Σήματα Συστήματα Ανάλυση ourier για σήματα και συνεχούς χρόνου Λυμένες ασκήσει ις Κνσταντίνος Κοτρόπουλος Τμήμα Πληροφορικής συστήματα Θεσσαλονίκη, Ιούνιος 3
Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό, όπς 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 αναδιαμόρφση του εκπαιδευτικού υλικού. Το έργο υλοποιείται στο πλαίσιο του Επιχειρησιακού Προγράμματος «Εκπαίδευση και Δια Βίου Μάθηση»» και συγχρηματοδοτείται από την Ευρπαϊκή Ένση (Ευρπαϊκό Κοιννικό Ταμείο) και από εθνικούς πόρους. Θεσσαλονίκη, Ιούνιος 3
Kef laio 5 An lush ourier gia s mata kai sust mata suneqoôc qrìnou 5.5 Lumènec ask seic Sthn Enìthta aut paratðjentai lumènec ask seic pou aforoôn thn Ôlh twn KefalaÐwn 4-5. 5.5.. Na prosdiorðsete thn epèktash twn akìloujwn shm twn se seir ourier: (a) Prionwt palmoseir Sq ma 5.5.(a) (b) x(t) =+cosπtcos(πt + π 4 ) (g) x(t), to eikonizìmeno sto Sq ma 5.5.(b). LÔsh: (a) To prionwtì sq ma èqei peritt summetrða. Wc ek toôtou a =kai a n = n =,,.. T/ b n = x(t)sinn tdt= t sin n( π T T/ )tdt= t sin t dt = sin t t cos t = = ()n n π ()n = ()n n =ρ, ρ =,,... = (5.5.) n =ρ +, ρ =,,,...
K. Kotrìpouloc: S mata-sust mata x(t) x(t) t 4 4 6 t (a) (b) Sq ma 5.5.: (a) Prionwt palmoseir. (b) S ma x(t). (b) x(t) = +cosπt cos (πt + π 4 ) =cos(πt + π 4 )+cosπt cos (πt + π 4 ) = cos πt sin πt + cos (πt +πt + π 4 )+cos(πt + π 4 πt) = cos πt sin πt + cos πt sin πt + + cos 8πt sin 8πt = cos 8πt + cos πt + 4 4 cos πt sin 8πt + sin πt + 4 4 sin πt. (5.5.) (g) AnagnwrÐzoume ìti T =4opìte a = T T x(t) dt = 4 4 x(t) dt = 3 4 (5.5.3) a n = 4 = = sin t x(t)cosn π 4 tdt= cos t dt + cos t dt + sin t = πn sin ( cos t dt + cos t dt ). (5.5.4)
K. Kotrìpouloc: S mata-sust mata 3 b n = x(t)sin( t T ) dt = 4 = cos t cos t = cos cos = cos = = n =ρ, ρ =,,... sin t dt + cos sin t dt ()n = ()n (5.5.5) 3 n =ρ +, ρ =,,,... 5.5.. EÐdame pwc h ènnoia thc idiosun rthshc eðnai exairetik qr simh sthn an lush G.Q.A susthm twn. To Ðdio mporeð na eipwjeð kai gia ta grammik qronometaballìmena sust mata. 'Estw èna tètoio sôsthma pou diegeðretai me eðsodo x(t) kai par gei èxodo y(t). Lème ìti èna s ma φ(t) eðnai mia idiosun rthsh tou sust matoc an φ(t) λφ(t) (5.5.6) ìpou λ eðnai genik mia migadik stajer pou lègetai idiotim pou antistoiqeð sthn φ(t). (a) Ac upojèsoume ìti mporoôme na anaparast soume thn eðsodo sto sôsthma mac wc grammikì sunduasmì idiosunart sewn ϕ k (t), oi opoðec èqoun antðstoiqec idiotimèc λ k : x(t) = + k= c k ϕ k (t). (5.5.7) Na ekfr sete thn èxodo y(t) tou sust matoc me ìrouc twn c k,ϕ k kai λ k. (b) Jewr ste to sôsthma pou qarakthrðzetai apì thn diaforik exðswsh y(t) =t d x(t) dt EÐnai autì to sôsthma grammikì? EÐnai qronik amet blhto? + t dx(t). (5.5.8) dt (g) DeÐxte ìti to sônolo twn sunart sewn ϕ k (t) =t k (5.5.9) eðnai idiosunart seic tou sust matoc. Poièc eðnai oi antðstoiqec idiotimèc?
4 K. Kotrìpouloc: S mata-sust mata (d) ProsdiorÐste thn èxodo tou sust matoc an: x(t) =t +3t + t4 + π. (5.5.) LÔsh: (a) φ k (t) λ k φ k (t) c k φ k (t) λ k c k φ k (t) omoiogèneia (5.5.) c k φ k (t) λ k c k φ k (t) epallhlða. (5.5.) k k (b) To sôsthma eðnai qronik metaballìmeno, diìti oi suntelestèc twn parag gwn den eðnai (g) stajerèc, lla qronikèc sunart seic. H arq thc epallhlðac isqôei giatð an tìte h èxodoc dðnetai apì thn y(t) = t d x(t) dt = t d x (t) dt + t dx(t) dt + t dx (t) dt } {{ } T{x (t)} x(t) =x (t)+x (t) (5.5.3) d = t x (t) dt + t d x (t) dt + d x (t) dx (t) + t dt dt + t dx (t) dt } {{ } T{x (t)} EÔkola mporeð na epalhjeuteð kai h arq thc omoiogèneiac. + dx (t) dt. (5.5.4) 'Ara T{t k } = t d dt tk + t d dt tk = t k(k ) t k + tk t k = k(k ) t k + kt k = {k k + k}t k = }{{} k idiotim t k. (5.5.5) φ k (t) k t k. (5.5.6) (d) t () t = t (5.5.7) 3 t 3 t =3t (5.5.8) t4 6 t4 =8t 4 (5.5.9) πt π =. (5.5.)
K. Kotrìpouloc: S mata-sust mata 5 Sunep c x(t) 3 t +8t 4 + t. (5.5.) 5.5.3. Na upologðsete to metasqhmatismì ourier twn akìloujwn shm twn me th qr sh tou orismoô kai me th qr sh twn idiot twn: (a) e at cos t u(t), a> { +cosπt an t < (b) x(t) = an t > (g) e +t u( t +). LÔsh: (a) Me th qr sh tou orismoô X() = = = = x(t) e jt dt = e at cos te jt dt e at cos t cos t j sin t dt e at cos t cos t dt j e at cos ( )tdt+ e at cos t sin t dt e at cos ( + )tdt j e at sin ( )tdt j e at sin ( + )tdt = a a +( ) + a a +( + ) = a + j( ) + a + j( + ) Me th qr sh twn idiot twn e at u(t) cos t j( ) a +( ) + j( + ) a +( + ). (5.5.) a + j π δ( )+δ( + ) (5.5.3) (5.5.4) opìte { } e at u(t) cos t = π a + j πδ( )+ a + j πδ( + ) = π π a + j( ) + π a + j( + ) = a + j( ) +. (5.5.5) a + j( + )
6 K. Kotrìpouloc: S mata-sust mata (b) +cosπt t < x(t) = t > (5.5.6) Me th qr sh tou orismoô X() = x(t) e jt dt = ( + cos πt) e jt dt = e jt dt + cos πt e jt dt = cos t dt j sin t dt + cos πt cos t dt j cos πt sin t dt sin t cos t = + j + cos (π + )t dt+ cos (π )t dt j sin ( π)t dt+ sin ( + π)t dt = sin sin ( ) cos cos ( ) + j + { sin (π + )t + (π + ) sin (π )t } + j { cos ( π)t cos ( + π)t } + (π ) ( π) ( + π) = sin + { } sin (π + ) sin (π + ) sin (π ) sin (π ) + + π + (π ) + j { cos ( π) cos ( π) π } cos ( + π) cos ( + π) + π = sin + sin (π + ) π + Me th qr sh twn idiot twn: + sin (π ) π. (5.5.7) 'Estw {g(t)} = G(). Tìte x(t) =(+cosπt) u(t +) u(t). (5.5.8) }{{}}{{} f(t) g(t) { } { } { } G() =e j u(t) e j u(t) =e j e j u(t). (5.5.9) Epeid ìmwc èqoume G() = { } u(t) = πδ()+ j e j e j πδ()+ =jsin πδ()+ = sin j j (5.5.3) (5.5.3)
K. Kotrìpouloc: S mata-sust mata 7 () = πδ()+π X() = ( G)() = π π δ( π)+δ( + π) G() πδ()+ +G() πδ( π)+g() πδ( + π) = (5.5.3) = G()+ G( λ)δ(λ π) dλ + G( λ)δ(λ + π) dλ = sin + λ=π G( λ) + G( λ) λ= π = sin sin ( π) sin ( + π) + +. (5.5.33) π + π (g) y(t) = e t u(t) y( t) = e t u( t) Y () = +j Y () =Y ( ) = j (5.5.34) (5.5.35) y( (t )) = e t u( t) Y 3 () =e j Y () =e j (5.5.36) j x(t) = e 3 e t u( t) =e 3 y( t +) Y () =e (3 j) j. (5.5.37) Me th qr sh tou orismoô Y () = = e e ( j)t j y(t) e jt dt = e +t e jt dt = e e ( j)t dt = e e( j) j = e3 j j. (5.5.38) 5.5.4. 'Estw X() o metasqhmatismìc ourier tou s matoc x(t) pou sqedi zetai sto Sq ma 5.5.. (a) BreÐte th f sh tou metasqhmatismoô X() (b) BreÐte to X() (g) BreÐte to X() d sin (d) UpologÐste to X() e j d (e) UpologÐste to X() d
8 K. Kotrìpouloc: S mata-sust mata x(t) 3 t Sq ma 5.5.: S ma x(t) thc 'Askhshc 5.5.4. (st) Sqedi ste ton antðstrofo metasqhmatismì ourier tou Re{X()}. Na ektelèsete ìlouc touc upologismoôc qwrðc analutikì upologismì thc X(), dhlad a- pofeôgontac thn exðswsh orismoô kai k nontac qr sh twn idiot twn kai twn pin kwn twn zeug n metasqhmatism n. LÔsh: Apì ta Sq mata 5.5.3(b) kai 5.5.3(d) prokôptei ìti to s ma x(t) analôetai wc: x(t) =x (t ) x (t ). (5.5.39) Gia to s ma x (t) kai to kajusterhmèno antðgrafì tou x (t ) èqoume A t <T x (t) = X () =A sin T =4 alloô A=,T = x (t ) e j 4 sin sin (5.5.4) (5.5.4) en gia ta s mata x (t) kai x (t ) paðrnoume sin x (t) (5.5.4) 'Ara x (t ) e j sin. (5.5.43) 4 X() =e j sin 4 sin (/) sin =4e j sin (/). /4 (5.5.44) Epomènwc
K. Kotrìpouloc: S mata-sust mata 9 x (t) x (t) t 3 t (a) (b) x (t) x (t) t t (g) (d) x (t) t (e) Sq ma 5.5.3: S mata (a) x (t), (b) x (t ), (g) x (t), (d) x (t ) kai (e) x (t ).
K. Kotrìpouloc: S mata-sust mata (a) X() =. (b) sin X() = lim 4 e j sin (/) = lim 4 e j = lim 4e j sin cos sin =4 lim 4 sin cos =8 =7.(5.5.45) (g) Antikajist ntac to metasqhmatismì X() sto olokl rwma prokôptei X() d = sin 4 sin (/) e j d. (/) (5.5.46) K noume th skèyh na anag goume ta oloklhr mata sthn (5.5.46) s> aut twn antistrìfwn metasqhmatism n ourier sin 4 e jt d (5.5.47) sin (/) (/) e jt d (5.5.48) ta opoða na upologðsoume kai akoloôjwc na p roume tic timèc touc gia t = : π π sin 4 e jt d sin (/) (/) e jt d = u(t +) u(t ) t= π (5.5.49) = ( t ) t= π. (5.5.5) 'Ara (d) Xekin ntac apì thn X()d =4π. (5.5.5) X() sin }{{} H() e j d (5.5.5) paðrnoume h(t) = H() e jt d =π {H()}. (5.5.53) AnagnwrÐzoume ìti h() eðnai to zhtoômeno olokl rwma, ìpou } { {H() = X() sin } = x(t) u(t +) u(t). (5.5.54) }{{} g(t)
K. Kotrìpouloc: S mata-sust mata H tim thc sunèlixhc sthn (5.5.54) gia t =isoôtai me to embadìn tou trapezðou, ìtan o orjog nioc palmìc èqei kèntro gia t =ìpwc faðnetai sto Sq ma 5.5.4. x(t) u(t +) u(t ) = + ( + ) = + 3 = 7 (5.5.55) opìte h() = π ( 7 )=7π. (5.5.56) x(λ),g(t λ) t t t+ 3 λ Sq ma 5.5.4: Upologismìc sunèlixhc sto er thma 5.5.4(d). (e) X() d = π x(t) dt = π x (t) dt + = π 4 dt + t 3 = π 8+ + 3 8 = π 8+ 3 3 = π + 8 3 x (t) dt + ( t) dt + 4 dt 4 tdt+ x (t) dt + t dt + 3 3 4 dt = t dt = x (t) dt +4 4 + 3 =π + 6π 3 = 76π 3. (5.5.57) (st) Re{X()} = X()+X () epeid gia to pragmatikì s ma x(t) isqôei x(t)+x( t) (5.5.58) X ( ) x (t) X () x ( t) =x( t). (5.5.59)
K. Kotrìpouloc: S mata-sust mata x(t)+x( t).5.5 3 3 t Sq ma 5.5.5: S ma x(t)+x( t). To apotèlesma sqedi zetai sto Sq ma 5.5.5. 5.5.5. Na brejeð o metasqhmatismìc ourier twn shm twn: (a) f(t) =cos t cos t (b) f(t) =g(t)cos t, ìpou G() eðnai gnwstìc (g) g(t) =f(at ), ìpou () eðnai gnwstìc. LÔsh: (a) cos t cos t cos t cos t () =πδ( )+δ( + ) (5.5.6) () =πδ( )+δ( + ) (5.5.6) π ( )() = π ( )(). (5.5.6) All ( )() = π δ(λ )+δ(λ + ) π δ( λ )+ +δ( λ + ) dλ { = π δ(λ )δ λ + +δ(λ )δ( λ + )+δ(λ + )δ( λ )+
K. Kotrìpouloc: S mata-sust mata 3 } +δ(λ + )δ( λ + ) dλ = π δ( )+δ( + )+δ( + )+ +δ( + + ) = π {δ ( + ) + δ ( ) + +δ +( ) + δ +( + ) }. (5.5.63) Sundu zontac tic (5.5.6) kai (5.5.63) paðrnoume cos t cos t π δ ( + ) + δ ( ) + +δ +( ) + δ +( + ). (5.5.64) (b) Epeid kai cos t () (5.5.64) g(t) G() (5.5.65) = π δ( )+δ()+δ()+δ( + ) = π δ()+δ( )+δ( + ) (5.5.66) tìte: g(t)cos t (G )() π π = G() δ()+δ( )+δ( + ) π = G()+G( )+G( + ) 4 = G()+ G( )+G( + ). (5.5.67) 4 An G() eðnai o metasqhmatismìc ourier tou Sq matoc 5.5.6, tìte o prokôptwn metasqhmatismìc sqedi zetai sto Sq ma 5.5.7. (g) Ja k noume qr sh twn idiot twn: g(t) =f(at ) = fa(t ) (5.5.68) a f(t) f(at) () (5.5.69) a ( a ) (5.5.7)
4 K. Kotrìpouloc: S mata-sust mata G() Sq ma 5.5.6: Metasqhmatismìc ourier G(). G( + 4 ) G() 4 G( ) Sq ma 5.5.7: Metasqhmatismìc ourier tou s matoc g(t)cos t. opìte h(z) =f(t ) }{{ a} z g(t) =h(az) H() =e j a () (5.5.7) a H( a )= a e j a ( ). (5.5.7) a 5.5.6. Na brejeð o metasqhmatismìc ourier tou sin t apì to gnwstì metasqhmatismì ourier tou e j t. LÔsh: sin t = ej t e j t j = πδ( ) πδ( + ) j = jπ δ( + ) δ( ). (5.5.73)
K. Kotrìpouloc: S mata-sust mata 5 () β A β β +β β +β Sq ma 5.5.8: sma (). 5.5.7. Na brejeð to s ma pou èqei metasqhmatismì ourier autìn tou Sq matoc 5.5.8. LÔsh: EÐnai fanerì ìti to () tou Sq matoc 5.5.8 eðnai to f sma tou diamorfwmènou kat pl toc s matoc basik c z nhc me metasqhmatismì ˆ (), pou sqedi zetai sto Sq ma 5.5.9. H analutik morf tou metasqhmatismoô ourier tou diamorfwmènou s matoc eðnai: ˆ () A β β Sq ma 5.5.9: sma ˆ () s matoc basik c z nhc.. () =A{ u( + β) u( β ) + u( + + β) u( + β) }. (5.5.74) 'Estw to zeôgoc An g(t) =cos t f(t) (). (5.5.75) π δ( + )+δ( ) (5.5.76)
6 K. Kotrìpouloc: S mata-sust mata tìte h apodiamìrfwsh pl touc (amplitude demodulation) sunðstatai se All r(t) =f(t) g(t) R() = ( G)(). (5.5.77) π R() = π ( G)() = π ( + )+( ) = ( + )+( ). (5.5.78) π Dhlad an f(t) = ˆf(t) cos t () autìn tou Sq matoc 5.5.8 (5.5.79) epeid () = ˆ ( )+ ˆ ( + ) (5.5.8) me antikat stash thc (5.5.8) sthn (5.5.78) paðrnoume: R() = ( + )+ ( ) = 4 ˆ ()+ ˆ ( + )+ ˆ ( )+ ˆ () = ˆ ()+ 4 ˆ ( + )+ 4 ˆ ( ). (5.5.8) Opìte an filtr rw to R() me èna katwdiabatì fðltro ja prokôyei to ˆ (). Apì to Sq ma 5.5.8 kai th (5.5.74) anagnwrðzw ìti: R() = ( + )+( ) = = A u( + β) u( β) ++ A u( + + β) u( + β) + A u( + β) u( β). (5.5.8) Sunep c apì tic (5.5.8) kai (5.5.8) tautopoi ìti ˆ () =A u( + β) u( β) ˆf(t) =A β sinc(βt) (5.5.83) π kai epomènwc ìpou sinc(x) = sin x x. f(t) = ˆf(t) cos t = Aβ π sinc(βt) cos t (5.5.84) 5.5.8. Na deðxete ìti an e jϕ(t) () (5.5.85)
K. Kotrìpouloc: S mata-sust mata 7 kai ϕ(t) eðnai pragmatik sun rthsh, tìte cos φ(t) sin φ(t) ()+ ( ) (5.5.86) () ( ). j (5.5.87) LÔsh: Apì th (5.5.85) paðrnoume () = {e jφ(t) } = = e jφ(t) e jt dt = e jφ(t) e jt dt = e j(φ(t)+t) (5.5.88) dt = e j(φ(t) t) dt (5.5.88) e jφ(t)+t dt e j(φ(t) θt) dt θ= = ( ) (5.5.89) opìte cos φ(t) = ejϕ(t) + e jφ(t) ()+ ( ) (5.5.9) sin φ(t) = ejφ(t) e jφ(t) () ( ). j j (5.5.9)