στο Αριστοτέλειο υλικού.

Σήματα Συστήματα Μετασχηματισμός Κωνσταντίνος Κοτρόπουλος Τμήμα Πληροφορικής Θεσσαλονίκη, Ιούνιος 203

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 203

Kef laio 8 Metasqhmatismìc 8. EujÔc metasqhmatismìc O dðpleuroc metasqhmatismìc orðzetai apì th sqèsh {x[n]} = X(z) = n= en o monìpleuroc metasqhmatismìc orðzetai apì th sqèsh U{x[n]} = X (z) = n=0 x[n] z n (8.) x[n] z n. (8.2) H sqèsh (8.) apoteleð th seir Laurent. O metasqhmatismìc orðzetai gia tic timèc ekeðnec tou migadikoô arijmoô z gia tic opoðec sugklðnei to jroisma. Oi timèc autèc exart ntai apì thn akoloujða x[n]. Sto migadikì epðpedo orðzoume tic perioqèc ekeð ìpou h dunamoseir (8.): sugklðnei den sugklðnei. Analìgwc me th morf thc akoloujðac pragmatopoieðtai h an lush thc sôgklishc. perioq ìpou h dunamoseir sugklðnei onom zetai perioq sôgklishc (region of convergence, ROC). Sto migadikì epðpedo (Sq ma 8.) o migadikìc arijmìc z parist netai apì th sqèsh H z = z e jθ, ìpou z kai θ eðnai oi polikèc suntetagmènec. Mia krðsimh ènnoia eðnai o monadiaðoc kôkloc, pou apoteleðtai apì ta shmeða ekeðna tou migadikoô epipèdou gia ta opoða

2 K. Kotrìpouloc: S mata-sust mata Im{z} z z θ Re{z} Sq ma 8.: Par stash z = z e iθ tou migadikoô arijmoô z. isqôei z =. O monadiaðoc kôkloc paðzei sto z- epðpedo to rìlo tou fantastikoô xona sto s- epðpedo tou metasqhmatismoô Laplace (dhlad, tou jω xona). Pr gmati gia z = e jω o dðpleuroc metasqhmatismìc (8.) tautðzetai me to metasqhmatismì Fourier D.Q., arkeð o monadiaðoc kôkloc na an kei sthn perioq sôgklishc tou metasqhmatismoô. H sqetik an lush sôgklishc parallhlðzetai me thn antðstoiqh pou ègine gia to metasqhmatismì Laplace. 8.. Aitiat akoloujða IsqÔei x[n] =0gia n<0, opìte o metasqhmatismìc dðnetai apì thn X(z) = x[n] z n. (8.3) n=0 'Estw h X(z) sugklðnei apolôtwc gia z = z, opìte x[n] z n <. (8.4) n=0 Epeid gia z > z èqoume z n < z n, èpetai ìti to jroisma thc dunamoseir c (8.3) sugklðnei gia k je z > z. Par deigma 8.. 'Estw x[n] =α n u[n], tìte X(z) = α n z n = (αz ) n = n=0 n=0 αz (8.5) gia αz < z > α. Lème ìti o metasqhmatismìc X(z) èqei pìlo sto z = α kai mhdenikì sto z =0.

K. Kotrìpouloc: S mata-sust mata 3 An R X eðnai h mikrìterh tim tou z gia thn opoða sugklðnei to jroisma thc dunamoseir c, tìte o X(z) sugklðnei gia z >R X. 8..2 AkoloujÐa dexi c pleur c 'Eqoume x[n] =0gia n<n. DiakrÐnoume duo peript seic: (a) n 0: IsqÔoun ìsa eip jhkan sthn perðptwsh thc aitiat c akoloujðac. (b) n < 0: O metasqhmatismìc dðnetai apì th sqèsh X(z) = n=n x[n] z n = n=n x[n] z n + x[n] z n. (8.6) O ìroc n=n x[n] z n eðnai peperasmèno jroisma kai sugklðnei gia z peperasmèno, en o ìroc n=0 x[n] z n antistoiqeð se jroisma dunamoseir c miac aitiat c akoloujðac, to opoðo sugklðnei gia z : z > z. n=0 To Sq ma 8.2 deðqnei ìti h perioq sôgklishc tou metasqhmatismoô miac akoloujðac dexi c pleur c eðnai to exwterikì enìc kuklikoô dðskou aktðnac z. An R X eðnai h mikrìterh tim tou z gia thn opoða sugklðnei to jroisma thc dunamoseir c (8.6), o X(z) sugklðnei gia z : z >= R X ektìc tou z =. Im{z} z Re{z} Sq ma 8.2: Perioq sôgklishc metasqhmatismoô akoloujðac dexi c pleur c.

4 K. Kotrìpouloc: S mata-sust mata 8..3 AkoloujÐa arister c pleur c 'Eqoume x[n] =0gia n >n 2, opìte o metasqhmatismìc dðnetai apì th sqèsh n 2 X(z) = x[n] z n = x[ m] z m. (8.7) n= m= n 2 To jroisma thc dunamoseir c (8.7) sugklðnei gia z <R X + ektìc tou z =0an n 2 > 0, ìpou R X + eðnai h megalôterh tim tou z gia thn opoða sugklðnei to jroisma thc dunamoseir c. H perioq sôgklishc eðnai to eswterikì enìc kuklikoô dðskou aktðnac R X + (Sq ma 8.3). Im{z} R X + Re{z} Sq ma 8.3: Perioq sôgklishc metasqhmatismoô akoloujðac arister c pleur c. 8..4 AmfÐpleurh akoloujða IsqÔei ìti x[n] 0gia n. O metasqhmatismìc dðnetai apì th sqèsh X(z) = x[n] z n = x[n] z n + x[n] z n. (8.8) n= n= O ìroc n= x[n]z n eðnai jroisma pou sugklðnei gia z <R X +, en o ìroc n=0 x[n]z n eðnai jroisma pou sugklðnei gia z >R X. 'Ara an R X <R X +, tìte h perioq sôgklishc eðnai ènac kuklikìc daktôlioc (Sq ma 8.4). AntÐjeta, an R X >R X +, tìte den up rqei perioq sôgklishc kai den orðzetai o metasqhmatismìc. n=0 8..5 AkoloujÐa peperasmènou m kouc O metasqhmatismìc dðnetai apì th sqèsh X(z) = n 2 n=n x[n] z n. (8.9)

K. Kotrìpouloc: S mata-sust mata 5 Im{z} R X Re{z} R X + Sq ma 8.4: Perioq sôgklishc metasqhmatismoô amfðpleurhc akoloujðac. To jroisma (8.9) sugklðnei arkeð x[n] < gia n n n 2. 'Etsi h perioq sôgklishc eðnai ìlo to migadikì epðpedo ektìc apì thn tim z = an n < 0 z =0an n 2 > 0. Epomènwc h perioq sôgklishc eðnai 0 < z < ektìc Ðswc apì tic timèc z =0kai z = (Sq ma 8.5). Im{z} Re{z} Sq ma 8.5: Perioq sôgklishc metasqhmatismoô akoloujðac peperasmènou m kouc. 8..6 SÔnoyh idiot twn perioq c sôgklishc metasqhmatismoô Apì thn an lush pou prohg jhke katèsth fanerì ìti oi idiìthtec pou isqôoun gia thn perioq sôgklishc tou metasqhmatismoô prokôptoun eôkola metagr fontac tic antðstoiqec

6 K. Kotrìpouloc: S mata-sust mata idiìthtec thc perioq c sôgklishc tou metasqhmatismoô Laplace antikajist ntac thn ènnoia thc lwrðdac sto s- epðpedo me aut n tou kuklikoô daktulðou sto z- epðpedo. Ac sunoyðsoume tic idiìthtec autèc gia lìgouc plhrìthtac, an kai eðnai pleonastikì: Idiìthta : H perioq sôgklishc tou metasqhmatismoô apoteleðtai apì èna kuklikì daktôlio me kèntro thn arq tou z- epipèdou. Idiìthta 2: H perioq sôgklishc den perièqei pìlouc. Idiìthta 3: An x[n] eðnai peperasmènhc di rkeiac, tìte h ROC eðnai olìklhro to z- epðpedo ektìc Ðswc twn z =0kai/ z =. Idiìthta 4: An x[n] eðnai s ma dexi c pleur c kai an o kôkloc z = r 0 an kei sthn ROC, tìte ìlec tic peperasmènec timèc tou z gia tic opoðec z >r 0 ja an koun epðshc sthn ROC. Idiìthta 5: An x[n] eðnai s ma arister c pleur c kai an o kôkloc z = r 0 an kei sthn ROC, tìte ìlec oi timèc tou z gia tic opoðec 0 < z <r 0 ja an koun epðshc sthn ROC. Idiìthta 6: An x[n] eðnai amfðpleuro s ma kai an o kôkloc z = r 0 an kei sthn ROC, tìte h ROC ja eðnai ènac kuklikìc daktôlioc sto z- epðpedo pou perièqei ton kôklo z = r 0. Idiìthta 7: An o metasqhmatismìc, X(z), tou s matoc x[n] eðnai rht sun rthsh tou z, tìte h ROC tou fr ssetai apì touc pìlouc ekteðnetai wc to peiro. Idiìthta 8: An o metasqhmatismìc, X(z), enìc s matoc dexi c pleur c x[n] eðnai rht sun rthsh tou z, tìte h ROC tou metasqhmatismoô eðnai h perioq tou z- epipèdou èxw apì ton piì apomakrusmèno pìlo, dhlad ektìc tou kôklou aktðnac Ðshc me to megalôtero mètro twn pìlwn tou X(z). Epiplèon an to x[n] eðnai aitiatì, tìte h ROC perièqei kai to z =. Idiìthta 9: An o metasqhmatismìc, X(z), enìc s matoc arister c pleur c x[n] eðnai rht sun rthsh tou z, tìte h ROC tou metasqhmatismoô eðnai h perioq tou z- epipèdou sto eswterikì tou kôklou kôklou aktðnac Ðshc me th mikrìtero mh-mhdenikì mètro twn pìlwn tou X(z) sumperilamb nontac endeqomènwc kai to z =0(ìpwc sthn perðptwsh antaitiat n shm twn).

K. Kotrìpouloc: S mata-sust mata 7 8.2 AntÐstrofoc metasqhmatismìc O antðstrofoc metasqhmatismìc upologðzetai apì pðnakec gnwst n metasqhmatism n tic idiìthtec tou metasqhmatismoô, ìpwc kai gia touc llouc metasqhmatismoôc, l.q. Fourier, Laplace thn exðswsh orismoô, ìmwc tìte apaiteðtai migadik olokl rwsh. Sth sunèqeia ex getai o tôpoc tou antðstrofou metasqhmatismoô. Ac xekin soume upologðzontac to olokl rwma C z k X(z) dz = C n= x[n] z n+k dz (8.0) ìpou C kleist kampôlh entìc thc perioq c sôgklishc tou X(z), h opoða perikleðei kai thn arq twn axìnwn. Enall soume thn jroish kai thn olokl rwsh sto dexð mèroc thc (8.0): z k X(z)dz = x[n] z n+k dz. (8.) C n= An qrhsimopoi soume to je rhma gia to olokl rwma Cauchy: k =0 z k dz = 2πj C 0 k 0 C (8.2) kai jèsoume n = k sthn (8.), tìte X(z) z k dz =2πjx[k]. (8.3) C Dhlad, o antðstrofoc metasqhmatismìc orðzetai wc x[n] = Δ X(z) z n dz (8.4) 2πj C h opoða sp nia qrhsimopoieðtai gia ton upologismì tou s matoc D.Q. x[n]. Antijètwc epilègoume mia apì tic ex c mejìdouc:. Qr sh jewr matoc upoloðpwn 2. An lush se jroisma merik n klasm twn 3. DiaÐresh poluwnômwn 4. Epèktash se dunamoseir. AkoloÔjwc ja analôsoume kajemi ap> autèc.

8 K. Kotrìpouloc: S mata-sust mata 8.2. Qr sh jewr matoc upoloðpwn MporoÔme na gr youme ìti x[n] = [ upìloipa (residuals) thc X(z)z n stouc pìlouc ] pou brðskontai sto eswterikì thc troqi c C. (8.5) An X(z) z n = A(z) (z z 0 ) s (8.6) ìpou A(z) den èqei rðzec sto z = z 0, tìte to upìloipo thc X(z)z n gia z = z 0 orðzetai wc ] Res [X(z) z n = z=z 0 d s A(z). (s )! dz z=z0 s (8.7) Par deigma 8.2. 'Estw tìte DiakrÐnoume tic peript seic: X(z) = X(z)z n =, z > α (8.8) αz zn zn = αz z α. (8.9) n>0: Eklègetai wc C kôkloc aktðnac megalôterhc tou α, o opoðoc perib llei ton pìlo z = α. 'Ara x[n] =Res [ X(z)z n ] z=α = 0! zn z=α = α n. (8.20) n<0: Up rqei pollaplìc pìloc sto 0. Gia na broôme to upìloipo sto mhdèn, epilègetai A(z) = z α (8.2) opìte Oi pìloi thc (8.22) eðnai X(z) z n = z =0 z = α z n (z α). (8.22) t xhc m = n hc t xhc. (8.23) Ta upìloipa thc (8.22) gia z =0dÐnontai apì thn [ ] Res = z n (z α) (m )! z=0 d m dz m [. (8.24) z α] z=0

K. Kotrìpouloc: S mata-sust mata 9 Ac analôsoume thn perðptwsh n =. Me b sh thn (8.22) ja èqoume X(z)z n n= = z z α = z(z α) (8.25) opìte dun mei thc (8.24) gia m = n = [ ] Res = [ ] = α. (8.26) z (z α) z=0 0! z α z=0 To upìloipo sto z = α eðnai: ra x[n] = α + α =0. [ ] Res z (z α) z=α = 0! [ ] = α. (8.27) z z=α To Ðdio ja isqôei gia n = 2, 3,... Sunep c, gia genikì n x[n] =α n u[n]. (8.28) 8.2.2 Mèjodoc an ptuxhc se merik kl smata AnalÔoume to metasqhmatismì X(z) se X(z) = N(z) D(z) (8.29) ìpou o bajmìc tou poluwnômou N(z) eðnai mikrìteroc tou bajmoô tou poluwnômou D(z). Tìte gia aploôc pìlouc X(z) = N k= A k z z k (8.30) ìpou z k eðnai oi pìloi thc X(z) kai ta A k eðnai ta upìloipa stouc pìlouc A k =(z z k ) X(z) z=zk. (8.3) An o bajmìc tou poluwnômou N(z) eðnai megalôteroc Ðsoc apì to bajmì tou D(z) tìte ekteloôme th diaðresh kai èqoume: X(z) =B m z m + B m z m + + B z + B 0 + ìpou m = bajmìc {N(z)} bajmìc {D(z)}. N k= A k z z k (8.32)

0 K. Kotrìpouloc: S mata-sust mata An o X(z) èqei pollaplì pìlo t xhc s sto z i tìte: me X(z) =B m z m + B m z m + + B z + B 0 + C l = N k= A k z z k + s l= C l (z z i ) l (8.33) d s l (s l)! dz [z z i] s X(z), l =, 2,...,s. s l (8.34) z=zi Par deigma 8.3. 'Estw x[n] akoloujða dexi c pleur c x[n] me metasqhmatismì pou dðnetai apì thn X(z) = Gia na broôme thn akoloujða x[n] parathroôme ìti z 2 (z α)(z b) = z 2 z 2 (α + b)z + αb. (8.35) X(z) z = z z 2 (α + b)z + αb = A z α + A 2 z b (8.36) ìpou A = A 2 = (z α)z z = = α (8.37) (z α)(z b) z=α (z b) z=α α b (z b)z = b (z α)(z b) z=b b α. (8.38) Sunep c Epomènwc X(z) z α =( α b ) z α +( b b α ) z b. (8.39) X(z) = α ( α b ) z z α +( b b α ) z z b (8.40) x[n] = α ( α b ) b αn u[n]+( b α ) bn u[n]. (8.4) 8.2.3 Mèjodoc suneqoôc diaðreshc Epidi ketai na grafeð o X(z) se jroisma seir c dun mewn tou z. Tìte h akoloujða x[n] dðnetai apì touc suntelestèc thc seir c. Par deigma 8.4. DÐnetai ìti EkteloÔme th diaðresh: X(z) = 3 5 2 z 3 2 z + = 3z2 5z 2 2 z 2 z 2 3z +. (8.42) 2 2

K. Kotrìpouloc: S mata-sust mata 3z 2 5 2 z z2 3 2 z + 2 3z 2 + 9 2 z 3 2 3+ 4 2 z + 6 4 z 2 k.o.k 4 2 z 3 2 4 2 z + 2 4 4 4 z + 6 4 z opìte x[n] =3, 2, 3 2,... 6 4 8 8 z 6 8 z 2 8.2.4 Epèktash se dunamoseir Efarmìzetai mìno sta aitiat s mata. Me th mèjodo aut anaptôssoume ton X(z) se dunamoseir, ìpwc h seir Taylor, opìte h akoloujða x[n] ja dðnetai apì touc suntelestèc twn dun mewn tou z : X(z) = p 0 + p z + + p n z n q 0 + q z + + q n z n = x[n] z n. (8.43) n=0 Apì ta ek tautìthtac Ðsa polu numa èqoume: p 0 = q 0 x[0] p = q 0 x[] + q x[0] (8.44). p n = q 0 x[n]+q x[n ] + + q n x[0] opìte arkeð na epilujeð to k tw trigwnikì sôsthma exis sewn (8.44) wc proc x[0], x[],...,x[n].

2 K. Kotrìpouloc: S mata-sust mata 8.3 Idiìthtec tou metasqhmatismoô 8.3. Grammikìthta An x[n] tìte X(z) me ROC: R X < z <R X + kai y[n] Y (z) me ROC: R Y < z <R Y +, Ax[n]+By[n] AX(z)+BY (z), me ROC: R < z <R + (8.45) ìpou h perioq sôgklishc R < z <R + eðnai toul qiston h tom twn epimèrouc perioq n sôgklishc. 8.3.2 Pollaplasiasmìc me ekjetik akoloujða An x[n] X(z) me ROC: R X < z <R X +, tìte α n x[n] X(α z), me ROC: α R X < z < α R X + (8.46) ìpou α pragmatikìc migadikìc. An o X(z) èqei pìlo sto z = z, tìte o X(α z) ja èqei pìlo sto z = αz. To Ðdio isqôei gia ta mhdenik tou X(z). An α eðnai pragmatikìc gðnetai metatìpish twn pìlwn kai twn mhdenik n kat m koc aktðnwn sto z- epðpedo. An α eðnai migadikìc, tìte gðnetai kai peristrof. 8.3.3 Diafìrish tou X(z) An x[n] X(z) me ROC: R X < z <R X +, tìte nx[n] z dx(z) dz, me ROC: R X < z <R X +. (8.47) 8.3.4 Metasqhmatismìc suzugoôc akoloujðac An x[n] X(z) me ROC: R X < z <R X +, tìte x [n] X (z ), me ROC: R X < z <R X +. (8.48) 8.3.5 Je rhma arqik c tim c An x[n] =0gia n<0, tìte x[0] = lim z X(z). (8.49)

K. Kotrìpouloc: S mata-sust mata 3 8.3.6 Sunèlixh akolouji n An x[n] X(z) me ROC: R = R X < z <R X + kai y[n] R Y < z <R Y +, tìte [x y](n) Y (z) me ROC: R 2 = X(z) Y (z), me ROC pou perièqei thn R R 2. (8.50) An ènac pìloc thc miac akoloujðac akur netai apì mhdenikì thc llhc, tìte h perioq sôgklishc thc X(z) Y (z) eðnai megalôterh. 8.3.7 Metatìpish akoloujðac Aut h idiìthta eðnai meg lhc shmasðac gia thn an lush grammik n susthm twn. x[n] 'Estw X(z). Me kajustèrhsh thc akoloujðac kat n 0 deðgmata, ìpou n 0 > 0, prokôptei h x[n n 0 ]. An h x[n] eðnai aitiat, dhlad x[n] =0gia n<0, tìte x[n n 0 ]=0gia n<n 0. 'Ara { } x[n n 0 ] = x[n n 0 ] z n n n0=m = n=n 0 m=0 x[m] z m n 0 = z n 0 X(z). (8.5) H perioq sôgklishc tou metasqhmatismoô thc metatopismènhc akoloujðac eðnai h perioq sôgklishc tou metasqhmatismoô thc arqik c akoloujðac me pijan prosj kh apaloif thc arq c tou apeðrou. H (8.5) isqôei ìqi mìno gia aitiatèc akoloujðec, all gia opoiad pote akoloujða. Monìpleuroc metasqhmatismìc Shmei nontai orismènec diaforèc sto monìpleuro metasqhmatismì akolouji n, ìpwc kai sto monìpleuro metasqhmatismì Laplace. Pr gmati; An x[n] eðnai aitiat akoloujða, tìte isqôei h (8.5). An h x[n] eðnai akoloujða dexi c pleur c, tìte o monìpleuroc metasqhmatismìc thc metatopismènhc akoloujðac dðnetai apì thn { } U x[n n 0 ] = n=0 x[n n 0 ] z n n n 0 =m = [ + = z n 0 x[m] z m + m=0 x[m] z n0 z m m= n 0 ] x[m] z m m= n 0

4 K. Kotrìpouloc: S mata-sust mata = z n 0 [ X (z)+ m= n 0 x[m]z m }{{} arqikèc sunj kec ]. (8.52) An h metatopismènh akoloujða eðnai thc morf c x[n + n 0 ], tìte o monìpleuroc metasqhmatismìc gia akoloujða dexi c pleur c eðnai { } [ n 0 U x[n + n 0 ] = z n 0 X (z) x[m] z ]. m (8.53) O PÐnakac 8. sunoyðzei tic idiìthtec tou dðpleurou metasqhmatismoô. Oi idiìthtec tou m=0 monìpleurou metasqhmatismoô paratðjentai ston PÐnaka 8.2. 8.4 MetasqhmatismoÐ merik n qarakthristik n shm twn (a) Gia to s ma monadiaðou deðgmatoc δ[n] o metasqhmatismìc eðnai: { } δ[n] =. (8.54) (b) Gia to s ma metatopismènou monadiaðou deðgmatoc δ[n k] o metasqhmatismìc eðnai: { } δ[n k] = z k. (8.55) (g) Gia to s ma u[n] o metasqhmatismìc eðnai: { } u[n] = z, z z > (8.56) Apìdeixh: IsqÔei X(z) = n=0 z n = n=0 z n. (8.57) 'Omwc to jroisma (8.57) eðnai gewmetrik prìodoc me lìgo z. 'Etsi, an z <, tìte X(z) = z, z >. (8.58) z (d) Gia thn akoloujða a n u[n] o metasqhmatismìc eðnai { } α n u[n] = z, z > α. (8.59) z α O PÐnakac 8.3 sugkentr nei orismèna sun jh zeôgh metasqhmatism n.

K. Kotrìpouloc: S mata-sust mata 5 PÐnakac 8.: Idiìthtec tou dðpleurou metasqhmatismoô. Idiìthta S ma Metasqhmatismìc Perioq sôgklishc e jω 0n x[n] X(e jω 0 z) R x[n] X(z) R g[n] G(z) R 2 Grammikìthta ax[n]+bg[n] ax(z)+bg(z) Toul qisto R R 2 Metatìpish sto qrìno x[n n 0 ] z n 0 X(z) R me pijan prosj kh diagraf thc arq c apeðrou Klim kwsh sto z- epðpedo z n 0 x[n] X( z z 0 ) z 0 R α n z[n] X(α z) α R : sônolo twn shmeðwn { α z} R Qronik anastrof x[ n] X(z ) R : sônolo twn shmeðwn {z } R SuzugÐa x [n] X (z ) R Diastol x (k) [n] X(z k ) R k : sônolo twn shmeðwn {z k } R Sunèlixh (x g)[n] X(z) G(z) Toul qisto R R 2 Pr th diafor x[n] x[n ] ( z ) X(z) Toul qisto R { z > 0} Parag gish sto z- e- nx[n] z dx(z) dz R pðpedo Suss reush n k= x[k] z X(z) Toul qisto An x[n] =0gia n<0, tìte Je rhma arqik c tim c R { z > } x[0] = lim z X(z)

6 K. Kotrìpouloc: S mata-sust mata PÐnakac 8.2: Idiìthtec tou monìpleurou metasqhmatismoô. Idiìthta S ma Monìpleuroc x[n] g[n] metasqhmatismìc X (z) G(z) Grammikìthta ax[n]+bg[n] ax (z)+bg(z) [ Metatìpish sto qrìno x[n n 0 ] z n 0 X (z)+ ] m= n 0 x[m] z m [ x[n + n 0 ] z n 0 X (z) ] n 0 m=0 x[m] z m Klim kwsh sto z- epðpedo e jω 0n x[n] X (e jω 0 z) z n 0 x[n] X ( z z 0 ) α n z[n] X (α z) Qronik anastrof x[ n] X (z ) Diastol x (k) [n] X (z k ) SuzugÐa x [n] X (z ) Sunèlixh x[n] =g[n] 0 gia n< (x g)[n] X (z) G(z) 0 Pr th diafor x[n] x[n ] ( z ) X (z) x[ ] dx (z) Parag gish sto z- epðpedo nx[n] z dz n Suss reush k=0 x[k] z X (z) Je rhma arqik c tim c x[0] = lim z X (z)

K. Kotrìpouloc: S mata-sust mata 7 PÐnakac 8.3: Sun jh zeôgh metasqhmatism n. S ma Metasqhmatismìc Perioq sôgklishc δ[n] 'Olo to z epðpedo u[n] z z > u[ n ] z z < δ[n m] z m 'Olo to z epðpedo ektìc z =0 (an m>0) z = (an m<0) a n u[n] az z >a a n u[ n ] az z <a na n u[n] az ( az ) 2 z >a na n u[ n ] [cos Ω 0 n] u[n] [sin Ω 0 n] u[n] [r n cos Ω 0 n] u[n] [r n sin Ω 0 n] u[n] az ( az ) 2 z <a [cos Ω 0 ] z [2 cos Ω 0 ] z + z 2 z > [sin Ω 0 ] z [2 cos Ω 0 ] z + z 2 z > [r cos Ω 0 ] z [2r cos Ω 0 ] z + r 2 z 2 z >r [r sin Ω 0 ] z [2r cos Ω 0 ] z + r 2 z 2 z >r

8 K. Kotrìpouloc: S mata-sust mata 8.5 Sqèseic metasqhmatism n, Laplace kai Fourier O metasqhmatismìc Laplace apoteleð epèktash tou metasqhmatismoô Fourier sthn perðptwsh shm twn suneqoôc qrìnou. 'Oson afor ta s mata diakritoô qrìnou, o metasqhmatismìc Fourier D.Q. (FT DT ) dðnetai apì tic sqèseic (eujôc kai antðstrofoc): X(Ω) = X(e jω ) = kai èqei ta ex c qarakthristik : n= x[n] = 2π 2π x[n] e jωn (8.60) X(Ω) e jωn dω (8.6) eðnai periodik sun rthsh thc suneqoôc metablht c Ω me perðodo 2π. h apìstash twn deigm twn eðnai ΔT =. An ΔT kai h perðodoc tìte eðnai X(e jωδt ) = n= x(nδt ) = 2π x(nδt ) e jωnδt (8.62) 2π ΔT X(e jωδt ) e jωnδt dω (8.63) Ω p = 2π ΔT en h suqnìthta F = ΔT. (8.64) O arijmìc twn deigm twn x(nδt ) eðnai peperasmènoc peiroc. Sthn pr xh ìmwc upologðzoume deðgmata tou metasqhmatismoô Fourier D.Q., dhlad to diakritì metasqhmatismì Fourier (DFT) thc akoloujðac se peperasmèno arijmì deigm twn me th qr sh twn algorðjmwn FFT, anex rthta an h akoloujða eðnai peperasmènh ìqi. Kat sunèpeia ston upologismì twn deigm twn tou DFT eis gontai dôo eðdh sfalm twn: sf lma epik luyhc, to opoðo. eðnai pio èntono stic uyhlèc suqnìthtec

K. Kotrìpouloc: S mata-sust mata 9 2. mei netai aux nontac ton arijmì twn deigm twn N isodônama mei nontac to b ma deigmatolhyðac ΔT an ΔT = T N kai T kratiètai stajerì. sf lma apokop c, ìtan to s ma diakritoô qrìnou den eðnai qronoperatì, opìte prèpei na pollaplasiasteð m> èna par juro sto pedðo tou qrìnou pou ja periorðsei ton arijmì twn deigm twn. An x[n] prokôptei apì deigmatolhyða qronoperatoô s matoc x(t) di rkeiac T, tìte ΔT = N T = f, ìpou f s eðnai h suqnìthta deigmatolhyðac kai to sf lma apokop c eðnai s mhdèn. 8.5. Sqèsh metasqhmatismoô kai metasqhmatismoô Fourier D.Q. An sthn perioq sôgklishc tou metasqhmatismoô brðsketai o monadiaðoc kôkloc, tìte mporoôme na metaboôme apì to metasqhmatismì ston metasqhmatismì Fourier D.Q. thc akoloujðac me antikat stash z = e jω, dhlad X(Ω) = X(e jω )=X(z) z=e jω = n= 8.5.2 Sqèsh metasqhmatism n kai Laplace x[n] e jωn (8.65) 'Otan eðnai gnwstìc o metasqhmatismìc miac akoloujðac x[n] mporoôme na metaboôme sto metasqhmatismì Laplace thc akoloujðac me antikat stash z = e s : X(s) =X(z) z=e s = n= x[n] e sn (8.66) Genikìtera ìtan ΔT o metasqhmatismìc Laplace akoloujðac deigm twn eðnai periodik sun rthsh me perðodo ω s = 2π ΔT sto fantastikì xona, epeid X(s + j 2π ΔT )= + n= x[n] e snδt e j2πn = X(s). (8.67) An mac ediafèrei o metasqhmatismìc, tìte mia met bash apì to s- epðpedo sto z- epðpedo eðnai h apeikìnish: z = e sδt = e (jω+σ)δt = e} σδt {{} e jωδt (8.68) z To Sq ma 8.6 epexhgeð thn apeikìnish e sδt = z. ParathroÔme pwc h antikat stash e sδt = z apeikonðzei k je lwrðda eôrouc ω s wc proc Im{s} tou s- epipèdou s> ìlo to z- epðpedo (dhlad, kataqrhstik lème periodik sun rthsh me perðodo jω s = j 2π T

20 K. Kotrìpouloc: S mata-sust mata eðnai mia pleiìtimh apeikìnish). Gia na katast soume thn apeikìnish monìtimh, deqìmaste thn apeikìnish mìno thc lwrðdac gia ω [ ωs, ωs 2 2 ) sto z- epðpedo. Apì thn (8.68) prokôptei Im{s} ωs 2 Im{z} Re{s} Re{z} ωs 2 (a) (b) Sq ma 8.6: Met bash apì to metasqhmatismì Laplace (a) sto metasqhmatismì (b). z < an σ<0 = an σ =0 > an σ>0, (8.69) dhlad :. To aristerì s- hmiepðpedo apeikonðzetai entìc tou monadiaðou kôklou. 2. O fantastikìc xonac tou s-epipèdou (jω xonac) apeikonðzetai sto monadiaðo kôklo (metasqhmatismìc Fourier D.Q.) 3. To dexð s- hmiepðpedo apeikonðzetai exwterik tou monadiaðou kôklou sto z- epðpedo. 4. Grammèc par llhlec proc to fantastikì xona tou s-epipèdou apeikonðzontai se omìkentrouc kôklouc me aktðna z = e σδt. 5. Grammèc par llhlec proc ton pragmatikì xona tou s epipèdou (σ xona) apeikonðzontai sto z- epðpedo se aktðnec me gwnða arg z = ωδt. 6. An s 0, tìte z. Dhlad, h arq twn axìnwn sto s- epðpedo apeikonðzetai sto z =sto z- epðpedo. 7. An ω metab lletai apì ωs se ωs, tìte arg z = ωδt metab lletai apì π wc π. 2 2

K. Kotrìpouloc: S mata-sust mata 2 8.6 An lush kai qarakthrismìc grammik n qronoamet blhtwn susthm twn qrhsimopoi ntac to metasqhmatismì O metasqhmatismìc paðzei ènan idiaðtera shmantikì rìlo sthn an lush kai thn anapar - stash twn grammik n qronoamet blhtwn susthm twn (G.Q.A.) diakritoô qrìnou (D.Q.) Apì thn idiìthta thc sunèlixhc sun goume ìti o metasqhmatismìc thc exìdou y[n] prokôptei wc Y (z) =H(z) X(z) (8.70) ìpou X(z) eðnai o metasqhmatismìc thc eisìdou x[n] kai H(z) eðnai o metasqhmatismìc thc kroustik c apìkrishc h[n], o opoðoc onom zetai sun rthsh sust matoc (system function) sun rthsh metafor c (transfer function). Gia z = e jω, h sun rthsh sust matoc ekfulðzetai sthn apìkrish suqnìthtac efìson bèbaia o monadiaðoc kôkloc an kei sthn perioq sôgklishc tou X(z). Sto Kef laio 8 eðdame ìti h sun rthsh sust matoc H(z) tan h idiotim tou sust matoc pou antistoiqeð sthn idiosun rthsh z n. Pollèc apì tic idiìthtec enìc sust matoc eðnai eujèwc sundedemènec me ta qarakthristik twn pìlwn, twn mhdenik n kai thc perioq c sôgklishc thc sun rthshc sust matoc, ìpwc staquologoôme sth sunèqeia. Idiìthta : 'Ena G.Q.A. sôsthma D.Q. eðnai aitiatì e n kai mìno e n h ROC tou H(z) eðnai to exwterikì enìc kuklikoô dðskou sumperilamb nontac kai to z =. Idiìthta 2: 'Ena G.Q.A. sôsthma D.Q. me rht sun rthsh sust matoc eðnai aitiatì e n kai mìno e n (a) h ROC eðnai to exwterikì enìc kôklou èxw apì ton pio apomakrusmèno pìlo kai (b) me to H(z) ekfrasmèno wc lìgo dôo poluwnômwn tou z, h t xh tou poluwnômou tou arijmht den uperbaðnei thn t xh tou poluwnômou tou paronomast. Idiìthta 3: 'Ena G.Q.A. sôsthma D.Q. eðnai eustajèc e n kai mìno e n h ROC tou H(z) perièqei to monadiaðo kôklo z =. Idiìthta 4: 'Ena G.Q.A. sôsthma D.Q. me rht sun rthsh sust matoc eðnai eustajèc e n kai mìno e n ìloi oi pìloi thc sun rthshc sust matoc H(z) keðntai sto eswterikì tou monadiaðou kôklou, dhlad èqoun mètro mikrìtero thc mon dac.

22 K. Kotrìpouloc: S mata-sust mata Kat' analogða proc to metasqhmatismì Laplace pou epitrèpei to gewmetrikì upologismì tou metasqhmatismoô Fourier suneqoôc qrìnou apì to di gramma pìlwn-mhdenik n, o metasqhmatismìc Fourier D.Q. mporeð na upologisteð gewmetrik c jewr ntac ta dianôsmata pìlwn kai mhdenik n sto z- epðpedo. Wstìso epeid sthn perðptwsh aut h rht sun rthsh prìkeitai na upologisteð p nw sto monadiaðo kôklo z=, jewroôme ta dianôsmata pou gontai apì touc pìlouc kai ta mhdenik kai katal goun s' èna shmeðo epð tou monadiaðou kôklou antð tou fantastikoô xona tou s-epipèdou. Par deigma 8.5. 'Estw prwtob jmio aitiatì G.Q.A. sôsthma D.Q. me kroustik apìkrish h[n] =a n u[n]. Apì to Par deigma 8. gnwrðzoume ìti H(z) = az = z, z > a. (8.7) z a Gia a < h ROC sumperilamb nei to monadiaðo kôklo kai epomènwc o metasqhmatismìc Fourier D.Q. thc kroustik c apìkrishc sugklðnei kai isoôtai me H(z) gia z = e j Ω, opìte h apìkrish suqnìthtac dðnetai apì thn H(e j Ω )=. (8.72) ae jω To Sq ma 8.7 deðqnei to di gramma pìlwn-mhdenik n thc H(z) sumperilamb nontac ta dianôsmata apì ton pìlo sto z = a kai to mhdenikì sto z =0proc èna shmeðo se gwnða Ω epð tou monadiaðou kôklou. To mètro thc apìkrishc suqnìthtac sth suqnìthta Ω eðnai o lìgoc tou m kouc tou dianôsmatoc v proc to m koc tou dianôsmatoc v 2. H f sh thc apìkrishc suqnìthtac eðnai h gwnða tou dianôsmatoc v wc proc ton pragmatikì xona meðon th gwnða tou dianôsmatoc v 2. Profan c to m koc tou dianôsmatoc v eðnai monadiaðo gia k je Ω. H gwnða tou dianôsmatoc v wc proc ton orizìntio xona eðnai Ω. Gia 0 <a<, to di nusma pou getai apì ton pìlo èqei to mikrìtero m koc gia Ω=0, en to m koc tou aux netai monìtona kaj c Ω metab lletai apì to 0 proc to π. Kat sunèpeia to mètro thc apìkrishc suqnìthtac ja megistopoieðtai gia Ω = 0 kai ja fjðnei monìtona kaj c to Ω aux nei apì to 0 proc to π. H gwnða tou dianôsmatoc pou getai apì ton pìlo xekin apì to 0 kai aux netai monìtona kaj c to o Ω aux nei apì to 0 proc to π. To mètro kai h f sh thc apìkrishc suqnìthtac sqedi zontai sta Sq mata 8.8 kai 8.9 antistoðqwc gia dôo timèc thc paramètrou a. To mètro thc paramètrou a paðzei rìlo parapl sio proc thn stajer qrìnou τ tou prwtob jmiou G.Q.A. sust matoc S.Q. ParathroÔme

K. Kotrìpouloc: S mata-sust mata 23 Im{z} v v 2 Ω + a Re{z} Sq ma 8.7: Di gramma pìlwn-mhdenik n gia to gewmetrikì upologismì thc apìkrishc suqnìthtac enìc prwtob jmiou G.Q.A. sust matoc D.Q. ìti to mètro sthn koruf thc apìkrishc suqnìthtac (dhlad gia Ω=0) elatt netai kaj c to a fjðnei proc to mhdèn. MporeÐ na deiqjeð ìti kaj c to a fjðnei proc to mhdèn, tìte h kroustik apìkrish aposbènnutai piì apìtoma kai h bhmatik apìkrish teðnei proc th mon da piì gr gora. 0 9 a=0.4 a=0.9 8 7 6 X(Ω) 5 4 3 2 0 - -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 Ω π Sq ma 8.8: Mètro thc apìkrishc suqnìthtac gia a =0.4 kai a =0.9.

24 K. Kotrìpouloc: S mata-sust mata 0.5 0.4 a=0.4 a=0.9 0.3 0.2 0. X(Ω) π 0-0. -0.2-0.3-0.4-0.5 - -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 Ω π Sq ma 8.9: F sh thc apìkrishc suqnìthtac gia a =0.4 kai a =0.9.