Σήματα Συστήματα Ανάλυση Fourier για σήματα και συστήματα συνεχούς χρόνου Περιοδικά Σήματα (Σειρά Fourier)

Σήματα Συστήματα Ανάλυση Fourier για σήματα και συστήματα συνεχούς χρόνου Περιοδικά Σήματα (Σειράά Fourier) Κωνσταντίνος Κοτρόπουλος Τμήμα Πληροφορικής Θεσσαλονίκη, Ιούνιος 3

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 3

Kef laio 4 An lush Fourier gia periodik s mata suneqoôc qrìnou 4. Eisagwg Sthn prohgoômenh enìthta, anaparist ntac thn eðsodo s> èna Grammikì Qronoamet blhto (G.Q.A.) sôsthma suneqoôc qrìnou san olokl rwma metatopismènwn monadiaðwn sewn (dhlad, sunart sewn δ(t τ)) me b rh exag game to olokl rwma thc sunèlixhc wc th sqèsh pou sundèei thn eðsodo (diègersh) me thn èxodo (apìkrish) tou G.Q.A. sust matoc. DeÐxame epðshc ìti h èxodoc enìc G.Q.A. sust matoc se mia aujaðreth eðsodo mporeð na upologisteð apì tic apokrðseic tou susthm toc se sunart seic metatopismènwn sewn. Shmei same ìti h apìkrish enìc G.Q.A. sust matoc se sun rthsh monadiaðac shc, pou onom sthke kroustik apìkrish, eðnai mia polô shmantik ènnoia. oôto giatð to olokl rwma thc sunèlixhc den mac parèqei mìno èna bolikì trìpo upologismoô thc apìkrishc enìc G.Q.A. susthm toc upì thn proôpìjesh ìti gnwrðzoume thn kroustik apìkris tou, all upodhloð ìti ta qarakthristik tou G.Q.A. sust matoc prosdiorðzontai pl rwc apì thn kroustik apìkrish. H an lush pou prohg jhke tan apotèlesma thc arq c thc upèrjeshc, ìpou jewr same tic sunart seic metatopismènwn monadiaðwn sewn wc stoiqei deic sunart seic. Antikeimenikìc skopìc thc enìthtac, pou sugkroteðtai apì to parìn kai to epìmeno kef laio, eðnai na anaptôxoume mia enallaktik anapar stash twn shm twn suneqoôc qrìnou sunart sei llwn stoiqeiwd n shm twn kai na ex goume wc epakìloujo mia llh majhmatik perigraf enìc G.Q.A. sust matoc. ShmeÐo ekkðnhshc ja eðnai p li h anapar stash enìc

K. Kotrìpouloc: S mata-sust mata s matoc san jroisma olokl rwma me b rh stoiqeiwd n shm twn, dhlad san grammikì sunduasmì stoiqeiwd n shm twn. a stoiqei dh s mata t ra ja eðnai ta fantastik ekjetik. 'Etsi gia men ta periodik s mata ja prokôyei h an ptuxh se seir Fourier, en gia ta mh-periodik s mata ja prokôyei o metasqhmatismìc Fourier pou ja melethjeð sto epìmeno kef laio. GÐnetai antilhptì ìti, exaitðac thc arq c thc upèrjeshc, h èxodoc enìc G.Q.A. sust matoc se opoiad pote eðsodo pou parðstatai wc grammikìc sunduasmìc stoiqeiwd n shm twn eðnai o grammikìc sunduasmìc twn epimèrouc apokrðsewn se kajèna ap> aut ta stoiqei dh s mata. Sthn prohgoômenh enìthta h sun rthsh thc monadiaðac shc epilèqjhke wc stoiqei dec s ma, diìti h apìkrish enìc G.Q.A. sust matoc ìtan diegeðretai apì tètoio s ma eisìdou eðnai h kroustik apìkrish. 'Ara prèpei na dikaiolog soume ìti h epilog twn fantastik n ekjetik n, kai genikìtera twn migadik n ekjetik n, wc stoiqeiwd n shm twn den eðnai aujaðreth. Pr gmati h apìkrish enìc G.Q.A. sust matoc s> èna migadikì ekjetikì, èqei mia apl morf kai epomènwc mac prosfèrei llh mia dunatìthta gia na analôsoume tic idiìthtec enìc G.Q.A. sust matoc. Ja deðxoume ìti:. a migadik ekjetik eðnai idiosunart seic (eigenfunctions) twn G.Q.A. susthm twn. O epimel c anagn sthc sto kousma tou ìrou idiosun rthsh ja eðnai se jèsh na a- nakalèsei apì th mn mh tou èna gn rimo suggenikì ìro apì th Grammik 'Algebra. EkeÐno, tou idiodianôsmatoc enìc tetragwnikoô pðnaka.. Ja deðxoume epiplèon ìti mia sullog M sunhmitìnwn hmitìnwn fantastik n ekjetik n sugkroteð mia b sh sto dianusmatikì q ro pou sugkrotoôn ta periodik s mata x(t) se qronikì di sthma miac periìdou, ac poôme gia t [,]. Dhlad, me thn prìodo thc enìthtac aut c, ja mporèsoume na apod soume ston poiotikì ìro stoiqei dec basikì s ma mia piì <<majhmatik >> qroi. EkeÐnh, thc sun rthshc b shc.

K. Kotrìpouloc: S mata-sust mata 3 4. Apìkrish enìc G.Q.A. sust matoc suneqoôc qrìnou se migadik ekjetik Wc basik ja mporoôsame na apokalèsoume ta s mata pou plhroôn tic ex c dôo poiotikèc prodiagrafèc.. MporoÔn na qrhsimopoihjoôn gia na kataskeu soun mia eureða kai qr simh t xh shm - twn.. H apìkrish enìc G.Q.A. sust matoc s> èna basikì s ma eðnai apl. Ja deðxoume ìti ta migadik ekjetik katarq n plhroôn thn idiìthta. 'Estw x(t) =e st, s C. H spoudaiìthta twn migadik n ekjetik n prokôptei apì to gegonìc ìti h apìkrish enìc G.Q.A sust matoc se eðsodo x(t) =e st eðnai + + y(t) = h(τ)x(t τ)dτ = h(τ)e s(t τ) dτ = e st{ + h(τ) e sτ } dτ = H(s)e st. } {{ } H(s) (4.) Lème ìti to migadikì ekjetikì e st eðnai mia idiosun rthsh tou G.Q.A. sust matoc kai h tim : H(s) = + h(τ) e sτ dτ (4.) eðnai h idiotim (eigenvalue) pou sqetðzetai me thn idiosun rthsh e st. H apìkrish enìc G.Q.A. sust matoc se diègersh e st deðqnetai parastatik sto Sq ma 4.. Ac upotejeð ìti e st G.Q.A. sôsthma H(s) e st Sq ma 4.: Apìkrish G.Q.A. sust matoc se migadikì ekjetikì. to s ma pou diegeðrei to G.Q.A. sôsthma analôetai se x(t) =a e s t + a e s t + a3 e s 3t. (4.3) An efarmìsoume thn exðswsh (4.) gia kajemi sunist sa tou ajroðsmatoc (4.3) paðrnoume tic ex c apokrðseic: a e s t a e s t a 3 e s 3t a H(s ) e s t a H(s ) e s t a 3 H(s 3 ) e s 3t. (4.4)

4 K. Kotrìpouloc: S mata-sust mata Opìte efarmìzontac thn arq thc upèrjeshc, h pl rhc apìkrish tou G.Q.A. sust matoc sth diègersh x(t) pou dðnetai apì thn (4.3) eðnai y(t) = 3 a k H(s k ) e s kt. (4.5) k= Dhlad, an gnwrðzoume tic idiotimèc H(s k ), tìte h apìkrish s> èna grammikì sunduasmì migadik n ekjetik n mporeð na upologisteð apeujeðac. Sto parìn kai sta epìmena dôo kef laia ja melet soume ta akìlouja probl mata: a) Anapar stash periodik n shm twn me ìrouc thc seir c Fourier (ajroðsmata fantastik n ekjetik n) b) Epèktash thc an lushc Fourier se mh-periodik s mata (metasqhmatismìc Fourier) g) GenÐkeush tou metasqhmatismoô Fourier sto metasqhmatismì Laplace. 4.. Istorik anadrom Efex c ja mac apasqol sei h an lush enìc periodikoô s matoc wc grammikoô sunduasmoô armonik n, dhlad fantastik n ekjetik n se suqnìthtec pou eðnai akèraia pollapl sia miac jemeli douc. Istorik h idèa qrhsimopoðhshc trigwnometrik n ajroism twn gia thn perigraf periodik n fainomènwn eðnai pan rqaia kai apodðdetai stouc arqaðouc BabulwnÐouc. Stouc neìterouc qrìnouc, o L. Euler to 748 qrhsimopoðhse trigwnometrik ajroðsmata kat th melèth thc kðnhshc miac pallìmenhc qord c. H idèa aut den ja kataxiwnìtan an den apodeiknuìtan ìti mia eurôterh t xh qr simwn shm twn ja mporoôse na analujeð me ton trìpo autì. H skut lh pèrase apì ton L. Euler ston D. Bernoulli, o opoðoc isqurðsthke ìti ìlec oi fusikèc kin seic miac pallìmenhc qord c ja mporoôsan na perigrafoôn wc jroisma armonik n to 753. 'Exi qrìnia argìtera, to 759, h kritik tou Euler, all kurðwc tou J. L. Lagrange, sunèteine sthn egkat lhyh twn trigwnometrik n seir n, epeid o Lagrange den mporoôse na fantasteð p c èna periodikì s ma me asunèqeiec (p.q. ènac periodikìc tetragwnikìc palmìc) ja mporoôse na analujeð wc jroisma hmitìnwn kai sunhmitìtwn se suqnìthtec pou eðnai akèraia pollapl sia miac jemeli douc, dhlad wc trigwnometrik seir. Misì ai na argìtera, kai par to eqjrikì klðma pou diamorf jhke gia tic trigwnometrikèc seirèc, o J. B. Fourier xanaqrhsimopoðhse tètoiec seirèc sth melèth tou probl matoc thc di doshc jermìthtac kai epexèteine th jewrða kai gia thn an lush mh-periodik n shm twn

K. Kotrìpouloc: S mata-sust mata 5 mèsw tou om numou metasqhmatismoô. H ergasða tou Fourier pou periègrafe th sqetik jewrða upobl jhke proc dhmosðeush gia pr th for to 87. Sun nthse ìmwc thn arnhtik kritik tou Lagrange kai par th jetik antimet pis thc apì touc Lacroix, Monge kai Laplace, dhmosieôthke met 5 olìklhra qrìnia. H al jeia eðnai pwc h austhr jemelðwsh thc jewrðac eðqe k poiec elleðyeic tic opoðec sumpl rwse argìtera o P. L. Dirichlet to 89. Wstìso, ìpwc ja exhghjeð argìtera, oi sunj kec Dirichlet perigr foun s mata pou spanðwc apant ntai sthn pr xh. Sta mèsa thc dekaetðac tou 96, h <<anak luyh>> tou gr gorou metasqhmatismoô Fourier apì touc Cooley kai uckey anazwpôrwse to endiafèron gia thn an lush Fourier, epeid, mei nontac to upologistikì kìstoc, prosèfere th dunatìthta enswm twshc thc an lushc se pollèc praktikèc efarmogèc, p.q. kwdikopoðhsh, an lush f smatoc, epexergasða eikìnac, omilðac k.o.k. Wstìso diapist jhke argìtera ìti h anak luyh twn Cooley-uckey den tan apokleistik. O Gauss kai argìtera ap> autìn, o Lanczos eðqan ter stia sumbol sthn an lush Fourier diakrit n shm twn kai susthm twn polô prin touc Cooley kai uckey. 4.3 ProapaitoÔmena gia thn exagwg thc seir c Fourier 'Ena periodikì s ma ikanopoieð th sqèsh : x(t + )=x(t), t. (4.6) Apì ta aploôstera periodik s mata eðnai oi hmitonoeideðc sunart seic kai ta fantastik ekjetik x(t) = cosω t x(t) = sin ω t x(t) = e jω t ìpou ω eðnai h jemeli dhc (kuklik ) suqnìthta metroômenh se rad/sec kai = π ω jemeli dhc perðodoc metroômenh se sec. Wc armonikèc orðzoume ta s mata (4.7) eðnai h φ k (t) =e jk ω t, k =, ±, ±,... (4.8) twn opoðwn h jemeli dhc suqnìthta eðnai akèraio pollapl sio thc ω. o prìblhma pou ja melet soume sto parìn kef laio eðnai h anapar stash enìc periodikoô s matoc x(t) me

6 K. Kotrìpouloc: S mata-sust mata perðodo wc grammikoô sunduasmoô armonik n x(t) = ParathroÔme ìti gia + k= a k e jkω t. (4.9) k =èqoume to stajerì ìro ìro dc k = ± prokôptoun oi basikèc sunist sec pou èqoun perðodo (pr tec armonikèc) k = ±: èqoume tic sunist sec hmðseiac periìdou (dipl siac suqnìthtac), dhlad tic deôterec armonikèc k.o.k., en k = ±N prokôptoun oi Niostèc armonikèc. H (4.9) orðzei thn epèktash tou periodikoô s matoc x(t) se seir Fourier. Antikeimenikìc mac skopìc eðnai na melet soume p c ja prosdiorðsoume touc suntelestèc a k sthn (4.9). Proc to parìn ac doôme, mèsa apì to Par deigma 4., parastatik pìso ploôsia s mata mporoôn na parjoôn prosjètontac armonikèc. Par deigma 4.. 'Estw 3 x(t) = a k e jk π t (4.) k= 3 me a =, a = a = 4, a = a = kai a 3 = a 3 =. ìte antikajist ntac sthn (4.) 3 paðrnoume x(t) = + 4 (ejπt + e jπt )+ (ej4πt + e j4πt )+ 3 (ej6πt + e j6πt ) = + cos πt +cos4πt + cos 6πt. (4.) 3 a Sq mata 4.(a)-(z) deðqnoun parastatik p c uperjètontac armonikèc to s ma x(t) gðnetai oloèna plousiìtero. Ja antimetwpðsoume thn (4.9) s> èna eurôtero plaðsio wc anapar stash enìc periodikoô s matoc san grammikoô sunduasmoô basik n shm twn, ìpou ta basik s mata ja antistoiqoôn se sunart seic b shc. dianusmatikì q ro (vector space). 'Omwc oi sunart seic b shc proôpojètoun èna

K. Kotrìpouloc: S mata-sust mata 7 3.5 3.5 3.5 3 3 3.5.5.5 x(t).5 x(t).5 x(t)+x(t).5.5.5.5 -.5 -.5 -.5 - - -.8 -.6 -.4 -...4.6.8 t - - -.8 -.6 -.4 -...4.6.8 t - - -.8 -.6 -.4 -...4.6.8 t (a) (b) (g) 3.5 3.5 3 3.5.5 x(t).5.5 x(t)+x(t)+x(t).5.5 -.5 -.5 - - -.8 -.6 -.4 -...4.6.8 t - - -.8 -.6 -.4 -...4.6.8 t (d) (e) 3.5 3.5 3 3 x3(t).5.5.5 x(t)+x(t)+x(t)+x3(t).5.5.5 -.5 -.5 - - -.8 -.6 -.4 -...4.6.8 t - - -.8 -.6 -.4 -...4.6.8 t (st) (z) Sq ma 4.: (a) O stajerìc ìroc x (t) =. (b) H pr th armonik x (t) = cos πt. (g) Upèrjesh stajeroô ìrou kai pr thc armonik c x (t)+x (t). (d) H deôterh armonik x (t) = cos 4πt. (e) Upèrjesh stajeroô ìrou, pr thc kai deôterhc armonik c x (t)+x (t)+x (t). (st) H trðth armonik x 3 (t) = cos 6πt. (z) Upèrjesh stajeroô ìrou kai twn tri n pr twn 3 armonik n x (t)+x (t)+x (t)+x 3 (t).

8 K. Kotrìpouloc: S mata-sust mata 4.3. DianusmatikoÐ Q roi o shmeðo ekkðnhshc gia na orðsoume thn ènnoia tou dianusmatikoô q rou eðnai o orismìc tou bajmwtoô pedðou (scalar field). 'Ena bajmwtì pedðo K eðnai èna sônolo arijm n pou eðnai kleistì wc proc thn prìsjesh kai ton pollaplasiasmì, dhlad to jroisma kai to ginìmeno opoiwnd pote dôo stoiqeðwn apì to K an kei epðshc sto K. ìso h prìsjesh ìso kai o pollaplasiasmìc eðnai telestèc dôo orism twn. 'Estw x, y, z K. ìte gia thn prìsjesh kai ton pollaplasiasmì isqôoun oi ex c idiìthtec: antimetajetik (commutative) x + y = y + x x y = y x (4.) prosetairistik (associative) (x + y)+z = x +(y + z) (x y) z = x (y z) (4.3) Ôparxh oudèterou stoiqeðou (null element) pou sumbolðzetai me Kgia thn prìsjesh kai Kgia ton pollaplasiasmì x +=x x =x. (4.4) IsqÔei epðshc h epimeristik (distributive) idiìthta tou pollaplasiasmoô wc proc thn prìsjesh x (y + z) =(x y)+(x z). (4.5) Epiplèon ìla ta stoiqeða tou K prèpei na èqoun ènan antðstrofo pou an kei sto K wc proc thn prìsjesh (pou sun jwc apokaleðtai antðjetoc) kai ìla ta stoiqeða tou K ektìc tou prèpei na èqoun ènan antðstrofo wc proc ton pollaplasiasmì. o sônolo twn pragmatik n arijm n R efodiasmèno me thn prìsjesh kai ton pollaplasiasmì eðnai èna pedðo. 'Alla pedða pou basðzontai sthn prìsjesh kai ton pollaplasiasmì eðnai ta sônola twn migadik n arijm n C kai twn rht n arijm n Q. Ja mac apasqol soun ta pedða R kai C sthn an lush shm twn kai susthm twn. 'Enac dianusmatikìc q roc mporeð na oristeð wc proc èna pedðo. Eidikìtera ènac dianusmatikìc q roc S sto pedðo twn pragmatik n arijm n R twn migadik n arijm n C eðnai èna sônolo stoiqeðwn x, y, z S pou lègontai dianôsmata gia ta opoða orðzontai:

K. Kotrìpouloc: S mata-sust mata 9 h prìsjesh dôo dianusm twn pou dðnei di nusma kai o pollaplasiasmìc enìc dianôsmatoc epð èna pragmatikì migadikì arijmì ex arister n pou dðnei di nusma. Dhlad, to S eðnai kleistì wc proc th dianusmatik prìsjesh kai ton pollaplasiasmì enìc dianôsmatoc epð èna pragmatikì migadikì arijmì ex arister n. H dianusmatik prìsjesh kai o pollaplasiasmìc enìc dianôsmatoc m> ènan pragmatikì migadikì arijmì ex arister n èqoun tic akìloujec idiìthtec:. Antimetajetik x, y S: x + y = y + x. (4.6). Prosetairistik x, y, z S: ( x + y)+ z = x +( y + z). (4.7) 3. 'Uparxh mhdenikoô dianôsmatoc x S: x + = x. (4.8) 4. 'Uparxh antðjetou dianôsmatoc x S y S: x + y =. (4.9) 5. Gia to oudètero stoiqeðo tou bajmwtoô pollaplasiasmoô R isqôei x S: x = x. (4.) 6. Epimeristikèc idiìthtec: x S,a,b R : a (b x) =(ab) x (4.) x S,a,b R : (a + b) x =(a x)+(b x) (4.) x, y S,a R : a ( x + y) =(a x)+(a y). (4.3) upikì par deigma enìc dianusmatikoô q rou eðnai to sônolo R n twn pragmatik n dianusm - twn st lhc megèjouc n x = x =(x,x,...,x n ) (4.4)

K. Kotrìpouloc: S mata-sust mata ìpou x i R, i =,,...,n kai eðnai o telest c anastrof c. EÐnai profanèc ìti antðstoiqa orðzetai ènac dianusmatikìc q roc p nw sto pedðo C. MporeÐ na deiqteð ìti sugkrotoôn dianusmatikoôc q rouc oi suneqeðc sunart seic x(t) gia t [t,t ] oi suneqeðc sunart seic φ(s) gia s = a + jb C, ìtan s C C. DianusmatikoÐ Upoq roi 'Ena mh kenì sônolo M Senìc dianusmatikoô q rou S eðnai ènac dianusmatikìc upoq roc ( gia suntomða apl c upoq roc) tou S, e n o periorismìc twn pr xewn sto M, kajist to sônolo M dianusmatikì q ro. Ikan kai anagkaða sunj kh gia na eðnai to M Su- poq roc tou S eðnai M. H tom dôo upoq rwn eðnai epðshc upoq roc. Gia par deigma upoq roi tou trisdi statou EukleÐdeiou dianusmatikoô q rou R 3 eðnai oi ex c: (a') to R 3 (b') èna opoiod pote epðpedo dia thc arq c (twn axìnwn) =(,, ) (g') mia opoiad pote eujeða dia thc arq c (d') h arq. 'Enac grammikìc sunduasmìc dianusm twn x i S, i =,,...,n, eðnai èna (dianusmatikì) jroisma thc morf c a x + a x +...+ a n x n ìpou a,a,...,a n R. (4.5) a dianôsmata x i S, i =,,...,n, lègontai grammik c exarthmèna an up rqoun a, a,..., a n R ìqi ìloi mhdèn tètoioi ste a x + a x +...+ a n x n =. (4.6) a dianôsmata x i S, i =,,...,n, lègontai grammik c anex rthta an h (4.6) alhjeôei mìno ìtan a = a =...= a n =. 'Estw X = { x, x,..., x n }. Par gon sônolo (span) tou X eðnai to sônolo twn grammik n sunduasm n aut n twn dianusm twn, dhlad span(x )={ y : y = a x + a x +...+ a n x n, a,a,...,a n R}. (4.7)

K. Kotrìpouloc: S mata-sust mata 'Enac grammikìc sunduasmìc dianusm twn mporeð na qrhsimopoihjeð gia na sqhmatðsei ènan aujaðreto upoq ro enìc dianusmatikoô q rou, epeid to span(x ) eðnai upoq roc tou S. AnagnwrÐzoume ìti h (4.9) se sunduasmì me to gegonìc ìti oi suneqeðc pragmatikèc sunart seic x(t) gia t [t,t ] sugkrotoôn èna dianusmatikì q ro ( ra kai oi suneqeðc periodikèc sunart seic an melethjoôn se èna di sthma miac periìdou) empðptei sth suz ths mac perð upoq rwn. Qr simec sunart seic se dianusmatikoôc q rouc 'Estw S o dianusmatikìc q roc twn suneq n sunart sewn miac anex rththc metablht c t (dhlad shm twn) gia t [t,t ] p nw sto R. 'Ola ta s mata pou ja qrhsimopoihjoôn sth sunèqeia ja an koun s> autì to dianusmatikì q ro. OrÐzoume tic akìloujec bajmwtèc sunart seic se dianusmatikoôc q rouc:. Eswterikì ginìmeno <x(t),y(t) >: EÐnai sun rthsh dôo shm twn x(t) kai y(t) gia t [t,t ] pou par gei th bajmwt tim <x(t),y(t) >= t t x(t) y(t) dt. (4.8) Perigr fei pìso moi zoun dôo s mata x(t) kai y(t), kat> analogða me thn probol enìc dianôsmatoc s> èna llo ston trisdi stato EukleÐdeio q ro. IsqÔoun oi akìloujec idiìthtec tou eswterikoô ginomènou: (a ) <x(t),y(t) > = <y(t),x(t) > (b ) <cx(t),y(t) > = c<y(t),x(t) >, c R (g ) <x(t)+y(t),z(t) > = <x(t),z(t) > + <y(t),z(t) > (d ) <x(t),x(t) >, ìpou h isìthta isqôei an x(t) =, t [t,t ]. o eswterikì ginìmeno enìc s matoc me ton eautì tou lègetai enèrgeia tou s matoc.. Apìluth tim mètro x(t) : EÐnai h tetragwnik rðza thc enèrgeiac enìc s matoc x(t) = <x(t),x(t) > = gia thn opoða isqôoun ta ex c t t x (t) dt (4.9) (a ) x(t), me thn isìthta na isqôei gia x(t) tautotik c mhdenikì gia t [t,t ]

K. Kotrìpouloc: S mata-sust mata (b ) a x(t) = a x(t), a R (omoiogèneia) (g ) x(t)+y(t) x(t) + y(t) (trigwnik anisìthta). 3. Apìstash dôo shm twn d(x(t),y(t)): EÐnai sun rthsh dôo shm twn pou lamb nei th bajmwt tim d(x(t),y(t)) = x(t) y(t) = t t (x(t) y(t)) dt. (4.3) Perigr fei pìso diafèroun dôo sunart seic. IsqÔoun oi akìloujec idiìthtec: (a ) d(x(t),y(t)), ìpou h isìthta isqôei an x(t) =y(t) (b ) d(x(t),y(t)) = d(y(t),x(t)) (antisummetrik ) (g ) d(x(t),y(t)) d(x(t),z(t)) + d(z(t),y(t)) (trigwnik anisìthta). Sto dianusmatikì q ro twn migadik n sunart sewn o orismìc tou eswterikoô ginomènou èqei wc ex c: <x(s),y(s) >= ìpou eðnai o telest c migadik c suzugðac. s s x(s) y (s) ds (4.3) B sh dianusmatikoô q rou 'Opwc eðnai gnwstì, èna sônolo grammik c anexart twn dianusm twn pou par goun to dianusmatikì q ro S eðnai b sh sto S. Dhlad, opoiod pote stoiqeðo tou S mporeð na anaparastajeð wc ènac monadikìc grammikìc sunduasmìc twn dianusm twn pou sugkrotoôn th b sh (efex c dianôsmata b shc). K je dianusmatikìc q roc diajètei mia b sh, molonìti den eðnai mða kai monadik en gènei. O arijmìc twn dianusm twn b shc orðzei th di stash (dimension) tou dianusmatikoô q rou. 'Eqontac orðsei thn ènnoia tou eswterikoô ginomènou sto dianusmatikì q ro twn suneq n shm twn x(t) p nw sto R gia t [t,t ] mporoôme na orðsoume th gwnða dôo shm twn wc (x(t),y(t)) = <x(t),y(t) > x(t) y(t) x(t), y(t). (4.3) Epomènwc dôo s mata lègontai orjog nia an <x(t),y(t) >=. (4.33)

K. Kotrìpouloc: S mata-sust mata 3 An ta mh-mhdenik dianôsmata x, x,..., x n eðnai an dôo orjog nia, tìte eðnai grammik c anex rthta. Sunep c mia sullog n sunart sewn ϕ (t), ϕ (t),..., ϕ n (t), t [t,t ], pou eðnai an dôo orjog niec kai epiplèon èqoun monadiaðo mètro, dhlad i = j <ϕ i (t),ϕ j (t) >= δ ij = i j (4.34) ìpou δ ij eðnai dèlta Kronecker, sugkrotoôn mia orjokanonik b sh. H apaðthsh gia monadiaðo mètro sunep getai mia klim kwsh (kanonikopoðhsh), h opoða mporeð eôkola na ikanopoihjeð genik c. MporeÐ na deiqjeð epðshc ìti kai h apaðthsh gia orjogwniìthta mporeð na apalunjeð mèsw thc diadikasðac Gram-Schmidt. Epomènwc h mình desmeutik idiìthta pou prèpei na plhroôn oi sunart seic b shc eðnai h grammik anexarthsða. Oi sunhmitonoeideðc sunart seic se di sthma miac periìdou me kat llhlh kanonikopoðhsh, ìpwc ja exet soume sth sunèqeia, sugkrotoôn mia orjokanonik b sh pou axiopoieðtai sthn epèktash se seir Fourier. Gia tic sunhmitonoeideðc sunart seic, jroisma, diafìrish kai o- lokl rwsh odhgoôn p li se sunhmitoeideðc sunart seic. Aut h idiìthta twn sunhmitonoeid n sunart sewn sun dei me ta poiotik qarakthristik enìc basikoô s matoc pou epikalest kame se prohgoômeno kef laio. Genikìtera, ta fantastik ekjetik se di sthma miac periìdou sugkrotoôn epðshc mia orjokanonik b sh me kat llhlh kanonikopoðhsh. Ektìc apì th seir Fourier llec di shmec seirèc eðnai oi: seirèc Legendre pou qrhsimopoioôn ta polu numa Legendre wc sunart seic b shc seirèc Laguerre pou qrhsimopoioôn ta om numa polu numa wc sunart seic b shc seirèc Walsh pou qrhsimopoioôn tic om numec sunart seic wc sunart seic b shc. Sth sunèqeia ja jemeli soume mia genik jewrða epèktashc enìc s matoc se seir, thc opoðac merik perðptwsh ja eðnai h epèktash se seir Fourier gia thn eôresh twn suntelest n a k sth (4.9). 4.3. Je rhma epèktashc se seir 'Estw x(t) = + n= a n ϕ n (t) (4.35)

4 K. Kotrìpouloc: S mata-sust mata ìtan oi sunart seic {ϕ n (t)} sugkrotoôn mia orjokanonik b sh sto dianusmatikì q ro twn shm twn x(t) gia t [t,t ]. Jèloume na broôme mia prosèggish ˆx(t) thc x(t) qrhsimopoi ntac peperasmèno pl joc ìrwn M< sthn (4.35) ˆx(t) = â n ϕ n (t). (4.36) Ac upotejeð ìti to M èqei epileqjeð. Poi eðnai h kalôterh prosèggish (4.36) thc (4.35)? Gia na apanthjeð to er thma, prèpei na oristeð to krit rio kalôterhc prosèggishc. 'Ena tètoio ja mporoôse na tan h apìstash metaxô twn x(t) kai ˆx(t) I = d(x(t), ˆx(t)) = t [x(t) â n ϕ n (t)] dt. (4.37) t Opìte prèpei na broôme ta â n, n = M,...,M, ste na elaqistopoieðtai to I, dhlad â n : I â n = n = M,...,M. (4.38) 'Eqoume I = = = Opìte t t x (t) dt t t x (t) dt t t x (t) dt â n t â n t t x(t) ϕ n (t) dt + t x(t)ϕ n (t) dt + â n <x(t),ϕ n (t) > + m= M m= M â n â m t t â n â m δ nm ϕ n (t) ϕ m (t) dt â n. (4.39) I â n ân,opt = <x(t),ϕ n (t) > +â n,opt = â n,opt = <x(t),ϕ n (t) > n = M,...,M. (4.4) o mikrìtero sf lma prosèggishc eðnai I min = t t x (t) dt â n,opt. (4.4) Epeid oi sunart seic {ϕ n (t)} sugkrotoôn mia orjokanonik b sh, to x(t) mporeð na anaparastajeð wc grammikìc sunduasmìc twn sunart sewn b shc, opìte â n,opt =<x(t),ϕ n (t) >=< + a m ϕ m (t), ϕ n (t) >= + m= m= a m <ϕ m (t),ϕ n (t) >= a n n (4.4)

K. Kotrìpouloc: S mata-sust mata 5 ìpou k name qr sh thc + m= a m δ mn = a n n. (4.43) H (4.4) orðzei to peperasmèno twn suntelest n. elik diatôpwsh tou jewr matoc: 'Estw S o dianusmatikìc q roc twn shm twn x(t) gia t [t,t ] p nw sto pedðo R kai {ϕ n (t)} mða orjokanonik b sh tou S. An x(t) anaptôssetai wc ex c N x(t) = a n ϕ n (t) (4.44) n= N ìpou N sun jwc eðnai peiro, jèloume na proseggðsoume to x(t) wc grammikì sunduasmì me peperasmèno pl joc (M +) sunart sewn orjokanonik c b shc, ste to mèso tetragwnikì sf lma I na eðnai el qisto. ìte arkeð oi suntelestèc thc prosèggishc na eklegoôn â n = a n =<x(t),ϕ n (t) > n = M,...,M (4.45) opìte h prosèggish dðnetai apì thn ˆx(t) = a n ϕ n (t) (4.46) kai to mèso tetragwnikì sf lma thc prosèggishc eðnai I = t t O deôteroc ìroc thc (4.47) ermhneôetai wc x (t) dt a n. (4.47) t t ˆx (t)dt = = t a n a m ϕ n (t)ϕ m (t) dt m= M m= M t a n a m δ nm = a n. (4.48) Parathr seic. Anisìthta Bessel: Epeid to mèso tetragwnikì sf lma prosèggishc eðnai mh arnhtikì èqoume I t x (t)dt a n. (4.49) t

6 K. Kotrìpouloc: S mata-sust mata. autìthta Parseval: An M = N =, tìte h enèrgeia tou s matoc dðnetai apì thn W = t t x (t) dt = + n= a n. (4.5) 3. Oi sunart seic b shc kaloôntai pl reic, ìtan lim η M = lim ( M M W a n)=, x(t). (4.5) Sun jwc ìlec oi b seic eðnai pl reic, opìte aux nontac to M, to l joc prosèggishc gðnetai mikrì. 4.3.3 Je rhma orjogwnik c arq c (Wiener-Kolmogorov) An oi suntelestèc â n kajistoôn el qisto to sf lma prosèggishc I = [ t t (x(t) ] â n ϕ n (t)) dt (4.5) dhlad, an h apìstash tou x(t) apì thn prosèggis tou ˆx(t) eðnai el qisth, tìte to s ma sf lmatoc e(t) =x(t) â n ϕ n (t) (4.53) eðnai orjog nio proc ìlec tic sunart seic ϕ n (t). IsqÔei kai to antðstrofo: An to s ma sf lmatoc eðnai orjog nio proc ìlec tic ϕ n (t), tìte h apìstash tou x(t) apì to ˆx(t) eðnai el qisth. Apìdeixh eujèoc: I = [ t (x(t) â m â m t t t [x(t) <x(t) ] â n ϕ n (t)) dt = â n ϕ n (t)]ϕ m (t) dt = m = M,...,M â n ϕ n (t),ϕ m (t) >= m = M,...,M. (4.54) Sthn apìdeixh den ègine qr sh ìti oi sunart seic {ϕ n (t)} M sugkrotoôn orjokanonik b sh. 'Ara to je rhma thc orjogwnik c arq c èqei genik isqô.

K. Kotrìpouloc: S mata-sust mata 7 4.3.4 Genik sumper smata 'Estw o dianusmatikìc q roc twn shm twn S kai jèloume na k noume grammik epèktash enìc s matoc x(t) S An oi sunart seic ϕ n (t) x(t) = â n ϕ n (t). (4.55) sugkrotoôn mia pl rh mh-pl rh orjokanonik b sh, k je suntelest c upologðzetai qrhsimopoi ntac thn (4.45) kai o upologismìc eðnai anex rthtoc apì ekeðnouc gia touc llouc suntelestèc, en an den sugkrotoôn orjokanonik b sh, gia na brejoôn oi suntelestèc arkeð na epilujeð to sôsthma exis sewn (4.54), opìte o upologismìc kajenìc suntelest den gðnetai anex rthta apì touc llouc. 4.4 Seir Fourier 4.4. Sunart seic b shc twn seir n Fourier 'Estw oi sunart seic ˆφ k (t) =coskω t, ẑ k (t) =e jkω t k =, ±, ±,..., (4.56) ìpou ω = π =πf. Eklègoume wc di sthma [t,t ] gia ton orismì tou dianusmatikoô q rou, to di sthma miac periìdou opìte an k n < ˆφ k (t), ˆφ n (t) > = en gia k = n = = = / < ˆφ k (t), ˆφ k (t) >= / / / cos kω t cos nω tdt (k + n)ω sin / { [ ] [ cos (k + n)ω t +cos (k n)ω t] } dt [ ] / (k + n)ω t + / / cos (kω t)dt = / / [ ] / sin (k n)ω t (k n)ω / (4.57) [ ] +coskω t dt =. (4.58)

8 K. Kotrìpouloc: S mata-sust mata An eklèxw tic sunart seic ϕ k (t) wc ϕ k (t) = cos kω t = ˆφ k (t) (4.59) tìte oi sunart seic {ϕ k (t)} eðnai sunart seic orjokanonik c b shc sto di sthma miac periìdou. An loga mporeð na deiqteð ìti eðnai sunart seic orjokanonik c b shc kai oi OmoÐwc ψ k (t) = sin kω t. (4.6) < ẑ k (t), ẑ m (t) > = = = / / / / e jkω t (e jmω t ) dt = cos[(k m)ω t] dt + j k = m k m. / / / / e j(k m)ω t dt sin[(k m)ω t] dt (4.6) Opìte arkeð na eklegoôn oi sunart seic z k (t) = e jkω t (4.6) ste na sugkrothjeð mia orjokanonik b sh fantastik n ekjetik n. 4.4. Exis seic an lushc kai sônjeshc thc seir c Fourier Sqedìn k je s ma x(t) mporeð na epektajeð se seir sto di sthma [t,t ] me t t = sunart sei twn sunart sewn b shc (4.59), (4.6) kai (4.6). An to s ma x(t) eðnai periodikì me perðodo, h epèktash isqôei gia k je t. H diatôpwsh sqedìn parapèmpei stic sunj kec Dirichlet pou ja exetastoôn argìtera. An ikanopoioôntai oi sunj kec Dirichlet, èna s ma x(t) analôetai se peiro jroisma twn parak tw morf n gia t t<t + : x(t) = a + (a n cos nω t + b n sin nω t) n= (trigwnometrik seir ) (4.63) x(t) = c n e jnω t (ekjetik seir ) (4.64) n=

K. Kotrìpouloc: S mata-sust mata 9 ìpou a = a n = b n = c n = t + t x(t) dt (4.65) t + t x(t) cosnω tdt, n (4.66) t + t x(t) sinnω tdt (4.67) t + t x(t) e jnω t dt. (4.68) Gia periodikì s ma x(t) me perðodo, oi (4.63) kai (4.64) isqôoun gia k je t lìgw tou x(t + )=x(t), t. Apìdeixh gia ekjetik seir Fourier Eklègontai oi sunart seic orjokanonik c b shc z n (t) = e jnω t. (4.69) Prosèggish x(t): ˆx(t) = ĉ n z n (t) = M Epeid oi {z n (t)} eðnai sunart seic orjokanonik c b shc ĉ n =<x(t),z n (t) >= t + x(t)zn(t) dt = t + t ĉ n e jnω t. (4.7) t x(t) e jnω t dt. (4.7) Antikajist ntac ta ĉ n sthn ˆx(t) pou dðnetai apì thn (4.7) paðrnoume ˆx(t) = t + [ x(t) e jnω ] t dt e jnω t t } {{} c n (4.7) ìpou anagnwrðzontai oi suntelestèc thc ekjetik c seir c Fourier (4.68) pou sqetðzontai me touc suntelestèc ĉ n pou problèpei h jewrða dia thc c n = ĉ n ĉ n = c n. (4.73) Mèso tetragwnikì sf lma: t + I = x (t) dt t } {{ } W ĉ n = W c n = W ( W c n ). } {{ } η M (4.74)

K. Kotrìpouloc: S mata-sust mata Mètro sf lmatoc: η M = W c n. (4.75) Oi sunart seic b shc (4.69) eðnai pl reic. Pr gmati (a') 'Eqoume η M+ = η M W [ c M + c M+ ]. (4.76) (b') IsqÔei ìti η M epit ssei, M. Epiprosjètwc, gia M h tautìthta tou Parseval + n= c n }{{} W. (4.77) ĉ n Epomènwc η M, M. M' lla lìgia h akoloujða {η M } sugkroteðtai apì jetikoôc pragmatikoôc arijmoôc mikrìterouc thc mon dac. (g') Epeid η M η M+ η M+ η M, h akoloujða {η M } den eðnai aôxousa. (d') 'Ara h akoloujða sugklðnei kai sunep c lim M η M =. Gia pl reic sunart seic b shc isqôei h tautìthta tou Parseval pou mac epitrèpei na ekfr soume thn enèrgeia se di sthma miac periìdou W kai th mèsh isqô P = W wc ex c: W = t + t x (t) dt = + c n P = W + = c n. (4.78) n= n= Gia trigwnometrik seir Fourier h (4.78) xanagr fetai gia thn enèrgeia se di sthma miac periìdou wc W = a 4 + (a n + b n) (4.79) n= kai gia th mèsh isqô P wc P = W = t + t x (t) dt = a 4 + (a n + b n). (4.8) n= Sto Par deigma 4.3 apodeiknôetai h exagwg twn (4.65) kai (4.66).

K. Kotrìpouloc: S mata-sust mata 4.4.3 Sunj kec Dirichlet Apì thn an lush pou prohg jhke, ègine fanerì ìti gia ta suneq periodik s mata (ta opoða sugkrotoôn dianusmatikì q ro an eklèxoume èna di sthma miac periìdou) h anapar stash se seir Fourier par gei èna mèso tetragwnikì sf lma pou mhdenðzetai gia epèktash peirwn ìrwn. An logh idiìthta sôgklishc thc seir c Fourier parathreðtai kai gia poll asuneq periodik s mata p.q. thn tetragwnik periodik palmoseir. 'Ena periodikì s ma x(t) epekteðnetai se seir Fourier se di sthma miac periìdou an isqôoun oi sunj kec Dirichlet :. o s ma x(t) eðnai monos manta orismèno se di sthma miac periìdou.. o s ma x(t) èqei peperasmèno arijmì asuneqei n peperasmènou megèjouc se di sthma miac periìdou. 3. o s ma x(t) parousi zei peperasmèno arijmì akrot twn se di sthma miac periìdou. 4. o s ma x(t) eðnai apolôtwc oloklhr simo se di sthma miac periìdou x(t) dt <. (4.8) An isqôoun oi sunj kec Dirichlet, tìte to x(t) isoôtai me thn epèktash se seir Fourier ektìc apì memonwmènec timèc tou t, stic opoðec to x(t) eðnai asuneqèc. Stic asunèqeiec, h seir Fourier sugklðnei sto hmi jroisma twn orðwn tou x(t) ekatèrwjen thc asunèqeiac, dhlad, an t eðnai shmeðo asunèqeiac h seir Fourier sugklðnei sto x(t )= [x(t+ )+x(t )]. (4.8) Oi sunj kec Dirichlet ikanopoioôntai apì sqedìn ìla ta <<qr sima>> s mata pou apant ntai sthn pr xh. oôto gðnetai katanohtì an exet soume tð eðdouc s mata parabi zoun tic sunj kec Dirichlet. 'Etsi h sunj kh 4 parabi zetai apì to periodikì s ma me perðodo = x(t) = t <t. (4.83) H sunj kh 3 parabi zetai apì to periodikì s ma me perðodo = x(t) = sin ( ) π t <t. (4.84)

K. Kotrìpouloc: S mata-sust mata H sunj kh parabi zetai apì to periodikì s ma me perðodo =8 t<4 x(t) = 4 t<6 6 t<7 4 7 t<7.5 8. (4.85) 4.4.4 Fainìmeno Gibbs Gia asuneq s mata sta shmeða asunèqeiac pèra apì to z thma thc sôgklishc parathreðtai kai to legìmeno fainìmeno Gibbs. En to ìrio tou ajroðsmatoc teðnei sto misu tou dexioô kai aristeroô orðou sto shmeðo asunèqeiac wc sunèpeia twn sunjhk n Dirichlet, gôrw apì to shmeðo asunèqeiac parathreðtai mia kum twsh me mia mèsh tim kat 9% megalôterh apì thn pragmatik tim tou s matoc. Epiplèon parathr jhke ìti:. H kum twsh eðnai anex rthth apì ton arijmì twn ìrwn thc seir c Fourier.. Me thn aôxhsh twn ìrwn thc seir c h kum twsh sugkentr netai kont sthn asunèqeia. o Sq ma 4.3 deðqnei to fainìmeno Gibbs parastatik gia thn periodik tetragwnik palmoseir. Seir Fourier periodik c tetragwnik c palmoseir c ( M =, M =3) Seir Fourier periodik c tetragwnik c palmoseir c ( M = 7, M = 9).8.8 x(t), ˆx(t).6.4 x(t), ˆx(t).6.4.. -. - -.5 - -.5.5.5 t -. - -.5 - -.5.5.5 t (a) (b) Sq ma 4.3: Epèktash se seir Fourier miac periodik c tetragwnik c palmoseir c ìtan to pl joc twn ìrwn thc epèktashc eðnai (a) M =, 3. (b) M =7, 9.

K. Kotrìpouloc: S mata-sust mata 3 4.4.5 Seirèc Fourier summetrik n shm twn H ekmet lleush twn summetri n dieukolônei touc upologismoôc, giatð. an x(t) eðnai rtiac summetrðac gia t [ /,/] b n = n (4.86) a n = 4 /. en an x(t) eðnai peritt c summetrðac gia t [ /,/] x(t) cosnω tdt (4.87) a n = n (4.88) b n = 4 / x(t) sinnω tdt. (4.89) Par deigma 4.. DÐnetai o periodikìc tetragwnikìc palmìc me jemeli dh perðodo kai di rkeia tou Sq matoc 4.4. O palmìc èqei thn ex c analutik perigraf : t < x(t) = < t <. Na exaqjoôn oi suntelestèc thc trigwnometrikhc seir c Fourier. (4.9) x(t) t + Sq ma 4.4: Periodikìc tetragwnikìc palmìc. LÔsh. Epeid o palmìc eðnai rtiac summetrðac oi suntelestèc thc seir c hmitìnwn eðnai mhdenikoð, dhlad b n =, n. Opìte mac endiafèroun oi suntelestèc a n. a = a n = x(t) dt = 4 x(t) dt = 4 dt = 4 (4.9) x(t)cosnω tdt= 4 cos nω tdt= 4 sin nω t nω = 4 sin nω = 4 nω nπ sin nω = sin nω, n. (4.9) nπ

4 K. Kotrìpouloc: S mata-sust mata An = 4 =4, opìte prokôptei summetrikìc tetragwnikìc palmìc, paðrnoume a n = sin(nω /4) nπ ìpou h sun rthsh sinc(x) orðzetai wc = sin(nπ/) nπ =sinc( nπ ) (4.93) sinc(x) = sin x x. (4.94) o Sq ma 4.5a deðqnei parastatik touc suntelestèc thc trigwnometrik c seir c Fourier tou periodikoô summetrikoô tetragwnikoô palmoô gia n =,,...,. ParathroÔme ìti oi suntelestèc deigmatolhptoôn th suneq perib llousa pou orðzei h sun rthsh sinc(x) gia x = nπ, me n Z+. Endeiktikèc timèc suntelest n thc seir c Fourier:.5.8.4 an, sinc(xπ/).6.4. cn,.5sinc(xπ/).3.. -. -. -.4 4 6 8 4 6 8 n, x -. - -5 - -5 5 5 n, x (a) (b) Sq ma 4.5: (a) Suntelestèc thc trigwnometrik c seir c Fourier tou periodikoô summetrikoô tetragwnikoô palmoô. (b) Suntelestèc thc ekjetik c seir c Fourier tou periodikoô summetrikoô tetragwnikoô palmoô. a = (4.95) a = π a 3 = sin(3π/) 3π/ (4.96) = 3π. (4.97) Oi suntelestèc thc ekjetik c seir c Fourier dðnontai apì tic sqèseic c = (4.98) kai sqedi zontai sto Sq ma 4.5b. c n = sinc(nπ )= a n, n = ±, ±,... (4.99)

K. Kotrìpouloc: S mata-sust mata 5 Par deigma 4.3. Na exaqjoôn oi sqèseic: a n = a = t + t t + t x(t) cosnω tdt x(t) dt gia thn trigwnometrik seir Fourier x(t) = a + ( ) a n cos nω t + b n sin nω t. H exagwg twn suntelest n b n af netai wc skhsh ston anagn sth. Apìdeixh n= (A) DeÐxame ìti oi sunart seic ϕ n (t) = ìpwc kai oi sunart seic ψ n (t) = cos nω t sugkrotoôn mia orjokanonik b sh, sin nω t. ParathroÔme epðshc ìti gia n m <ϕ n (t),ψ m (t) > = = = = = }{{} n m = t + t t + t t + t cos nω t sin mω tdt= (4.) cos nω t sin mω tdt= [ ] sin(n m)ω t +sin(n + m)ω t dt = { } sin(n m)ω tdt+ sin(n + m)ω tdt = { [ ] n m cos(n m)ω t + [ + ] } n + m cos(n + m)ω t = { ( [ n m ) cos(n m) π ] + +( [ n + m ) cos(n + m) π ] =. (4.) An n = m apì thn (4.) paðrnoume = cos nω t sin nω tdt= sin nω tdt= sin nω tdt= cos nω t =. (4.) 'Ara <ϕ n (t),ψ m (t) >= n, m Z + (4.3)

6 K. Kotrìpouloc: S mata-sust mata Opìte ìtan endiaferìmaste gia touc suntelestèc thc seir c sunhmitìnwn agnooôme touc ìrouc tou ajroðsmatoc epèktashc thc x(t) pou emplèkoun hmðtona. (B) H jewrða problèpei ìti: â n =<x(t),ϕ n (t) > n (4.4) kai ìti x(t) = dc ìroc + = dc ìroc + â n ϕ n (t) =dc ìroc + n= n= ( ) â n cos nω t â n cos nω t. (4.5) n= UpologÐzontac thn (4.4) èqoume: â n =<x(t),ϕ n (t) >= x(t) cosnω tdt. (4.6) Antikajist ntac sthn (4.5): x(t) = dc ìroc + n= [ = dc ìroc + n= [ ] x(t) cosnω tdt cos nω t ] x(t) cosnω tdt cos nω t. }{{} a n n UpoleÐpetai na prosdioristeð o dc ìroc. An jèsoume ìpou n =sthn ϕ n (t) = cos nω t t (4.7) paðrnoume En isqôei ϕ (t) = <ϕ (t),ϕ n (t) >= n t. (4.8) parathroôme ìti <ϕ (t),ϕ (t) >= 'Ara prèpei na kanonikopoihjeð h ϕ (t) se: dt =. ϕ (t) = =, t (4.9)

K. Kotrìpouloc: S mata-sust mata 7 gia na apoteleð stoiqeðo orjokanonik c b shc twn sunhmitoeid n sunart sewn. ìte â =<x(t),ϕ (t) >= opìte o dc ìroc sthn trigwnometrik seir eðnai [ a ] =â ϕ (t) = x(t)dt a = x(t)dt. (4.) x(t) dt. (4.) 4.5 Idiìthtec thc seir c Fourier suneqoôc qrìnou H anapar stash enìc periodikoô s matoc suneqoôc qrìnou se seir Fourier èqei èna shmantikì arijmì idiot twn pou mac epitrèpoun afenìc na katano soume se b joc th fusik shmasða miac tètoiac anapar stashc pou anadeiknôei to suqnotikì perieqìmeno tou s matoc kai afetèrou mac dieukolônoun sthn exagwg thc seir c gia poll s mata. Sthn an ptuxh ja qrhsimopoi soume th sômbash x(t) FS a k (4.) gia na dhl soume ìti èna periodikì s ma x(t) me perðodo kai jemeli dh (kuklik ) suqnìthta ω = π èqei suntelestèc ekjetik c seir c Fourier a k, dhlad isqôei to zeôgoc exis sewn an lushc kai sônjeshc: a k = x(t) e jkω t dt (4.3) x(t) = a k e jkω t. (4.4) k= Efex c ja anaferìmaste sthn ekjetik seir Fourier. O PÐnakac 4. sunoyðzei tic idiìthtec thc seir c Fourier. 4.5. Grammikìthta 'Estw x(t) kai y(t) dôo periodik s mata me perðodo pou èqoun suntelestèc seir c Fourier a k kai b k antistoðqwc. K je grammikìc sunduasmìc twn x(t) kai y(t) eðnai epðshc periodikì s ma me perðodo kai suntelestèc seir c Fourier pou dðnontai apì th z(t) =Ax(t)+By(t) FS c k = Aa k + Bb k, A, B C. (4.5) H apìdeixh thc (4.5) eðnai apl kai sthrðzetai sthn efarmog thc exðswshc an lushc (4.3).

8 K. Kotrìpouloc: S mata-sust mata PÐnakac 4.: Idiìthtec thc seir c Fourier suneqoôc qrìnou. Idiìthta Periodikì S ma Suntelestèc seir c x(t) y(t) periodik me perðodo kai jemeli dh suqnìthta ω = π Fourier a k b k Grammikìthta Ax(t)+By(t) Aa k + Bb k Qronik metatìpish x(t t ) a k e jkω t Metatìpish suqnìthtac e jmω t x(t) a k M SuzugÐa x (t) a k Qronik anastrof x( t) a k Qronik klim kwsh Periodik sunèlixh Pollaplasiasmìc Diafìrish Olokl rwsh Suzug c summetrða gia pragmatik s mata Pragmatik s mata rtiac summetrðac Pragmatik s mata peritt c summetrðac x(α t), α> (periodikì me perðodo α ) x(τ) y(t τ) dτ t x(t) y(t) dx(t) dt x(t) dt (peperasmènhc tim c kai periodikì mìno an a =) x(t) R x(t) R: x(t) =x( t) x(t) R: x(t) = x( t) a k a k b k + l= a l b k l jkω a k ( ) a jkω k 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 AposÔnjesh se rtio kai perittì mèroc pragmatikoô s matoc a k kajar c fantastikoð kai peritt c summetrðac x e (t) = (x(t)+x( t)) Re{a k } x o (t) = (x(t) x( t)) j Im{a k } autìthta Parseval gia periodik s mata x(t) dt = + k= a k

K. Kotrìpouloc: S mata-sust mata 9 4.5. Qronik metatìpish H qronik metatìpish enìc periodikoô s matoc x(t) pou èqei perðodo den alloi nei thn periodikìthta tou s matoc. Epomènwc to prokôpton s ma y(t) =x(t t ) mporeð na analujeð se seir Fourier me suntelestèc b k = x(t t ) e jkω t dt τ=t t = x(τ) e jkω (τ+t ) dτ = e jkω t a k. (4.6) Dhlad x(t t ) FS e jkω t a k (4.7) pou upodhloð ìti ìtan èna s ma metatopðzetai sto qrìno to mètro twn suntelest n thc seir c Fourier paramènei analloðwto. 4.5.3 Qronik anastrof H qronik anastrof enìc periodikoô s matoc den alloi nei thn periodikìthta tou s matoc, ra to qronik c anastrammèno s ma mporeð na analujeð se seir Fourier me suntelestèc ac poôme b k. Ac xekin soume apì th (4.4) h opoða eðnai tautìthta antikajist ntac ìpou t to t: x( t) = apì thn opoða prokôptei ìti k= a k e jkω ( t) m= k = m= a m e jmω t (4.8) x( t) FS b k = a k (4.9) pou upodhloð ìti h qronik anastrof odhgeð se anastrof tou deðkth thc seir c Fourier. An x(t) eðnai s ma rtiac summetrðac, tìte abðasta prokôptei ìti a k = a k, opìte h akoloujða twn suntelest n Fourier èqei rtia summetrða. AntistoÐqwc an to s ma èqei peritt summetrða, dhlad x(t) = x(t), tìte a k = a k, opìte h peritt summetrða antigr fetai apì thn akoloujða twn suntelest n thc seir c Fourier. 4.5.4 Qronik klim kwsh H qronik klim kwsh enìc s matoc x(t) pou èqei perðodo kat par gonta α R + odhgeð s> èna periodikì s ma me perðodo α kai jemeli dh suqnìthta αω. Ac xekin soume apì thn

3 K. Kotrìpouloc: S mata-sust mata exðswsh sônjeshc antikajist ntac ìpou t to α t. ìte: x(αt)= a k e jkω (αt) = a k e jk(αω ) t k= k= (4.) pou upodhloð ìti oi suntelestèc thc seir c paramènoun analloðwtoi, all anafèrontai se armonikèc me jemeli dh suqnìthta αω. 4.5.5 Pollaplasiasmìc sto pedðo tou qrìnou Pollaplasi zontac dôo periodik s mata x(t) kai y(t) thc idðac periìdou me suntelestèc seir c Fourier a k kai b k antistoðqwc, prokôptei periodikì s ma me thn aut perðodo tou opoðou oi suntelestèc thc seir c Fourier, ac poôme c k, ex gontai wc ex c x(t)y(t) = m=k+l = ( k= m= )( a k e jkω t ( k= l= a k b m k ) e jmω t ) b l e jlω t = k= l= a k b l e j (k+l) ω t (4.) ap> ìpou prokôptei ìti x(t)y(t) FS c k = m= a m b k m. (4.) H (4.) upodhloð ìti oi suntelestèc thc seir c Fourier tou ginomènou dôo periodik n shm twn sto pedðo tou qrìnou prokôptoun wc sunèlixh twn akolouji n twn suntelest n Fourier twn dôo epimèrouc shm twn. 4.5.6 SuzugÐa kai suzug c summetrða Xekin ntac apì thn (4.4) paðrnoume ( x (t) = k= ) a k e jkω t = k= a k e jkω t m= k = m= a m e jmω t (4.3) opìte x (t) FS a k. (4.4) Shmantikèc idiìthtec summetrðac prokôptoun gia pragmatikì periodikì s ma x(t), giatð x(t) pragmatik c tim c x(t) =x (t) FS a k = a k a k = a k (4.5)

K. Kotrìpouloc: S mata-sust mata 3 opìte lème ìti h akoloujða twn suntelest n Fourier èqei suzug summetrða (conjugate symmetry). oôto praktik shmaðnei ìti a R Re{a k } = Re{a k } Im{a k } = Im{a k } (4.6) a k = a k a k = a k ìpou c upodhloð th f sh tou migadikoô arijmoô c C. Epomènwc oi akoloujðec twn pragmatik n mer n twn suntelest n thc ekjetik c seir c Fourier, twn suntelest n sunhmitìnwn thc trigwnometrik c seir c Fourier kai twn mètrwn twn suntelest n thc ekjetik c seir c Fourier èqoun rtia summetrða, en oi akoloujðec twn fantastik n mer n twn suntelest n thc ekjetik c seir c Fourier, twn suntelest n hmitìnwn thc trigwnometrik c seir c Fourier kai twn f sewn twn suntelest n thc ekjetik c seir c Fourier èqoun peritt summetrða. Epiplèon o dc ìroc pou eðnai h mèsh tim tou s matoc se di rkeia miac periìdou eðnai pragmatikìc arijmìc, ìpwc anamenìtan. An to s ma x(t) eðnai pragmatikì kai rtiac summetrðac, tìte x(t) pragmatikì rtiac summetrðac FS a k = a k epeid x(t) =x( t) a k = a k epeid x(t) =x (t) a k = a k. (4.7) epomènwc h akoloujða twn suntelest n thc seir c Fourier eðnai pragmatik akoloujða rtiac summetrðac. OmoÐwc mporeð na deiqjeð ìti an x(t) eðnai pragmatikì s ma peritt c summetrðac, tìte h akoloujða twn suntelest n thc seir c Fourier eðnai kajar c fantastik akoloujða peritt c summetrðac. Sthn teleutaða perðptwsh o dc ìroc eðnai mhdenikìc, a =. 4.5.7 autìthta tou Parseval Apì th genik jewrða thc epèktashc se seir Fourier prokôptei x(t) dt = a k (4.8) k= pou upodhloð ìti h mèsh isqôc se di sthma miac periìdou tou periodikoô s matoc x(t) eðnai Ðsh me to jroisma twn tetrag nwn twn mètrwn twn suntelest n thc seir c Fourier. o

3 K. Kotrìpouloc: S mata-sust mata tetr gwno tou mètrou tou k-stoô suntelest isoôtai me th mèsh isqô thc k-sthc armonik c sunist sac tou s matoc x(t). Pr gmati a k e jkω t dt = a k dt = a k. (4.9) Par deigma 4.4. Jewr ste to periodikì s ma y(t) me jemeli dh perðodo 4, tou opoðou mia perðodoc sqedi zetai sto Sq ma 4.6. Na prosdiorðsete touc suntelestèc thc seir c Fourier.8.6.4. y(t) -. -.4 -.6 -.8 - - -.5 - -.5.5.5 t Sq ma 4.6: MÐa perðodoc tou periodikoô s matoc y(t). tou s matoc autoô qrhsimopoi ntac tic idiìthtec thc seir c Fourier kai to dedomèno ìti oi suntelestèc thc seir c Fourier tou s matoc x(t) tou Sq matoc 4.7 eðnai sin( kπ ) an k a k = kπ an k =. (4.3) Apì th sôgkrish twn Sqhm twn 4.6 kai 4.7 prokôptei ìti y(t) =x(t ). (4.3) 'Estw b k oi suntelestèc thc seir c Fourier tou s matoc x(t ). Apì thn idiìthta thc qronik c metatìpishc èqoume b k = a k e jk ( π) 4 = a k e jk π = a k ( ) k. (4.3) Oi suntelestèc thc seir c Fourier tou stajeroô s matoc z(t) = eðnai an k c k = an k =. (4.33)

K. Kotrìpouloc: S mata-sust mata 33.8.6 x(t).4. - -.5 - -.5.5.5 t Sq ma 4.7: Mia perðodoc tou periodikoô s matoc x(t). 'Estw d k oi suntelestèc thc seir c Fourier tou s matoc y(t). Efarmìzontac thn idiìthta thc grammikìthtac prokôptei ìti a k ( ) k d k = a an k an k = a k ( ) k an k = an k =. (4.34) ExÐsou ègkurh eðnai h lôsh pou sthrðzetai sth sqèsh y(t) = x(t). Par deigma 4.5. Jewr ste to s ma h(t) me jemeli dh perðodo 4 tou opoðou mia perðodoc sqedi zetai sto Sq ma 4.8. Na prosdiorðsete touc suntelestèc thc seir c Fourier tou s matoc autoô qrhsimopoi ntac tic idiìthtec thc seir c Fourier. 'Estw e k oi suntelestèc thc seir c Fourier tou s matoc h(t). o s ma y(t) tou Sq matoc 4.6 anagnwrðzoume ìti eðnai h par gwgoc tou s matoc h(t) tou Sq matoc 4.8. Epomènwc apì th sqetik idiìthta thc seir c Fourier èqoume Opìte gia k, e k = d k jkπ =( )k d k = jk π 4 e k. (4.35) a k jkπ sin(π k ) =( )k, k (4.36) j (k π) en gia k =, o suntelest c e mporeð na prosdioristeð brðskontac to embadì k tw apì thn kampôlh se mi perðodo tou h(t) kai diair ntac me th di rkeia thc periìdou e =. (4.37)

34 K. Kotrìpouloc: S mata-sust mata.9.8.7.6 h(t).5.4.3.. - -.5 - -.5.5.5 t Sq ma 4.8: Mia perðodoc tou periodikoô s matoc h(t). Par deigma 4.6. Upojèste ìti sac dðnetai h akìloujh plhroforða gia èna s ma x(t):. o s ma x(t) eðnai pragmatikì.. o s ma x(t) eðna periodikì me perðodo =6kai èqei suntelestèc ekjetik c seir c Fourier a k. 3. a k =gia k >. 4. o s ma me suntelestèc ekjetik c seir c Fourier b k = e jπk a k èqei peritt summetrða. 5. 6 6 x(t) dt =. Na breðte to s ma x(t). Apì to dedomèno 3 prokôptei ìti πt j x(t) =a + a e 3 + a e j πt 3. (4.38) Apì to dedomèno 4 prokôptei ìti to s ma me suntelestèc ekjetik c seir c Fourier b k, ac poôme y(t), eðnai èna metatopismèno antðgrafo tou s matoc x( t), epeid to teleutaðo s ma anagnwrðzoume ìti èqei suntelestèc seir c Fourier a k. H kajustèrhsh eðnai t = 3, epeid e jkω t = e jk π 6 3 = e j πk. (4.39) 'Ara y(t) =x( (t 3)) = x( 3 t). Epeid h qronik anastrof kai h qronik metatìpish den mporoôn na all xoun thn periodikìthta tou s matoc, to dedomèno 5 isqôei kai gia to s ma

K. Kotrìpouloc: S mata-sust mata 35 x( 3 t). Opìte epeid up rqei antistoiqða èna proc èna metaxô twn suntelest n a k kai b k, oi mìnoi mh-mhdenikoð suntelestèc Fourier ja eðnai oi b kai b. Kat sunèpeia h tautìthta tou Parseval epit ssei: b + b =. (4.4) o s ma y(t) eðnai peritt c summetrðac, ra b = b kai b eðnai kajar c fantastikoð arijmoð. 'Ara b = ± j. (4.4) Opìte Sunep c x(t) = cos( πt 3 ) x(t) =cos(πt 3 ). a = e j π b = jb = jb = a = e j π b = jb =. (4.4) 4.6 Seir Fourier kai grammik qronoamet blhta sust mata Se prohgoômenh enìthta eðdame ìti h apìkrish enìc G.Q.A. sust matoc se diègersh pou analôetai se grammikì sunduasmì migadik n ekjetik n apokt mia exairetik c apl morf. Dhlad, an diegeðroume èna G.Q.A. sôsthma me eðsodo x(t) =e st, tìte h apìkrish tou eðnai y(t) =H(s) e st ìpou H(s) = + h(τ) e sτ dτ, s C (4.43) kai h(t) eðnai h kroustik apìkrish tou sust matoc. Sthn enìthta aut ja mac apasqol sei mia eidikìterh perðptwsh shm twn gia ta opoða Re{s} =, dhlad x(t) =e jωt. 'Ena tètoio s ma parist nei èna fantastikì ekjetikì sth suqnìthta ω. H par stash (4.43), tìte exeidikeôetai se H(jω)= + h(t) e jωt dt (4.44) kai onom zetai apìkrish suqnìthtac (frequency response). Ac upojèsoume ìti èna G.Q.A. sôsthma diegeðretai apì èna periodikì s ma me anapar - stash seir c Fourier x(t) = + k= a k e jkω t (4.45)

36 K. Kotrìpouloc: S mata-sust mata kai ìti h kroustik apìkrish tou G.Q.A. sust matoc eðnai h(t). ìte h apìkrish tou sust matoc ja eðnai y(t) = + k= a k H(jkω ) e jkω t. (4.46) Sun goume ìti y(t) eðnai epðshc periodikì s ma me jemeli dh perðodo Ðdia m> ekeðnh tou x(t). Oi suntelestèc thc seir c Fourier thc apìkrishc (4.46) eðnai a k H(jkω ). Epomènwc to G.Q.A. sôsthma epifèrei tropopoðhsh kajenìc suntelest thc seir c Fourier thc diegèrsewc xeqwrist pollaplasi zont c ton me thn tim thc apìkrishc suqnìthtac sth sugkekrimènh suqnìthta (dhlad armonik ). Sthn epìmenh enìthta meletoôme efarmog thc an lushc pou prohg jhke se prwtob jmia fðltra epilektik suqnot twn. a fðltra aut perigr fontai apì mia prwtob jmia grammik diaforik exðswsh me stajeroôc suntelestèc kai ulopoioôntai p.q. me prwtob jmia hlektrik kukl mata RC, ìpwc to eikonizìmeno sto Sq ma 4.9. + R + + vs(t) AC ic(t) C vc(t). Sq ma 4.9: Prwtob jmio kôklwma RC. 4.6. Katwdiabatì fðltro RC 'Estw v s (t) h t sh thc phg c pou diegeðrei to kôklwma. kukl matoc RC Ac epilèxoume wc apìkrish tou tou Sq matoc 4.9 thn t sh sta kra tou puknwt v c (t). Apl efarmog tou nìmou pt shc t sewn tou Kirchoff odhgeð sthn akìloujh èkfrash C t i c (τ) dτ + Ri c (t) =v s (t) (4.47) ìpou to reôma pou diarrèei ton puknwt sqetðzetai me thn t sh sta kra tou puknwt dia thc i c (t) =C dv c(t). (4.48) dt Antikajist ntac thn (4.48) sthn (4.47) prokôptei h diaforik exðswsh pou perigr fei to kôklwma: RC dv c(t) dt + v c (t) =v s (t). (4.49)

K. Kotrìpouloc: S mata-sust mata 37 Sth sunèqeia upojètoume ìti epib lletai mia kajar c hmitonoeid c diègersh sto kôklwma v s (t) =e jωt. ìte h apìkrish eðnai v c (t) =H(jω)e jωt opìte antikajist ntac sthn (4.49) prokôptei kai RC jω H(jω) e jωt + H(jω) e jωt = e jωt (4.5) H(jω)= +RC jω = jωc R +. (4.5) jωc H apìkrish suqnìthtac den eðnai tðpote llo to lìgo thc sônjethc antðstashc tou puknwt proc th sônjeth antðstash thc sundesmologðac thc antðstashc kai tou puknwt se seir. o mètro thc apìkrishc suqnìthtac kai h f sh dðnontai analutik apì tic sqèseic H(jω) = +ω R C (4.5) H(jω) = arctan( ω RC) (4.53) kai sqedi zontai sta Sq mata 4.a kai 4.b wc proc ω gia RC =. O xonac twn suqnot twn mporeð na jewrhjeð ìti bajmonomeðtai se ωrc genikìtera. ParathroÔme ìti gia.5.9.4.8.3 H(jω).7.6.5.4.77 H(jω)/π.. -. -..3 -.3. -.4. -5-4 -3 - - 3 4 5 ωrc (a) H(jω) -.5-5 -4-3 - - 3 4 5 ωrc (b) H(jω) Sq ma 4.: Apìkrish suqnìthtac prwtob jmiou katwdiabatoô fðltrou RC. (a) Mètro (b) F sh. ω èqoume H(jω), en gia megalôterec suqnìthtec to mètro thc apìkrishc suqnìthtac fjðnei kaj c ω aux netai. 'Ara to fðltro autì epitrèpei th dièleush twn qamhl n suqnot twn, en exasjenðzei tic uyhlèc suqnìthtec. Lème ìti to aplì kôklwma RC eðnai èna mh-idanikì katwdiabatì fðltro (lowpass filter). Gia th suqnìthta ω = RC parathroôme

38 K. Kotrìpouloc: S mata-sust mata ìti H(j RC ) =.77. (4.54) Dhlad h apìkrish suqnìthtac pèftei sto.77 thc tim c thc gia ω =. Epeid log ( ) = 3 (metriètai se decibel, db) onom zoume th suqnìthta ω = RC suqnìthta apokop c 3 db (cutoff frequency). o di sthma suqnot twn ω [, RC ] kaleðtai eôroc z nhc (bandwidth). PrwjÔstera, lème ìti h kroustik apìkrish tou kukl matoc RC eðnai h(t) = RC e kai h bhmatik apìkris tou dðnetai apì th sqèsh RC t u(t) (4.55) s(t) =[ e RC t ] u(t). (4.56) Oi dôo apokrðseic sqedi zontai sto Sq ma 4. gia τ =. O xonac tou qrìnou mporeð na jewrhjeð ìti bajmonomeðtai se t τ genikìtera. o mègejoc τ = RC èqei idiaðterh fusik τ.9.8.7 τh(t) τe.6.5.4.3 s(t).8.6.4 e....5.5.5 3 3.5 4 4.5 5 τ = RC t τ (a) h(t).5.5.5 3 3.5 4 4.5 5 τ = RC t τ (b) s(t) Sq ma 4.: (a) Kroustik apìkrish kai (b) bhmatik apìkrish tou prwtob jmiou katwdiabatoô fðltrou. shmasða. Lègetai stajer qrìnou (time-constant). ParathroÔme ìti h() = τ h(τ) = e h() = τe (4.57) s(τ) = e