n. Ja perioristoôme, kat' arq n, se migadik ekjetik thc morf c z n = e jωn (7.5) dhlad tètoia me z =. S mata thc morf c (7.5) eðnai fantastik ekjetik. Ja melet soume thn epèktash se seir ourier twn periodik n shm twn D.Q. (diakrit seir ourier) to metasqhmatismìc ourier D.Q. wc epèktash thc diakrit c seir c ourier. H melèth apoblèpei sthn an deixh twn omoiìthtwn kai ton entopismì twn diafor n metaxô twn ergaleðwn D.Q. kai twn antistoðqwn ergaleðwn S.Q. 7.2 Diakrit Seir ourier 'Ena s ma D.Q. eðnai periodikì ìtan Z + : x[n] =x[n + ]. (7.6) To s ma e j n eðnai periodikì s ma D.Q. me perðodo, epeid e j (n+) = e j n e j = e j n. (7.7) Ta fantastik ekjetik me perðodo dðnontai apì th sqèsh φ k [n] =e jk n. (7.8) Ta s mata aut èqoun suqnìthtec pou eðnai pollapl siec thc jemeli douc suqnìthtac kai epomènwc eðnai armonikèc. En ìla ta s mata S.Q. φ k (t) =e jkω t (7.9) eðnai diakekrimèna, up rqoun mìno diaforetik s mata sto sônolo {φ k [n] =e jkωn }, e- peid ta fantastik ekjetik pou diafèroun sth suqnìthta kat pollapl sia tou eðnai tautìshma. 'Ontwc e j(ω+r)n = e jωn e jrn = e jωn (7.)

7 K. Kotrìpouloc: S mata-sust mata 5 φ k+r [n] =e j(k+r) n = e jk n e jrn = φ k [n]. (7.) Epomènwc to k prèpei na metab lletai se di sthma tim n eôrouc, p.q. k =,,..., k =3, 4,..., +2, k.o.k. 'Etsi to jroisma sthn epèktash se seir ourier ja prèpei na perioristeð se prosjetèouc x[n] = a k φ k [n] = a k e jk n. (7.2) k=<> k=<> H (7.2) orðzei th diakrit seir ourier, ìpou a k eðnai oi suntelestèc thc seir c ourier Prosdiorismìc twn suntelest n thc diakrit c seir c ourier Prìblhma: Dojèntoc enìc periodikoô s matoc D.Q. x[n] me perðodo na prosdioristoôn oi suntelestèc thc diakrit c seir c ourier a k ste x[n] = k=<> a k φ k [n], φ k [n] =e jk n. (7.3) To prìblhma isodunameð me eôresh thc lôshc tou sunìlou twn grammik n exis sewn x[] = k=<> x[] = k=<> a k a k e j k. (7.4) x[ ] = a k e j k( ) k=<> gia diadoqikèc timèc tou n, n =,,...,. sôsthma exis sewn me agn stouc a k, gia k =< >. To sôsthma twn exis sewn (7.4) eðnai MporeÐ na deiqjeð ìti oi exis seic eðnai grammik c anex rthtec (apodeðxte to), ra to sôsthma èqei mða kai monadik lôsh wc proc a k. Sth sunèqeia ja deðxoume ìti mporeð na upologisteð mia kleist sqèsh gia touc suntelestèc a k me ìrouc twn deigm twn x[n], ste na mh qrei zetai na katafeôgoume se epðlush sust matoc exis sewn. Proc toôto bohj h tautìthta: n= { k =, ±,±2,... e jk n = alloô (7.5)

8 6 K. Kotrìpouloc: S mata-sust mata dhlad, to jroisma twn tim n enìc fantastikoô ekjetikoô se di sthma miac periìdou eðnai mhdèn, ektìc an to fantastikì ekjetikì eðnai stajer. Ac prospaj soume na ermhneôsoume gewmetrik thn (7.5) gia =6me th bo jeia tou Sq matoc 7.. EÔkola gðnetai antilhptì ìti to jroisma ìlwn twn fasik n dianusm twn eðnai mhdèn ektìc an k =, 6, 2,...Gia k je n o grammikìc sunduasmìc twn fasik n dianusm twn (èna apì k je gr fhma) me touc suntelestèc a k dðnei to x[n], dhlad x[n] = k=<> a ke jk n. Parathr ste ìti o lìgoc e jk (n+) e jk n = e jk (7.6) den exart tai apì to n, opìte èqoume jroisma ìrwn gewmetrik c proìdou. GnwrÐzoume ìti to jroisma ìrwn gewmetrik c proìdou me lìgo a dðnetai apì thn Gia k =, ±,±2... èqoume n= a n = { a= a a a. (7.7) en en gènei > n= e jk n = e jk e jk e j kn = (7.8) k =, ±,±2,... = alli c. (7.9) Ac pollaplasi soume kai ta dôo mèrh thc (7.3) me e jr n kai ac ajroðsoume gia n =< n=<> x[n] e jr n = a k k=<> n=<> n=<> e j(k r) n = e jr n k=<> a k e jk n = { ar k r = λ, λ Z alli c. (7.2) Dialègoume to k na metab lletai se di sthma tim n eôrouc pou perièqei to r. Dhlad, k r< λ =. (7.2) k r = λ All λ =sunep getai k = r. Epomènwc to dexð mèroc thc (7.2) eðnai mh-mhdenikì gia k = r, en mhdenðzetai gia k r. Opìte lônontac wc proc a k paðrnoume a k = x[n] e jk n k =,...,. (7.22) n=<>

9 K. Kotrìpouloc: S mata-sust mata 7 Im Im n=2 n= n=,4 n=3 2 o 6 o n= n=,3 Re Re n=4 n=5 (a) e jk( 6 )n k= = e j( 6 n) n=2,5 (b) e jk( 6 )n k=2 = e j 3 n Im Im n=2,5 n=,3,5 8 o 24 o n=,2,4 n=,3 Re Re n=,4 (g) e jk( 6 )n k=3 = e jπn (d) e jk( 6 )n k=4 = e j 4π 3 n Im Im n=4 n=5 n=3 6 o n= Re 36 o n=,,2,...,5 Re n=2 n= (e) e jk( 6 )n k=5 = e j5 6 n (st) e jk( 6 )n k=6 = e jn Sq ma 7.: Gia na upologisteð to deðgma x[n] arkeð na sqhmatðsoume to grammikì sunduasmì twn fasik n dianusm twn φ k [n] =e jk 6 n me suntelestèc a k, gia k =< 6 >=, 2,...,6.

10 8 K. Kotrìpouloc: S mata-sust mata SunoyÐzoume ìti h h exðswsh sônjeshc thc diakrit c seir c ourier eðnai x[n] = en h exðswsh an lushc dðnetai apì thn a k = k=<> n=<> ìpou a k eðnai oi fasmatikoð suntelestèc thc akoloujðac x[n]. a k e jk n (7.23) x[n] e jk n (7.24) Shmantik parat rhsh: 'Estw k {,,..., } sthn exðswsh sônjeshc. Tìte x[n] =a φ [n]+a φ [n]+ + a φ [n]. (7.25) An t ra dialègame k {, 2,..., } tìte eðnai exðsou ègkurh h epèktash x[n] =a φ [n]+ + a φ [n]+a φ [n] (7.26) opìte prèpei kai arkeð a φ [n] =a φ [n]. (7.27) All φ [n] =φ [n] (7.28) opìte a = a kai genikìtera a k = a k+. (7.29) Dhlad, oi suntelestèc a k epanalamb nontai periodik me perðodo. Sunep c. H anapar stash seir c ourier D.Q. eðnai peperasmènh kai apoteleðtai apì ìrouc. 2. H (7.23) orðzetai wc jroisma se opoiod pote aujaðreta epilegmèno sônolo diadoqik n tim n tou k. Par deigma 7.. 'Estw x[n] =sinω n. (7.3) Up rqoun treic diaforetikèc peript seic pou exart ntai apì to an o lìgoc Ω akèraioc eðnai lìgoc akeraðwn

11 K. Kotrìpouloc: S mata-sust mata 9 rrhtoc arijmìc. H anapar stash seir c ourierautoô tou s matoc orðzetai mìno stic pr tec duo peript seic. Gia thn perðptwsh pou o lìgoc Ω eðnai akèraioc, dhlad ìtan Ω = (7.3) to x[n] eðnai periodikì s ma me perðodo. Tìte x[n] = 2j (ej n e j n ) (7.32) ra a = a = 2j kai oi upìloipoi suntelestèc eðnai mhdenikoð. Apì tic idiìthtec twn suntelest n prokôptei ìti a + = 2j kai a = 2j opìte sthn perðptws mac oi suntelestèc ja eðnai ìpwc sto Sq ma 7.2. Mìno mia perðodoc a,a,...,a qrhsimopoieðtai sthn exðswsh sunjèsewc. 2j ak 2j k Sq ma 7.2: Suntelestèc diakrit c seir c ourier tou s matoc x[n] = sin n. Gia thn perðptwsh pou = Ω m Ω = m (7.33) ìpou kai m den èqoun koinoôc par gontec, to s ma eðnai p li periodikì me perðodo, opìte epekteðnetai se seir ourier: x[n] = 2j ejm n 2j e jm n. (7.34) 'Ara a m = 2j, a m = 2j kai oi upìloipoi suntelestèc se di sthma miac periìdou eðnai mhdèn. L.q. gia =5kai m =3èqoume a = a =, a 5 3 = a 2 = 2j, a 3 = 2j, a 4 =.

12 K. Kotrìpouloc: S mata-sust mata Par deigma 7.2. 'Estw periodikì s ma me perðodo x[n] = + sin n +3cos n +cos(4π n + π ). (7.35) 2 Gia na upologisteð h diakrit seir ourier arkeð na efarmosteð h tautìthta tou Euler x[n] = + ] [e j n e j n +3 ] [e j n + e j n + ] [e j(2 n+ π 2 ) + e j(2 n+ π 2 ) 2j 2 2 = +( j )ej n +( 3 2 2j )e j n + j 2 ej2 n j 2 e j2 n. (7.36) AnagnwrÐzoume ìti a = a = 3 2 j 2 a = a a 2 = j 2 a 2 = a 2 (7.37) en oi upìloipoi suntelestèc eðnai mhdenikoð. Genikìtera gia k je pragmatik akoloujða x[n] isqôei a k = a k. (7.38) H diakrit seir ourier èqei tic idiìthtec pou parallhlðzontai proc tic antðstoiqec idiìthtec thc seir c ourier suneqoôc qrìnou kai sunoyðzontai ston PÐnaka 7.. Par deigma 7.3. 'Estw h periodik tetragwnik palmoseir D.Q. di rkeiac 2 + deigm twn tou Sq matoc 7.3. Sq ma 7.3: Periodik tetragwnik palmoseir di rkeiac 2 +deigm twn kai periìdou. Oi suntelestèc thc diakrit c seir c ourier dðnontai apì thn a k = n= e jk n. (7.39)

13 K. Kotrìpouloc: S mata-sust mata PÐnakac 7.: Idiìthtec thc diakrit c seir c ourier. Idiìthta Periodikì S ma Suntelestèc seir c x[n] y[n] periodik me perðodo kai jemeli dh suqnìthta Ω = ourier a k b k Grammikìthta Ax[n]+By[n] Aa k + Bb k Qronik metatìpish x[n n ] a k e jk n Metatìpish suqnìthtac e jm n x[n] a k M SuzugÐa x [n] a k Qronik anastrof x[ n] a k Periodik sunèlixh Pollaplasiasmìc r=<> x[n] y[n] x[r]y[n r] a Qronik klim kwsh x (m) [n] m a k (periodikì me perðodo m) k b k l=<> a l b k l Suzug c summetrða gia pragmatik s mata Pragmatik s mata rtiac summetrðac Pragmatik s mata peritt c summetrðac Pr th Diafor x[n] x[n ] e jk a k n Trèqon 'Ajroisma k= x[k] (peperasmènhc tim c kai ( ) a e jk k periodikì mìno an a =) x[n] R x[n] R: x[n] =x[ n] x[n] R: x[n] = x[ n] AposÔnjesh se rtio kai perittì mèroc pragmatikoô s matoc a k = a k Re{a k } =Re{a k } Im{a k } = Im{a k } a k = a k a k = a k a k pragmatikoð kai rtiac summetrðac a k kajar c fantastikoð kai peritt c summetrðac x e [n] = (x[n]+x[ n]) Re{a 2 k } x o [n] = (x[n] x[ n]) jim{a 2 k } Tautìthta Parseval gia periodik s mata n=<> x[n] 2 = k=<> a k 2

14 2 K. Kotrìpouloc: S mata-sust mata K noume thn allag metablht c m = n +, opìte an k, ±,±2,... a k = 2 m= = ejk = ejk e jk (m ) = ejk e jk (2 +) e jk e jk jk e 2 2 m= e jk m (e jk e jk ) ( jk e 2 jk e 2 = exp { jk[ ] = An k =, ±,±2,... tìte ) } 2j sin(k 2 + ) 2 2j sin( k ) 2 sin(k ) sin( k 2 ). (7.4) a k = 2 +. (7.4) H èkfrash (7.4) gia touc suntelestèc thc seir c ourier gr fetai pio sunoptik a k = sin (2 +) Ω 2 sin Ω 2 Ω= k (7.42) opìte oi suntelestèc thc seir c ourier anagnwrðzontai wc deðgmata thc perib llousac thc sun rthshc suneqoôc metablhthc Ω sin(2 +) Ω 2 sin Ω 2 (7.43) pou lamb nontai me omoiìmorfh deigmatolhyða tou diast matoc tim n [, ) thc metablht c Ω. To Sq ma 7.4 deðqnei parastatik touc suntelestèc thc seir c ourier gia periodikì tetragwnikì palmì 5 mh-mhdenik n deigm twn kai periìdou. An h perðodoc tou tetragwnikoô palmoô auxhjeð se =2, tìte ja prokôyei piì pukn deigmatolhyða thc perib llousac (7.43), ìpwc faðnetai sto Sq ma 7.5. H perib llousa tou Sq matoc 7.5 èqei thn Ðdia morf m> aut tou Sq matoc 7.4, all to misì Ôyoc ekeðnhc. Ac sugkrðnoume th seir ourier tou periodikoô tetragwnikoô palmoô diakritoô qrìnou me thn antðstoiqh seir ourier tou periodikoô tetragwnikoô palmoô suneqoôc qrìnou. Oi suntelestèc thc seir c ourier suneqoôc qrìnou tan ParathroÔme ìti: a k = sin kω T kπ = ω T π sinc(kω T ), sinc(x) = sin x x. (7.44)

15 K. Kotrìpouloc: S mata-sust mata a k π Ω Sq ma 7.4: Suntelestèc a k apì th deigmatolhyða thc perib llousac (7.43) gia =kai 2 +=5. a k π Ω Sq ma 7.5: Suntelestèc a k apì th deigmatolhyða thc perib llousac (7.43) gia =2. Sth diakrit seir ourier h sunarthsiak morf thc perib llousac eðnai, me thn eureða ènnoia, p li tôpou sinc, all me k pwc diaforetik orismènh th sun rthsh sinc(x) aut th for. H apaðthsh gia periodik seir suntelest n odhgeð sthn tropopoðhsh tou orismoô thc sun rthshc sinc(x) se sinc(x) = sin βx sin x. (7.45) H seir ourier S.Q. eggu tai th bèltisth anakataskeu tou periodikoô s matoc S.Q. an p roume peirouc ìrouc sthn epèktash. Gia na mei soume ta l jh anakataskeu c, kai epomènwc na sugklðnei h seir, arkoôse na paðrnoume oloèna kai perissìterouc ìrouc sthn epèktash. All me thn aôxhsh tou arijmoô twn ìrwn parathroôsame to fainìmeno Gibbs stic asunèqeiec. H melèth thc antðstoiqhc perðptwshc sta s mata D.Q. katadeiknôei ìti den up rqoun probl mata sôgklishc oôte fainìmeno Gibbs.

16 4 K. Kotrìpouloc: S mata-sust mata Pr gmati; gia periodikì s ma D.Q. x[n] me perðodo èstw to anakataskeuasmèno periodikì s ma D.Q. gia arketèc timèc tou M. M ˆx[n] = a k e jk n (7.46) k= M Ac jewrhjeð ìti eðnai perittìc arijmìc, l.q. =9. Tìte mporeð na deiqteð ìti gia M =4h anakataskeu eðnai tèleia. 'Ara den up rqoun probl mata sôgklishc oôte fainìmeno Gibbs. ToÔto ofeðletai sto gegonìc ìti h periodik akoloujða D.Q. prosdiorðzetai apì peperasmèno arijmì, paramètrwn, tic timèc thc akoloujðac sto di sthma miac periìdou. H exðswsh an lushc thc seir c ourier metasqhmatðzei autì to sônolo twn paramètrwn se èna isodônamo sônolo, tic timèc twn suntelest n ourier kai h exðswsh sônjeshc mac lèei p c na anakataskeu soume to arqikì s ma diakritoô qrìnou. Epomènwc gia perittì, an p roume M = 2 ìrouc, tìte to jroisma perièqei akrib c ìrouc kai ˆx[n] =x[n]. An eðnai rtioc, arkeð na upologðsoume to anakataskeuasmèno periodikì s ma D.Q. mèsw thc M ˆx[n] = a k e jk n (7.47) gia M =, opìte p li ˆx[n] =x[n]. 2 k= M+ Par deigma 7.4. 'Estw h akìloujh plhroforða gia thn akoloujða x[n]:. H x[n] eðnai periodik me perðodo = n= x[n] = n=2 ( )n x[n] =. 4. H x[n] èqei thn el qisth isqô (enèrgeia an perðodo), ìtan ikanopoioôntai oi sqèseic -3. a prosdiorðsete thn akoloujða x[n]. Apì thn tautìthta tou Parseval h isqôc thc akoloujðac eðnai P = 5 a k 2. (7.48) k=

17 K. Kotrìpouloc: S mata-sust mata 5 Apì thn plhroforða 2 èqoume 5 x[n] =2 a = n= n= Apì thn plhroforða 3 antloôme ìti opìte a 3 = 6 x[n]e jk n =6,k= = 6 7 ( ) n x[n] = n=2 5 x[n]e j3 6 n = 6 n= 5 ( ) n x[n] n= 5 x[n](e jπ ) n = 6 n= 5 x[n] = 2 6 = 3. (7.49) n= 5 ( ) n x[n] = 6. (7.5) 'Ara apì tic sqèseic (7.49) kai (7.5) prosdiorðsthkan oi suntelestèc a kai a 3. n= H isqôc kajðstatai el qisth mìno ìtan oi upìloipoi suntelestèc eðnai mhdenikoð, dhlad a = a 2 = a 4 = a 5 =. Opìte x[n] =a + a 3 e jπn = ( )n. (7.5) Seir ourier kai G.Q.A. sust mata An to periodikì s ma D.Q. pou diegeðrei èna G.Q.A. sôsthma analujeð se seir ourier D.Q. tìte x[n] = tìte h èxodoc tou sust matoc ja eðnai: ìpou y[n] =(x h)[n] = k=<> ξ= = k=<> = k=<> H( k )= h[ξ] { a k a k e jk n (7.52) k=<> ξ= a k e jk (n ξ) h[ξ] e jk ξ }{{} k H(Ω) Ω= =H( k ) } e jk n a k H( k ) ejk n (7.53) ξ= h[ξ] e jk ξ (7.54) eðnai h tim thc apìkrishc suqnìthtac gia Ω= k. Wc apìkrish suqnìthtac enìc G.Q.A. sust matoc D.Q. orðzoume th sun rthsh H(Ω) = h[n] e jωn. (7.55)

18 6 K. Kotrìpouloc: S mata-sust mata Epomènwc h èxodoc tou G.Q.A. sust matoc eðnai epðshc periodik me perðodo aut n thc diegèrsewc x[n]. Apì thn (7.53) sun getai ìti oi suntelestèc thc diakrit c seir c ourier thc exìdou eðnai b k = a k H( k ). (7.56) H èkfrash (7.53) èqei nìhma ìtan h apìkrish suqnìthtac eðnai kal orismènh kai fragmènh. Tètoiec kal c sumperiferìmenec apokrðseic suqnìthtac èqoun ta eustaj sust mata p.q. h[n] =a n u[n] me a <. An a >, tìte to sôsthma kajðstatai astajèc kai den orðzetai apìkrish suqnìthtac. Par deigma 7.5. 'Estw h[n] = a n u[n] <a< (7.57) x[n] = cos n = 2 ej n + 2 e j n (7.58) tìte H( k )= a n e jk n = n= n= H èxodoc tou G.Q.A. sust matoc eðnai (ae jk ) n = ae jk. (7.59) An ekfr soume ton par gonta y[n] = 2 H( ) ej n + 2 H( = e j 2 ae j n + 2 ae j ) e j n ae j se polikèc suntetagmènec, dhlad ae j e j n. (7.6) = re jθ (7.6) tìte prokôptei ìti y[n] =r cos ( n + θ). (7.62) iltr risma Se pollèc efarmogèc mac endiafèrei na metab lloume ekousðwc ta sqetik pl th twn suqnotik n sunistws n enìc s matoc, dhlad ta pl th twn fasmatik n suntelest n thc diakrit c seir c ourier tou s matoc, apaleðfontac k poiec suqnotikèc sunist sec enisqôontac k poiec llec. Ta G.Q.A. sust mata pou all zoun th morf tou f smatoc enìc s matoc kaloôntai G.Q.A. fðltra. Sto D.Q., ta G.Q.A. fðltra brðskoun eureðec efarmogèc. Sun jwc

19 K. Kotrìpouloc: S mata-sust mata 7 ulopoioôntai me epexergastèc genikoô eidikoô skopoô gia na epexergastoôn s mata S.Q. pou èqoun uposteð deigmatolhyða (p.q., omilða) qronoseirèc, ìpwc dhmografik dedomèna, timèc qrhmatisthriak n deikt n k.o.k. To aploôstero fðltro D.Q. eðnai to G.Q.A. sôsthma pou upologðzei ton arijmhtikì mèso twn deigm twn thc eisìdou y[n] = k= x[n k]. (7.63) Den eðnai dôskolo na ex goume thn kroustik apìkrish tou arijmhtikoô mèsou h[n] = k= δ[n k]. (7.64) H apìkrish suqnìthtac tou arijmhtikoô mèsou mporeð na deiqjeð ìti eðnai H(Ω) = e jω 2 sin Ω 2. sin Ω (7.65) Anapar stash mh-periodik n shm twn: O metasqhmatismìc ourier diakritoô qrìnou Oi suntelestèc thc seir c ourier gia periodik s mata eðnai deðgmata miac perib llousac kai kaj c h perðodoc thc akoloujðac aux nei, tìte ta deðgmata pukn noun kai to di sthma metaxô touc smikrônetai. Gia ta s mata S.Q. o metasqhmatismìc ourier enìc mh-periodikoô s matoc x(t) proèkuye apì to periodikì s ma x(t) pou èqei wc pr th perðodo to dosmèno s ma. Sto ìrio kaj c T prokôptei ìti x(t) x(t), en h seir ourier S.Q. teðnei sto metasqhmatismì ourier S.Q. Ja akolouj soume an logh prosèggish. XekinoÔme apì mða mh-periodik akoloujða x[n] peperasmènhc di rkeiac, dhlad, 2 : x[n] =, n / [, 2 ] (7.66) ìpwc gia to s ma sto Sq ma 7.6. Kataskeu zoume to s ma x[n] pou èqei wc pr th perðodo to s ma x[n] kai kat llhlh perðodo. Gia, parathroôme ìti x[n] x[n] gia k je peperasmèno n. All to x[n] wc periodikì s ma epekteðnetai se seir ourier D.Q.: x[n] = a k e jk n (7.67) k=<>

20 8 K. Kotrìpouloc: S mata-sust mata x[n] 2 n Sq ma 7.6: Mh-periodikì s ma D.Q. x[n]. ìpou oi suntelestèc thc seir c ourier dðnontai apì thn a k = x[n] e jk n. (7.68) n=<> Epeid x[n] =x[n] gia n [, 2 ] kai lìgw thc (7.66) h (7.68) xanagr fetai wc a k = OrÐzoume thn perib llousa 2 x[n] e jk n = n= + x[n] e jk n. (7.69) X(Ω) = X(e jω )= tìte oi suntelestèc a k dðnontai apì th + x[n] e jωn (7.7) a k = X(Ω) Ω= k = X(kΩ ), me Ω = kai k =< >. (7.7) Epomènwc, oi suntelestèc a k eðnai an logoi proc isapèqonta deðgmata thc perib llousac. Antikajist ntac sthn (7.67) kai k nontac qr sh thc sqèshc = Ω (7.72) paðrnoume x[n] = 'Otan, tìte k=<> X(kΩ )e jkωn = k=<> X(kΩ ) e jkω n Ω. (7.73) x[n] x[n] gia n [, 2 ] (7.74) Ω dω (7.75) kω Ω suneq c metablht (7.76) dω (7.77) k=<>

21 K. Kotrìpouloc: S mata-sust mata 9 kai h (7.73) ermhneôetai wc arijmhtikìc upologismìc tou oloklhr matoc sthn èkfrash x[n] = X(Ω) e jωn dω (7.78) me qr sh tou kanìna tou trapezðou. Mènei na sqoli soume giatð h olokl rwsh sthn (7.78) prèpei na ekteðnetai se di sthma eôrouc. Profan c h perib llousa X(Ω) wc grammikìc sunduasmìc fantastik n ekjetik n eðnai trigwnometrik sun rthsh, ra eðnai periodik me perðodo. To ginìmeno X(Ω) e jωn ja eðnai epðshc periodik sun rthsh me perðodo. Epomènwc orj c to olokl rwma upologðzetai se opoiod pote di sthma eôrouc. To zeôgoc twn exis sewn pou orðzei to metasqhmatismì ourier D.Q. eðnai X(Ω) x[n] = = x[n] e jωn (7.79) X(Ω) e jωn dω (7.8) opìte shmei noume x[n] T DT X(Ω). Sunep c ta mh-periodik s mata mporoôn na ekfrastoôn san grammikìc sunduasmìc fantastik n ekjetik n pou eðnai apeirost kont sth suqnìthta kai èqoun pl th X(Ω) dω. H sun rthsh X(Ω) lègetai f sma tou s matoc D.Q. x[n] kai mac lèei p c to s ma x[n] aposuntðjetai stic di forec suqnìthtec Ω. O periorismìc gia s mata periorismènhc di rkeiac mporeð na arjeð kai oi exis seic tou metasqhmatismoô na isqôoun. Oi sunj kec sôgklishc tou ajroðsmatoc (7.79) eðnai x[n] < x[n] 2 <. (7.8) Mia ousi dhc diafor me to metasqhmatismì ourier S.Q. eðnai ìti o metasqhmatismìc ourier D.Q. eðnai periodik sun rthsh me perðodo. Parathr ste ìti eðnai sun rthsh thc suneqoôc anex rththc metablht c Ω. Sto di sthma Ω [, ), oi qamhlèc suqnìthtec sto x[n] katwdiabatì X(Ω) n Ω Sq ma 7.7: Entopismìc qamhl n suqnot twn sto di sthma Ω [, ).

22 2 K. Kotrìpouloc: S mata-sust mata x[n] anwdiabatì X(Ω) n π Ω Sq ma 7.8: Entopismìc uyhl n suqnot twn sto di sthma Ω [, ). metasqhmatismì ourier D.Q. emfanðzontai perð to (Sq ma 7.7), en oi uyhlèc suqnìthtec perð to π (Sq ma 7.8). Par deigma 7.6. Gia to s ma D.Q. x[n] =a n u[n], a < (7.82) o metasqhmatismìc ourier D.Q. eðnai X(Ω) = a n u[n] e jωn = (ae jω ) n = n=. (7.83) ae jω To mètro tou metasqhmatismoô ourier sqedi zetai sto Sq ma 7.9(a), en h f sh tou metasqhmatismoô ourier paratðjetai sto Sq ma 7.9(b) gia <a<. An <a< tìte to π 4 a arctan( a a 2 ) X(Ω) X(Ω) +a arctan( a a 2 ) π 2 π Ω (a) Mètro π 4 π 2 π Ω (b) sh Sq ma 7.9: Metasqhmatismìc ourier D.Q. tou s matoc x[n] =a n u[n] gia <a<. mètro kai h f sh tou metasqhmatismoô ourier eðnai ìpwc sto Sq ma 7..

23 K. Kotrìpouloc: S mata-sust mata 2 π 4 +a arctan( a a 2 ) X(Ω) X(Ω) a arctan( a a 2 ) π 2 π Ω (a) Mètro π 4 π 2 π Ω (b) sh Sq ma 7.: Metasqhmatismìc ourier D.Q. tou s matoc x[n] =a n u[n] gia <a<. Par deigma 7.7. 'Estw to s ma D.Q. x[n] =a n, a <. (7.84) X(Ω) = m= n = = + + n= a n e jωn = a n e jωn + ae jω + ( + n= m=+ a n e jωn + a m e jωm = ) ae jω = + n= a n e jωn a n e jωn + + m= a m e jωm a 2 2a cos Ω + a 2. (7.85) ParathroÔme ìti a2 lim X(Ω) = Ω ( a) = +a 2 a (7.86) a2 lim X(Ω) = Ω π ( + a) = a 2 +a. (7.87) Epomènwc gia <a< to s ma x[n] eðnai katwdiabatì. Sthn perðptwsh aut to mètro tou metasqhmatismoô ourier D.Q. eðnai ìpwc autì tou Sq matoc 7.. Par deigma 7.8. 'Estw tetragwnikìc palmìc di rkeiac 2 +deigm twn n x[n] = n >. (7.88)

24 22 K. Kotrìpouloc: S mata-sust mata +a a X(Ω) a +a π 2 π Ω Sq ma 7.: Mètro metasqhmatismoô ourier D.Q tou s matoc x[n] =a n, gia <a<. O metasqhmatismìc ourier D.Q dðnetai apì thn X(Ω) = e jωn = sin Ω( + sin Ω n= 2 2 ) (7.89) pou apoteleð to diakritì an logo thc sinc. To mètro tou metasqhmatismoô ourier D.Q. sqedi zetai sto Sq ma 7.2(a) gia =kai sto Sq ma 7.2(b) gia = X(Ω) X(Ω) 2 π 2 + Ω 4 π 2 + π 2 π 2 π π 2 + π Ω 2 π (a) = (b) =2 Sq ma 7.2: Mètro metasqhmatismoô ourierd.q tou tetragwnikoô palmoô di rkeiac 2 + deigm twn. Den up rqei prìblhma sôgklishc sthn exðswsh sônjeshc tou metasqhmatismoô ourier diakritoô qrìnou, giatð h olokl rwsh gðnetai se peperasmèno di sthma. ParomoÐwc den up rqei fainìmeno Gibbs.

25 K. Kotrìpouloc: S mata-sust mata 23 Par deigma 7.9. Gia to s ma x[n] = δ[n] o metasqhmatismìc ourier D.Q. eðnai X(Ω) = δ[n] e jωn =. (7.9) An orðsoume wc tìte gia W π paðrnoume: ˆx[n] = W e jωn dω= W sin πn πn n = n sin Wn πn (7.9) = δ[n]. (7.92) 7.4 Periodik s mata kai metasqhmatismìc ourierd.q. MporoÔn na kajierwjoôn shmantikèc sqèseic metaxô thc anapar stashc se seir ourier periodik n shm twn kai tou metasqhmatismoô ourier mh-periodik n shm twn pou apoblèpoun sthn antimet pish twn akìloujwn erwthm twn:. P c h epèktash se seir ourier periodik n shm twn mporeð na apokthjeð apì to metasqhmatismì ourier thc pr thc periìdou thc akoloujðac? 2. P c h epèktash se seir ourier periodik n shm twn mporeð na sumperilhfjeð sto plaðsio tou metasqhmatismoô ourier ermhneôontac to metasqhmatismì ourier enìc periodikoô s matoc san trèno sewn sto pedðo thc suqnìthtac? 7.4. Metasqhmatismìc ourier D.Q. thc pr thc periìdou 'Estw x[n] periodikì s ma me perðodo. Ac jewr soume ìti to s ma x[n] anaparist mia perðodo tou x[n]. Tìte gia aujaðreto M x[n] an M n M + x[n] = alloô. (7.93) Oi suntelestèc thc diakrit c seir c ourier tou x[n] eðnai ìpou x[n] T DT a k = n=<> x[n] e jk n = X(k ) (7.94) X(Ω). O metasqhmatismìc ourier D.Q. X(Ω) exart tai apì thn epilog tou M sth (7.93), en ta deðgmat tou, X( k ), den exart ntai apì to M. ToÔto faðnetai sto Par deigma 7..

26 24 K. Kotrìpouloc: S mata-sust mata x[n] x2[n] - n (a) x [n] - M M + n (b) x 2 [n] Sq ma 7.3: S mata x [n] gia =6 kai x 2 [n] gia =6 kai M=2 sto Par deigma 7.. Par deigma 7.. 'Estw èna periodikì trèno sewn D.Q. me perðodo Tìte a k = = n=<> n= x[n] = + k= x[n] e jk n = δ[n k]. (7.95) M+ n=m x[n] e jk n δ[n] e jk n n= =. (7.96) An epileqjeð M =, tìte to s ma thc pr thc periìdou èqei metasqhmatismì ourier D.Q. en an <M< paðrnoume x [n] =δ[n] x 2 [n] =δ[n ] Sta deðgmata suqnìthtac Ω=k/ èqoume: T DT X (Ω) = (7.97) T DT X 2 (Ω) = e jω. (7.98) X ( k ) = (7.99) X 2 ( k ) = k e j = (7.) dhlad oi duo metasqhmatismoð paðrnoun tic Ðdiec timèc sta deðgmata suqnìthtac anexart twc thc epilog c tou M.

27 K. Kotrìpouloc: S mata-sust mata Metasqhmatismìc ourier D.Q. periodik n shm twn antastik ekjetik 'Estw x[n] =e jω n. Sto S.Q. èqoume e jω t δ(ω ω ). (7.) Sto D.Q. perimènoume an logo apotèlesma, all epeid e jω n = e j(ω +r)n, r Z (7.2) prokôptei ìti o metasqhmatismìc ourier D.Q. tou x[n] =e jω n dðnetai apì thn X(Ω) = + l= δ(ω Ω l). (7.3) To Sq ma 7.4 deðqnei to metasqhmatismì ourier D.Q. enìc fantastikoô ekjetikoô. Ja epalhjeôsoume thn orjìthta thc (7.3). 'Eqoume X(Ω) e jωn dω = = + l= [ + l= δ(ω Ω l)e jωn dω ] e j(ω+l)n δ(ω Ω l) dω [ = e jω n δ(ω Ω ) dω + }{{} + l= l ] δ(ω Ω l) dω } {{ } = e jω n. (7.4) X(Ω)... Ω Ω Ω + Ω Sq ma 7.4: Metasqhmatismìc ourier D.Q. tou fantastikoô ekjetikoô e jω n.

28 26 K. Kotrìpouloc: S mata-sust mata AujaÐreto periodikì s ma Genikìtera èna periodikì s ma mporeð na analujeð wc x[n] =b e jω n + b 2 e jω 2n + + b M e jω M n (7.5) opìte o metasqhmatismìc ourier D.Q. tou periodikoô s matoc x[n] prokôptei wc + X(Ω) = b δ(ω Ω l)+b 2 l= l= δ(ω Ω 2 l)+ + +b M δ(ω Ω M l). (7.6) l= ParathroÔme ìti: O X(Ω) eðnai èna periodikì trèno palm n me pr th perðodo pou sugkroteðtai apì kroustikoôc palmoôc sta Ω, Ω 2,...,Ω M. O X(Ω) den prokôptei apì thn exðswsh orismoô tou metasqhmatismoô (jumhjeðte ìti ta periodik s mata den eðnai apolôtwc ajroðsima), all eðnai apìdosh metasqhmatismoô ourier D.Q. wc sunèpeia twn idiot twn tou metasqhmatismoô. 'Estw O X(Ω) eðnai ègkuroc metasqhmatismìc ourier D.Q. akìma kai an Ω den eðnai thc morf c m. O antðstrofoc metasqhmatismìc ourier D.Q. (7.5) eðnai ègkuroc mìno ìtan Ω m = m. x[n] = k=<> a k e jk n (7.7) periodikì s ma me perðodo. Gia na upologðsoume to metasqhmatismì ourier D.Q. arkeð na ektelèsoume ta akìlouja b mata:. Epèlexe k =,,...,. 2. Jèse Ω =, Ω 2 =, Ω 3 =2( ),..., Ω =( ) sthn (7.6) 3. Tìte o metasqhmatismìc ourier D.Q. eðnai X(Ω) = + k= a k δ(ω k ). (7.8)

29 K. Kotrìpouloc: S mata-sust mata 27 Par deigma 7.. 'Estw tìte x[n] = X(Ω) = + k= + k= δ[n k] (7.9) δ(ω k ) (7.) epeid a k =. ParathroÔme ìti en dôo diadoqikoð palmoð sto qrìno apèqoun deðgmata, sth suqnìthta apèqoun Diakritìc metasqhmatismìc ouriergia s mata peperasmènhc di rkeiac O metasqhmatismìc ourier D.Q. eðnai periodik sun rthsh thc suneqoôc metablht c Ω gegonìc pou ton kajist dôsqrhsto stouc arijmhtikoôc upologismoôc. Gia na jerapeôsoume aut thn eggen adunamða tou metasqhmatismoô ourier D.Q. orðzoume ek kataskeu c to diakritì metasqhmatismì ourier (Discrete ourier Transform) DT gia s mata peperasmènhc di rkeiac wc ex c: a. 'Estw tètoioc ste x[n] =gia n>. b. Kataskeu zoume èna periodikì s ma x[n] me perðodo. Tìte x[n] =x[n] n. (7.) g. Oi suntelestèc thc seir c ourier tou periodikoô s matoc x[n] eðnai X[k] = n= opìte h exðswsh sônjeshc thc seir c ourier eðnai x[n] = k= x[n] e jk n k =,,..., (7.2) X[k] e jk n n =,,...,. (7.3) To zeôgoc twn exis sewn (7.2) kai (7.3) sugkroteð to diakritì metasqhmatismì ourier (DT). Oi gr goroi metasqhmatismoð ourier (T) eðnai apodotikoð algìrijmoi upologismoô tou DT. O DT klhronomeð pollèc apì tic idiìthtec tou metasqhmatismoô ourier diakritoô qrìnou. Exart tai apì thn epilog tou. Ja melethjeð se b joc sthn Yhfiak EpexergasÐa Shm twn.

30 28 K. Kotrìpouloc: S mata-sust mata 7.5 Idiìthtec tou metasqhmatismoô ourier DiakritoÔ Qrìnou Up rqoun pollèc omoiìthtec kai arketèc diaforèc me thn perðptwsh suneqoôc qrìnou. 'Otan h exagwg kai h ermhneða miac idiìthtac eðnai tautìshmh me thn perðptwsh suneqoôc qrìnou, parajètoume apl c thn idiìthta. Epiprosjètwc o metasqhmatismìc ourier D.Q. diathreð sten sqèsh me th diakrit seir ourier. Pollèc apì tic idiìthtec tou metasqhmatismoô an gontai stic idiìthtec thc seir c ourier. Sthn enìthta aut qrhsimopoioôme touc sumbolismoôc: X(Ω) = X(e jω )={x[n]} x[n] = {X(Ω)} x[n] T DT X(Ω) antð x[n] X(Ω). (7.4) 7.5. Periodikìthta O metasqhmatismìc ourierd.q eðnai periodik sun rthsh wc proc Ω me perðodo. ToÔto den isqôei ston metasqhmatismì ourier S.Q Grammikìthta x [n] x 2 [n] X (Ω) X 2 (Ω) Migadik suzugða ax [n]+bx 2 [n] ax (Ω) + bx 2 (Ω) (7.5) Apìdeixh Xekin ntac apì ton orismì { } x [n] = x[n] + X(Ω) x [n] X ( Ω) (7.6) { + } x [n] e jωn = x[n] e jωn = X ( Ω). }{{} X( Ω)

31 K. Kotrìpouloc: S mata-sust mata Qronik anastrof Apìdeixh x[n] X(Ω) x[ n] X( Ω). (7.7) { } x[ n] = = + + m= x[ n] e jωn n= m = + m= x[m] e j( Ω)m = X( Ω). x[m] e jωm Idiìthtec summetrðac An x[n] eðnai pragmatik akoloujða, tìte { Re{X(Ω)} =Re{X( Ω)} X(Ω) = X ( Ω) (7.8) Im{X(Ω)} = Im{X( Ω)}. IsqÔei epðshc ìti X(Ω) eðnai rtia sun rthsh wc proc Ω, en X(Ω) eðnai peritt sun rthsh wc proc Ω kai x e [n] x o [n] Re{X(Ω)} (7.9) jim{x(ω)}. (7.2) Apìdeixh Epeid x[n] eðnai pragmatik akoloujða èpetai x[n] =x [n] (7.6) X(Ω) = X ( Ω). (7.2) An X(Ω) = Re{X(Ω)} + jim{x(ω)} analôontac thn (7.2) paðrnoume opìte prokôptei h sqèsh (7.8). Re{X(Ω)} + jim{x(ω)} = Re{X( Ω)} jim{x( Ω)} (7.22) Qrhsimopoi ntac tic (7.7) kai (7.8) o metasqhmatismìc ourier D.Q. thc rtiac sunist sac tou x[n], x e [n], prokôptei wc x[n]+x[ n] x e [n] = {x e [n]} = [ ] X(Ω) + X( Ω) 2 2 [ ] = Re{X(Ω)} + jim{x(ω)} + Re{X( Ω)} + jim{x( Ω)} 2 (7.8) = Re{X(Ω)}. (7.23)

32 3 K. Kotrìpouloc: S mata-sust mata OmoÐwc apodeiknôetai h (7.2) Qronik metatìpish kai metatìpish suqnìthtac x[n] e jω n x[n] X(Ω) x[n n ] e jωn X(Ω) (7.24) X(Ω Ω ). (7.25) Diafìrish kai 'Ajroish Gia to s ma thc pr thc diafor c x[n] x[n ] prokôptei me efarmog thc idiìthtac thc qronik c metatìpishc ìti x[n] x[n ] ( e jω ) X(Ω). (7.26) 'Estw y[n] = n m= x[m]. ParathroÔme ìti y[n] y[n ] = x[n] Y (Ω) e jω Y (Ω) = X(Ω) Y (Ω) = X(Ω). (7.27) e jω H (7.27) den eðnai akrib c, giatð den eggu tai thn periodikìthta tou Y (Ω). Prèpei na prostejeð ènac ìroc pou antikatroptðzei th mèsh tim pou prokôptei apì to jroisma, ìpwc sthn perðptwsh suneqoôc qrìnou H akrib c sqèsh eðnai n m= t x[m] x(τ)dτ jω X(ω)+πX()δ(ω). e X(Ω) + πx() jω k= Par deigma 7.2. An efarmìsoume thn (7.28) gia to s ma u[n] paðrnoume u[n] e + π + jω k= δ(ω k). (7.28) δ(ω k). (7.29) Epeid wc gnwstì x[n] =δ[n] X(Ω) = (7.3)

33 K. Kotrìpouloc: S mata-sust mata 3 ja epalhjeôsoume ìti ìntwc katal goume sthn (7.3) an xekin soume apì thn (7.29) kai efarmìsoume tic idiìthtec tou metasqhmatismoô ourier D.Q. Pr gmati x[n] =δ[n] = u[n] u[n ] X(Ω) = e + π + δ(ω k) jω e jω e X(Ω) = + π k= jω πe jω + k= + k= δ(ω k) π δ(ω k) + k= Klim kwsh sto qrìno kai sth suqnìthta e jk δ(ω k) =. (7.3) 'Estw x[n] X(Ω). (7.32) An y[n] =x[ n], tìte sômfwna me thn idiìthta thc qronik c anastrof c Y (Ω) = X( Ω). En sto suneq qrìno isqôei x(at) a X(ω a ) (7.33) stic akoloujðec den orðzontai deðgmata x[an] an a<, dhlad, gia mh-akèraiec timèc tou a. L.q. to s ma x[2n] uponoeð ìti paðrnoume upìyh k je deôtero deðgma tou x[n], dhlad ta rtia deðgmata tou x[n]. 'Estw k jetikìc akèraioc, tìte orðzoume to s ma x (k) [n] = { x[n/k], an n mod k =, an n mod k (7.34) ìpou n mod k =upodhloð ìti to n eðnai pollapl sio tou k, en n mod k upodhloð ìti to n den eðnai pollapl sio tou k. 'Estw k =3kai x[n] autì tou Sq matoc 7.5(a). To s ma x (3) [n] sqedi zetai sto Sq ma 7.5(b). O metasqhmatismìc ourier D.Q. tou x (k) [n] prokôptei wc X (k) (Ω) = = r= x (k) [n] e jωn n=rk = r= x (k) [rk] e jωrk x[r] e j(kω)r = X(kΩ). (7.35) 'Ara x (k) [n] X(kΩ). (7.36)

34 32 K. Kotrìpouloc: S mata-sust mata x[n] 2 x(3)[n] n n (a) (b) Sq ma 7.5: (a) S ma x[n] (b) S ma x (3) [n]. ParathroÔme ìti h antðstrofh sqèsh metaxô di rkeiac sto qrìno kai eôrouc sth suqnìthta isqôei kai p li. An apl nei èna s ma kai << epibradônetai sto qrìno>>, tìte o metasqhmatismìc ourier D.Q. sumpièzetai. O metasqhmatismìc ourier D.Q X(kΩ) eðnai periodikì s ma wc proc Ω me perðodo k. x (3) [n] Ta Sq mata 7.6 kai 7.7 deðqnoun parastatik ta zeôgh x (2) [n] X (3) (kω) gia èna tetragwnikì palmì di rkeiac pènte deigm twn. X (2) (kω) kai.5.5 x[n].5 x(2)[n] n n 4 4 X(Ω) 2 X(2)(Ω) π - π π 2 π Ω -2 -π -π/2 π/2 π Ω (a) (b) Sq ma 7.6: ZeÔgh: (a) x[n] X(Ω) kai (b) x (2) [n] X (2) (Ω).

35 K. Kotrìpouloc: S mata-sust mata 33.5 x(3)[n] n 4 X(3)(Ω) π - 3 π 3 π 2 3 π Ω Sq ma 7.7: ZeÔgoc x (3) [n] X (3) (Ω) Diafìrish sth suqnìthta An x[n] X(Ω), tìte dx(ω) dω = d dω x[n] e jωn = x[n] jn e jωn = { jnx[n]}. (7.37) Opìte prokôptei nx[n] j dx(ω) dω. (7.38) 7.5. Tautìthta tou Parseval x[n] 2 = X(Ω) 2 dω. (7.39) To aristerì mèroc thc sqèshc (7.39) eðnai h enèrgeia tou s matoc x[n]. H sun rthsh X(Ω) 2 kaleðtai puknìthta f smatoc enèrgeiac. Gia periodik s mata mac endiafèrei h isqôc (enèrgeia se mia perðodo). Tìte qrhsimopoioôme touc suntelestèc thc seir c ourier n=<> 7.5. Idiìthta thc sunèlixhc E n y[n] =(x h)[n], tìte x[n] 2 = ìpou X(Ω) = {x[n]} kai H(Ω) = {h[n]}. k=<> a k 2. (7.4) Y (Ω) = X(Ω)H(Ω) (7.4)

36 34 K. Kotrìpouloc: S mata-sust mata Par deigma 7.3. 'Estw h[n] =δ[n n ], tìte An x[n] X(Ω), tìte H(Ω) = + y[n] =(x h)[n] =x(n n ) δ[n n ] e jωn = e jωn. (7.42) e jωn X(Ω) = {x[n n ]}. (7.43) Par deigma 7.4. 'Estw h[n] = a n u[n] x[n] = b n u[n] tìte gia y[n] =(h x)[n] paðrnoume Y (Ω) = H(Ω)X(Ω) = H(Ω) = X(Ω) = ae jω (7.44) be jω (7.45) ( ae Ω )( be jω ). (7.46) Gia na upologðsoume ton antðstrofo metasqhmatismì anaptôssoume se merik kl smata. DiakrÐnoume dôo peript seic An a b, tìte ìpou A = y[n] = a a b kai B = a a b an u[n] Y (Ω) = b a b. 'Ara A ae + B Ω be jω (7.47) b a b bn u[n] = a b [ ] a n+ u[n] b n+ u[n]. (7.48) An a = b, tìte Y (Ω) = ( ae jω ) = j d ( 2 a ejω dω ae jω ). (7.49) GnwrÐzoume ìti a n u[n] Opìte apì thn idiìthta diafìrishc sth suqnìthta na n u[n] j d ( ) dω ae jω kai thn idiìthta metatìpishc sto qrìno prokôptei (n +) a an+ u[n +]. (7.5) ae Ω j a ejω d ( dω ae jω (7.5) ). (7.52)

37 K. Kotrìpouloc: S mata-sust mata 35 'Ara o zhtoômenoc antðstrofoc metasqhmatismìc eðnai y[n] =(n +)a n u[n] (7.53) diìti gia n = n +=, molonìti u[n +]. H apìkrish suqnìthtac enìc G.Q.A. sust matoc D.Q., H(Ω), paðzei ton Ðdio rìlo me ekeðnh tou sust matoc S.Q. AxÐzei na shmeiwjeð ìti den èqei apìkrish suqnìthtac opoiod pote G.Q.A. sôsthma D.Q. To sôsthma me kroustik apìkrish h[n] =2 n u[n] den èqei apìkrish suqnìthtac. 'Ena sôsthma eðnai eustajèc fragmènhc eisìdou-fragmènhc exìdou an h kroustik apìkrish eðnai apolôtwc ajroðsimh, dhlad h[n] < (7.54) gegonìc pou eggu tai sôgklish tou {h[n]}. Epomènwc ta eustaj G.Q.A. sust mata D.Q. èqoun kal c orismènh H(Ω) Periodik Sunèlixh Gia periodikèc akoloujðec to jroisma thc sunèlixhc den sugklðnei. 'Etsi orðzetai ènac nèoc telest c h periodik sunèlixh dôo akolouji n pou eðnai periodikèc me koin perðodo ỹ[n] =( x x 2 )[n] = x [m] x 2 [n m]. (7.55) m=<> Parastatik èstw ta s mata tou Sq matoc 7.8. To qronik c antestrammèno s ma x 2 [ m] kai ta qronik c antestrammèna kai metatopismèna s mata x 2 [ m] kai x 2 [2 m] sqedi zontai sto Sq ma 7.9. ParathroÔme ìti ta s mata x 2 [ m] kai x 2 [2 m] prokôptoun apì kuklikèc metatopðseic deigm twn thc pr thc periìdou tou s matoc x 2 [ m]. IsqÔei ỹ[n + ] =ỹ[n], epeid prìkeitai gia periodik akoloujða me perðodo. Gia thn periodik sunèlixh isqôei an logh idiìthta, ìpwc kai gia thn mh-periodik, dhlad, an {a k }, {b k }, kai {c k } eðnai oi suntelestèc thc diakrit c seir c ourier twn x [n], x 2 [n] kai ỹ[n] antistoðqwc, tìte c k = a k b k. (7.56) H pio shmantik qr sh thc idiìthtac aut c gia touc suntelestèc thc seir c ourier eðnai h efarmog thc mazð me to diakritì metasqhmatismì ourier (DT) ston upologismì thc

38 36 K. Kotrìpouloc: S mata-sust mata x2[m] x[m] 2 m (a) x 2 [m] m (b) x [m] Sq ma 7.8: Periodik s mata x 2 [m] kai x [m]. mh-periodik c (dhlad, grammik c) sunèlixhc duo akolouji n peperasmènhc di rkeiac. 'Estw x [n] = ektìc tou diast matoc n, x 2 [n] = ektìc tou diast matoc n 2. (7.57) 'Estw y[n] h mh-periodik sunèlixh twn x [n] kai x 2 [n]. Tìte: y[n] =(x x 2 )[n] = ektìc tou diast matoc n + 2. (7.58) An epilèxoume ènan opoiod pote akèraio + 2, kataskeu soume duo s mata me perðodo wc ex c x [n] = x [n] n< x 2 [n] = x 2 [n] n< (7.59) kai upologðsoume thn periodik sunèlixh twn x [n] kai x 2 [n] ỹ[n] = x [m] x 2 [n m] (7.6) m=<> tìte y[n] =ỹ[n] n. (7.6) Sunep c katal goume ston ex c algìrijmo upologismoô thc mh-periodik c (grammik c) sunèlixhc. ParagemÐzoume tic akoloujðec x [n] kai x 2 [n] me mhdenik gia na kataskeu soume tic periodikèc akoloujðec x [n] kai x 2 [n] epilègontac thn perðodo, ste + 2.

39 K. Kotrìpouloc: S mata-sust mata 37 x2[ m] x2[ m] 2 m m (a) x 2 [ m] (b) x 2 [ m] x2[2 m] m (g) x 2 [2 m] Sq ma 7.9: To qronik c anestrammèno s ma x 2 [ m] kai kuklikèc metatopðseic twn deigm twn thc pr thc periìdou tou. Gia + 2 h mh-periodik sunèlixh twn x [n] kai x 2 [n] isoôtai me thn periodik sunèlixh twn x [n] kai x 2 [n]. Opìte arkeð:. Upologismìc twn DTs X [k] kai X2 [k] twn x [n] kai x 2 [n]. 2. Pollaplasiasmìc twn DTs gia ton upologismì tou DT thc y[n] Ỹ [k] = X [k] X 2 [k]. (7.62) 3. Upologismìc tou antðstrofou DT thc Ỹ [k]. To apotèlesma eðnai h epijumht sunèlixh y[n].

40 38 K. Kotrìpouloc: S mata-sust mata Idiìthta thc diamìrfwshc 'Estw y[n] =x [n]x 2 [n]. An x [n] ourier D.Q. tou s matoc y[n] dðnetai apì thn Y (Ω) = y[n] e jωn = x [n] x 2 [n] e jωn = = X (θ) Par deigma 7.5. 'Estw Tìte [ X (Ω) kai x 2 [n] X 2 (Ω), tìte o metasqhmatismìc x 2 [n] e j(ω θ)n ] X (Ω) = dθ = x 2 [n] { } X (θ) e jθn dθ e jωn X (θ) X 2 (Ω θ) dθ. (7.63) x [n] =e jπn =( ) n. (7.64) + r= δ(ω (2r +)π). (7.65) O metasqhmatismìc ourier X (Ω) sqedi zetai sto Sq ma 7.2. Sto di sthma Ω [, ) X (Ω) π π π 3π Ω Sq ma 7.2: Metasqhmatismìc ourier X (Ω). èqoume X (θ)x 2 (Ω θ) =X 2 (Ω θ) δ(θ π) =X 2 (Ω π) δ(θ π) (7.66) opìte Y (Ω) = X 2 (Ω π) δ(θ π) dθ = X 2 (Ω π). (7.67) Pollaplasiasmìc epð ( ) n èqei wc apotèlesma thn enallag qamhl n kai uyhl n fasmatik n perioq n sto f sma thc akoloujðac eisìdou. Pr gmati an X 2 (Ω) eðnai o metasqhmatismìc ourier tou Sq matoc 7.2, tìte o metasqhmatismìc ourier Y (Ω) èqei th morf tou Sq matoc O PÐnakac 7.2 sunoyðzei tic idiìthtec tou metasqhmatismoô ourier D.Q.

41 K. Kotrìpouloc: S mata-sust mata 39 PÐnakac 7.2: Idiìthtec tou metasqhmatismoô ourier D.Q. Idiìthta Mh-periodikì S ma Metasqhmatismìc ourier x[n] X(Ω) = X(e jω ) y[n] Y (Ω) = Y (e jω ) Grammikìthta Ax[n]+By[n] AX(e jω )+BY(e jω ) Qronik metatìpish x[n n ] e jωn X(e jω ) Metatìpish suqnìthtac e j Ω n x[n] X(e j(ω Ω ) ) SuzugÐa x [n] X (e jω ) Qronik anastrof x[ n] X(e jω ) Diastol sto qrìno x (k) [n] X(e jkω ) Sunèlixh x[n] y[n] X(e jω ) Y (e jω ) Pollaplasiasmìc x[n] y[n] X(ejθ )Y (e j(ω θ) )dθ Pr th diafor x[n] x[n ] ( e jω )Q(e jω ) n Suss reush k= x[k] e jω X(e jω ) + πx(e j ) + k= δ(ω k) Diafìrish sth suqnìthta nx[n] j dx(ejω ) dω X(e jω )=X (e jω ) Suzug c summetrða gia pragmatik s mata Pragmatik s mata rtiac x[n] R x[n] R: x[n] =x[ n] Re{X(e jω )} =Re{X(e jω )} Im{X(e jω )} = Im{X(e jω )} X(e jω ) = X(e jω ) X(e jω )= X(e jω ) X(e jω ) pragmatik kai summetrðac Pragmatik s mata peritt c summetrðac rtiac summetrðac x[n] R: x[n] = x[ n] X(e jω ) kajar c fantastik kai peritt c AposÔnjesh se rtio kai perittì mèroc pragmatikoô s matoc summetrðac x e [n] = 2 (x[n]+x[ n]) x o [n] = 2 (x[n] x[ n]) Re{X(ejΩ )} j Im{X(e jω )} Tautìthta Parseval gia mh-periodik s mata + x[n] 2 = X(ejΩ ) 2 dω

42 4 K. Kotrìpouloc: S mata-sust mata X 2 (Ω) π π Ω Sq ma 7.2: Metasqhmatismìc ourier X 2 (Ω). Y (Ω) π π Ω Sq ma 7.22: Metasqhmatismìc ourier Y (Ω). 7.6 Duadikèc Idiìthtec 7.6. Diakrit seir ourier ParathroÔme ìti den up rqei summetrða metaxô thc exðswshc an lushc kai thc exðswshc sônjeshc tou metasqhmatismoô ourier D.Q. Wstìso up rqei tètoia summetrða ston orismì thc diakrit c seir c ourier.pr gmati, èstw mia tautìthta metaxô dôo periodik n akolouji n f[m] kai g[m] thc Ðdiac periìdou f[m] = r=<> g[r] e jr m. (7.68) An antikatast soume k = m kai r = n tìte f[k] = n=<> g[n] e jk n (7.69)

43 K. Kotrìpouloc: S mata-sust mata 4 opìte sun goume ìti oi periodikèc akoloujðec g[n] kai f[k] apoteloôn èna zeôgoc diakrit c seir c ourier An jèsoume sthn (7.68) m = n kai r = k paðrnoume opìte f[n] = g[n] DS f[k]. (7.7) k=<> g[ k] ejk n (7.7) f[n] DS g[ k]. (7.72) Ac prospaj soume na ermhneôsoume ta apotelèsmata thc an lushc. Oi suntelestèc seir c ourier enìc periodikoô s matoc x[n], a k, eðnai periodik akoloujða. Wc periodik akoloujða mporoôn na epektajoôn se seir ourier. H duadik idiìthta epit ssei oi suntelestèc thc seir c ourier thc periodik c akoloujðac a k prèpei na eðnai oi timèc x[ n], an logec twn arqik n tim n tou s matoc, all anestrammènec sto qrìno. Praktikèc sunèpeiec thc duadik c idiìthtac eðnai ta zeôgh twn idiot twn: x[n n ] e jm n x[n] DS a k e jk n (7.73) DS a k M (7.74) kai x[r]y[n r] r=<> x[n]y[n] DS a k b k (7.75) DS l=<> a l b k l (7.76) 'Estw x[n] DS a k. H apìdeixh thc (7.74) èqei wc ex c. a k a k M a k M x[ n] e jm n x[n] e jm n DS x[ n] duadik idiìthta DS x[ n] e jm n idiìthta thc metatìpishc DS x[ n] e jm n grammikìthta DS a ( k) M duadik idiìthta DS a k M qronik anastrof (7.77)

44 42 K. Kotrìpouloc: S mata-sust mata GÐnetai fanerì ìti axiopoi ntac th duadik idiìthta epitugq netai el ttwsh twn upologism n kai dipl axiopoðhsh tou pðnaka twn idiot twn thc diakrit c seir c ourier. Par deigma 7.6. 'Estw x[n] = sin[( n )( + 2 )] sin( n 2 ). (7.78) AnagnwrÐzoume ìti prìkeitai gia thn akoloujða twn suntelest n thc diakrit c seir c ouriermiac periodik c tetragwnik c palmoseir c di rkeiac 2 + kai periìdou. 'Ara oi suntelestèc thc seir c ourier thc x[n] ja eðnai epð ta deðgmata thc tetragwnik c palmoseir c anestrammèna sto qrìno. Lìgw summetrðac den epèrqetai kami metabol x[n] =4 5 =2 n Sq ma 7.23: Suntelestèc seir c ourier thc akoloujðac x[n] pou orðzetai sthn (7.78) gia =2kai = Metasqhmatismìc ourier D.Q. kai seir ourier S.Q. Up rqei epðshc duadikìthta metaxô metasqhmatismoô ourierd.q. kai thc seir c ouriers.q., ìpwc prokôptei apì thn antiparabol twn exis sewn orismoô twn. x[n] = X(Ω) = + X(Ω) e jωn dω x(t) = x[n] e jωn + k= a k e jkω t a k = T T x(t) e jkω t dt 'Estw f(u) periodik sun rthsh suneqoôc metablht c me perðodo kai mh-periodik akoloujða g[m] pou sqetðzetai me thn f(u) dia thc sqèsewc f(u) = + m= g[m] e jum. (7.79)

45 K. Kotrìpouloc: S mata-sust mata 43 An u =Ωkai m = n, tìte f(ω) eðnai o metasqhmatismìc ourier D.Q. thc g[n] g[n] T DT f(ω) (7.8) opìte 'Estw u = t kai m = k. Tìte h (7.79) xanagr fetai wc g[m] = f(u) e jum du. (7.8) f(t) = + k= g[ k] e jtk. (7.82) H f(t) eðnai periodik me perðodo = T, ra ω =. Epomènwc h g[ k] eðnai h akoloujða twn suntelest n thc seir c ourier S.Q. thc f(t) f(t) S g[ k] (7.83) Ac prospaj soume na ermhneôsoume kai p li ta apotelèsmata thc an lushc. To x[n] eðnai s ma diakritoô qrìnou me metasqhmatismì ourier D.Q. X(Ω). O X(Ω) eðnai periodik sun rthsh wc proc Ω me perðodo, ra epekt simh se seir ourier S.Q. me ω =. Oi suntelestèc ourier S.Q. thc X(Ω) ja eðnai h arqik akoloujða anestrammènh sto qrìno. Praktik sunèpeia thc duadik c idiìthtac eðnai ta zeôgh twn idiot twn gia thn periodik sunèlixh sth seir ourier S.Q. kai thc diamìrfwshc sto metasqhmatismì ourier D.Q. x (r) x 2 (t r) dr a[n]b[n] S a k b k (7.84) T DT X (θ) X 2 (Ω θ) dθ. (7.85) Par deigma 7.7. (a) 'Estw periodikì s ma suneqoôc qrìnou x(t) me perðodo kai suntelestèc seir c ourier S.Q. k a k = alloô. (7.86)

46 44 K. Kotrìpouloc: S mata-sust mata To s ma diakritoô qrìnou a k eðnai ènac mh-periodikìc tetragwnikìc palmìc. Apì ton metasqhmatismì ourier D.Q. tou tetragwnikoô palmoô sun goume ìti to arqikì s ma den eðnai llo apì to sunepeða thc duadikìthtac. x(t) = sin( + 2 )t sin(t/2) (7.87) (b) An t ra èqoume to metasqhmatismì ourier D.Q. X(Ω) pou orðzetai sto di sthma π Ω π wc Ω <W X(Ω) = W< Ω π (7.88) tìte o X(Ω) ja epekteðnetai se seir ourier S.Q. wc periodikìc tetragwnikìc palmìc S.Q. Oi suntelestèc thc seir c ourier ja dðnontai apì thn sin(kω T ) kπ T =W ω = = sin(kw). (7.89) kπ 'Ara wc apotèlesma thc duadikìthtac to s ma x[n] pou èqei to dosmèno metasqhmatismì ourier D.Q. (7.88) eðnai ìpou sinc(x) = sin x x. x[n] = sin Wn πn = W π sinc(wn) (7.9) O PÐnakac 7.3 sunoyðzei tic duadikèc idiìthtec.

47 K. Kotrìpouloc: S mata-sust mata 45 PÐnakac 7.3: Duadikèc idiìthtec. ErgaleÐo PedÐo Qrìnou PedÐo Suqnìthtac Seir ourier x(t) = + a k e jkω t a k = T T x(t) e jkω t dt periodik sun rthsh suneqoôc metablht c mh-periodik sun rthsh diakrit c metablht c Metasqhmatismìc ourier () (2) x(t) = + X(ω)ejωt dω X(ω) = + e jωt dt mh-periodik sun rthsh suneqoôc metablht c mh-periodik sun rthsh suneqoôc metablht c (3) (4) Suneq c Qrìnoc ErgaleÐo PedÐo Qrìnou PedÐo Suqnìthtac Seir ourier x[n] = k=<> a k e jk( )n a k = n=<> x[n] e jk( )n periodik sun rthsh diakrit c metablht c periodik sun rthsh diakrit c metablht c Metasqhmatismìc ourier (5) (6) x[n] = X(Ω) ejωn dω X(Ω) = + x[n] e jωn mh-periodik sun rthsh periodik sun rthsh suneqoôc diakrit c metablht c metablht c (7) (8) Diakritìc qrìnoc ZeÔgh duadik n idiot twn (3) (4) (5) (6) (7) (2) () (8)