Ανάλυση ασκήσεις. συστήματα

Σήματα Συστήματα Ανάλυση Fourier για σήματα και διακριτού χρόνου Λυμένες ασκήσεις Κωνσταντίνος Κοτρόουλος Τμήμα Πληροφορικής συστήματα Θεσσαλονίκη, Ιούνιος 3

Άδειες Χρήσης Το αρόν εκαιδευτικό υλικό υόκειται σε άδειες χρήσης Creative Commons. Για εκαιδευτικό υλικό, όως εικόνες, ου υόκειται σε άλλου τύου άδειας χρήσης, η άδεια χρήσης αναφέρεται ρητώς. Χρηματοδότηση Το αρόν εκαιδευτικό υλικό έχει ανατυχθεί στα λαίσια του εκαιδευτικού έργου του διδάσκοντα. Το έργο «Ανοικτά Ακαδημαϊκά Μαθήματα στο Αριστοτέλειο Πανειστήμιο Θεσσαλονίκης»» έχει χρηματοδοτήσει μόνο τη αναδιαμόρφωση του εκαιδευτικού υλικού. Το έργο υλοοιείται στο λαίσιο του Ειχειρησιακού Προγράμματος «Εκαίδευση και Δια Βίου Μάθηση»» και συγχρηματοδοτείται αό την Ευρωαϊκή Ένωση (Ευρωαϊκό Κοινωνικό Ταμείο) και αό εθνικούς όρους. Θεσσαλονίκη, Ιούνιος 3

Kef laio 7 An lush Fourier gia s mata kai sust mata diakritoô qrìnou 7.7 Lumènec ask seic 7.7.. UpologÐste to Metasqhmatismì Fourier D.Q. twn akìloujwn shm twn me th qr sh twn idiot twn tou: (a) ( )n un +3] (b) n u n] (g) r n sin(ω nt)] un] r < (d) r n cos(ω nt) (e) (n +)α n un] (a) IsqÔei ( )n un] ( ) 3 {( )n+3 un +3]} e j (7.7.) ( e j3 ) 3. (7.7.) e j

K. Kotrìpouloc: S mata-sust mata (b) (g) xn] n un] ( )n un] x n] n u n] X( ) X() (7.7.3) e j. (7.7.4) ej r n sin(ω nt )]un] r n j ejω Tn e jωtn ]un] ] e jωtn r n un] e jωtn r n un] j ] X( ω T ) X( + ω T ) j j re ] j( ω T ) re j(+ω T ) (7.7.5) ìpou X(). (7.7.6) re j (d) Gia na upologðsoume to metasqhmatismì Fourier D.Q. tou s matoc yn] r n cos(ω nt) (7.7.7) xekinoôme apì to zeôgoc metasqhmatismoô Fourier D.Q. pou apodeðxame se efarmog : xn] r n, r < X() re +. j (7.7.8) rej IsqÔei epðshc cos(ω nt ) ] e jωtn + e jω Tn (7.7.9) opìte (e) Gia to s ma Y () { re + j( ω T ) re + j( ω T ) } re + j(+ω T ) re j(+ω T ). (7.7.) (n +)α n un] α < (7.7.) xekin me apì to zeôgoc metasqhmatismoô Fourier D.Q. xn] α n un] X() αe j α <. (7.7.)

K. Kotrìpouloc: S mata-sust mata 3 Epiplèon èqoume nxn] j dx() d j ( α( j)e j ) ( αe j ) αje j j ( αe j ) α e j ( αe j ). (7.7.3) 'Ara nα n un]+α n un] αe j ( αe j ) + αe j αe j + αe j ( αe j ) ( αe j ). (7.7.4) 7.7.. Sto prìblhma autì meletoôme orjog nia diakrit s mata. DÔo s mata diakritoô qrìnou φ k n] kai φ m n] lègontai orjog nia sto di sthma (, ) e n A φ k n]φ k k m mn] n k m. (7.7.5) An h tim thc stajer c A k A m eðnai, tìte ta s mata lègontai orjokanonik. (a) Jewr ste ta s mata φ k n] δn k] k, ±, ±,...,± (7.7.6) DeÐxte ìti ta s mata aut eðnai orjokanonik sto di sthma (,). Apìdeixh n φ k n]φ mn] n m k δn k]δn m] δ mk alloô (7.7.7) ìpou m kai k kinoôntai sto di sthma (,). (b) Jewr ste ta s mata φ k n] e jk( )n k,,...,. (7.7.8) DeÐxte ìti ta s mata aut eðnai orjog nia s> opoiod pote di sthma m kouc. Apìdeixh n e jk( )n e jm( )n n e j(k m)( )n k m, ±,... alli c. (7.7.9)

4 K. Kotrìpouloc: S mata-sust mata Epeid ìmwc k,,..., m,,..., k m k m, ±,... k m. (7.7.) (g) DeÐxte ìti e n M xn] α i φ i n] (7.7.) ìpou φ i n] eðnai orjog nia sto di sthma (, ), tìte i n xn] M α i A i. (7.7.) i Apìdeixh n xn] xn]x n] n { M }{ M } α i φ i n] αl φ l n] n M i M i i l M α i αl φ i n] φ l n] l n M α i αl A i δ il l M α i αi A i i M α i A i. (7.7.3) i (d) 'Estw φ i n], i,,...,m, èna sônolo orjogwnðwn sunart sewn p nw sto di sthma (, ) kai xn] dosmèno s ma. Upojèste ìti epijumoôme na proseggðsoume to xn] san grammikì sunduasmì twn φ i n], dhlad ˆxn] ìpou α i eðnai stajeroð suntelestèc. 'Estw M α i φ i n] (7.7.4) i DeÐxte ìti e n epijumoôme na elaqistopoihjeð to en] xn] ˆxn]. (7.7.5) E n en] (7.7.6)

K. Kotrìpouloc: S mata-sust mata 5 tìte α i A i n xn] φ i n]. (7.7.7) Apìdeixh E n en] e n] n (xn] ˆxn])(x n] ˆx n]) xn] ˆxn] x n] xn]ˆx n]+ ˆxn]ˆx n] n n n n M ] xn] α i φ i n] x n] n n i M M M xn] αi φ i n]+ α i αl φ i n] φ l n] n i n i l M xn] α i φ i n] x n] n i n M M αi xn] φ i n]+ α i A i. (7.7.8) i n i AntikajistoÔme α i b i + jc i E M xn] (b i + jc i ) φ i n] x n] n i n M M (b i jc i ) xn] φ i n]+ (b i + c i )A i. (7.7.9) i n i 'Omwc prèpei kai arkeð ϑe ϑb i ϑe ϑc i j n φ i n] x n] n φ i n] x n]+j n xn] φ i n]+b i A i (7.7.3) n xn] φ i n]+c i A i (7.7.3) sunep c b i A i c i A i j n {φ i n] x n]+xn] φ i n]} A i Re n {φ i n] x n] xn] φ i n]} { n xn] φ i n] } (7.7.3)

6 K. Kotrìpouloc: S mata-sust mata A i j { {xn] φ i n] φ i n] x n]} } Im xn] φ i n]. (7.7.33) A i n n Epomènwc α i b i + jc i A i n xn] φ i n]. (7.7.34) (e) Efarmìste to apotèlesma gia ton upologismì twn suntelest n, ìtan φ i n] δn i]. (7.7.35) α i A i n xn] δn i] xn] δn i] xi] n i, ±, ±,...,±. (7.7.36) 7.7.3. (a) 'Estw xn] pragmatikì periodikì s ma me perðodo kai migadikoôc suntelestèc thc seir c Fourier α k. An h kartesian morf twn α k anaparðstatai wc α k b k + jc k, ìpou b k kai c k eðnai pragmatikoð arijmoð, na deðxte ìti α k α k. α k xn] e jk( )n n<> α k xn] e jk( )n n<> { xn] n<> n<> xn] cos { cos k( k( )n ] )n ] + j + j sin k( ]} } )n n<> xn] sin k( ] )n Re{α k } jim{α k } α k α k. (7.7.37) (b) Poi eðnai h sqèsh metaxô twn b k kai b k? Poi eðnai h antðstoiqh sqèsh metaxô twn c k kai c k? α k α k b k b k c k c k. (7.7.38)

K. Kotrìpouloc: S mata-sust mata 7 (g) Upojèste ìti eðnai rtioc. DeÐxte ìti a / eðnai pragmatikìc arijmìc. α k α / xn] e jk( )n n<> n<> n<> n<> xn] e j ( )n xn] e jn ( ) n xn] pragmatikìc. (7.7.39) (d) DeÐxte ìti h akoloujða xn] mporeð na ekfrasteð wc trigwnometrik seir Fourier thc morf c: gia perittì arijmì xn] α + xn] {α + α / ( ) n } + ( )/ k ( )/ k b k cos( kn ) c k sin( kn ] ) b k cos( kn ) c k sin( kn ] ) (7.7.4) (7.7.4) an eðnai rtioc arijmìc. xn] k<> α k e jk( )n k ( ) α k e α + jk( )n } {α k e jk( )n + α k e jk( )n k A k B {}}{{}} k { α + (b k + jc k ) cos k( )n +j sin k( k )n +(b k jc k ) cos k( ]} )n j sin k( )n

8 K. Kotrìpouloc: S mata-sust mata α + {(b k A k c k B k )+j(b k B k c k A k ) k +(b k A k c k B k ) j(b k B k c k A k )} o.e.d. (7.7.4) Gia rtio to eðnai perittìc, opìte xn] k + α k e jk( k + α k e jk( )n )n + α + (l k) {α + α / ( ) n } + α k e jk( )n + α / ( ) n k α l e jl( )n + α k e jk( )n l k } {α + α / ( ) n } + {α k e jk( )n + α k e jk( )n {α + α / ( ) n } + k k { b k cos k( ] )n c k sin k( ]} )n. (7.7.43) 7.7.4. 'Estw mh-grammikì sôsthma (Sq ma 7.7.) tou opoðou h èxodoc dðnetai apì th sqèsh ìpou upodhloð to mètro. M max n k ( )k x n k] (7.7.44) Epeid to M eðnai Ðso me th mègisth tim kajìlo to qrìno, prokôptei ìti eðnai stajer. 'Estw xn] e j n. Gia diaforetikèc epilogèc tou, to M ja eðnai mia stajer pou exart tai apì to, dhlad, gi> aut n thn oikogèneia eisìdwn M M( ). a prosdiorðste an to M M( ) eðnai periodikì s ma wc proc. An nað, na breðte thn perðodì tou. xn] Mh-grammikì sôsthma M Sq ma 7.7.: SÔsthma thc 'Askhshc 7.7.4.

K. Kotrìpouloc: S mata-sust mata 9 M( ) max n max n k ej n ( )k e j (n k) ( )k e j k k (7.7.45) Epeid to e j n eðnai periodikì me perðodo èpetai ìti M( )M( + ) eðnai periodikì me thn Ðdia perðodo. 7.7.5. 'Ena G.Q.A. sôsthma èqei apìkrish suqnìthtac H() j 3 e < 3 6 3 6. (7.7.46) H eðsodoc sto sôsthma eðnai mia periodik palmoseir monadiaðwn sewn me perðodo 6, dhlad, BreÐte thn èxodo tou sust matoc. xn] k δn +6k]. (7.7.47) xn] X() 6 k k δ( + k ) 6 δ( + k ). (7.7.48) 6 ParathroÔme ìti kai o metasqhmatismìc Fourier D.Q. tou s matoc eisìdou eðnai periodikìc me perðodo 6 kai ìti mèsa sto di sthma < ja up rqoun 6 monadiaðec seic sto pedðo thc suqnìthtac stic timèc k k, k, ±, ±,... k.o.k. (7.7.49) 6 To G.Q.A. sôsthma eðnai èna katwdiabatì fðltro (Sq ma 7.7.) pou apomakrônei ìlec tic seic ektìc ekeðnwn gia k, ± stic opoðec prosjètei mia diafor f shc e 3j k. 'Etsi Y () { δ() + e j 6 6 δ( + 6 6 )+e j } 6 6 δ( 6 ). (7.7.5)

K. Kotrìpouloc: S mata-sust mata H(e j ) 4 6 6 6 4 6 Sq ma 7.7.: Apìkrish suqnìthtac tou sust matoc kai metasqhmatismìc Fourier D.Q. thc diegèrsewc. 'Ara h èxodoc eðnai yn] 6 Y ()e jn d { δ() + e j 6 6 δ( + { 6 j(n+ +e 6 ) 6 { +e 6 { +e 6 6 j( 6 j( 6 6 n+ 6 6 ) + e n 6 6 ) + e } 6 6 )+e j 6 δ( 6 ) e jn d } 6 j(n + e 6 ) 6 j( 6 j( 6 +cos( n 6 6 6 ) 6 + 8 cos(n 6 3 8 ] n 6 6 ) } n 6 6 ) } ). (7.7.5) 7.7.6. Mia sun jhc arijmhtik pr xh eðnai h diafor pr thc t xhc yn] (xn]) xn] xn ] (7.7.5) ìpou xn] eðnai h eðsodoc kai yn] eðnai h èxodoc tou sust matoc. (a) DeÐxte ìti to sôsthma eðnai G.Q.A. Grammikìthta (αx n]+bx n]) αx n]+bx n] (αx n ] + bx n ])

K. Kotrìpouloc: S mata-sust mata α (x n]) + b (x n]). (7.7.53) Qronik ametablhtìthta (xn ]) xn ] xn ] yn ]. (7.7.54) (b) BreÐte thn kroustik apìkrish tou sust matoc. Jètoume wc eðsodo xn] δn]. Tìte h èxodoc tou G.Q.A. sust matoc eðnai h kroustik apìkrish: yn] hn] δn] δn ]. (7.7.55) (g) a sqedi sete thn apìkrish suqnìthtac tou sust matoc (mètro kai f sh). H(e j ) e j (7.7.56) Mètro: H(e j ) ( cos ) + j sin ( cos ) +sin +cos + sin cos cos. (7.7.57) To mètro thc apìkrishc suqnìthtac sqedi zetai sto Sq ma 7.7.3. H(e j ) Sq ma 7.7.3: Mètro apìkrishc suqnìthtac H(e j ).

K. Kotrìpouloc: S mata-sust mata F sh: H(e j ) ( cos ) + j sin (7.7.58) ( ) sin H(e j ) arctan. (7.7.59) cos H f sh thc apìkrishc suqnìthtac sqedi zetai sto Sq ma 7.7.4 H(e j ) 4 4 Sq ma 7.7.4: F sh apìkrishc suqnìthtac H(e j ). (d) DeÐxte ìti e n xn] (f g)n], tìte (xn]) (fn] gn]) fn] (gn]). (7.7.6) DeÐxame ìti (xn]) (x h)(n) xn] {δn] δn ]} (7.7.6) ra {(f g)n]} (f g) h]n] f (g h)]n] f (h g)]n] (f h) g]n] o.e.d. (7.7.6) (e) BreÐte thn kroustik apìkrish tou sust matoc pou ja mporoôse na sundejeð se sundesmologða seir c me èna sôsthma diafor c pr thc t xhc gia na anakt sei thn eðsodo. Dhlad breðte to h i n] ìpou h i n] (xn]) xn].

K. Kotrìpouloc: S mata-sust mata 3 Jèloume h i n] hn] δn] sto pedðo thc suqnìthtac H i () H() H i (). (7.7.63) e j All to sôsthma pou èqei tètoia apìkrish suqnìthtac eðnai h monadiaða bhmatik sun rthsh h i n] un]. (7.7.64) 7.7.7. 'Estw X(e j ) o metasqhmatismìc Fourier tou s matoc xn] pou faðnetai sto Sq ma 7.7.5. a ektelèsete touc upologismoôc pou akoloujoôn qwrðc na upologðsete rht c to metasqhmatismì X(e j ). xn].5.5 -.5 - -5-4 -3 - - 3 4 5 6 7 8 9 n Sq ma 7.7.5: S ma xn]. (a) X(e j ) (b) X(e j ) (g) X(ej )d (d) a prosdiorðsete kai na sqedi sete to s ma pou èqei metasqhmatismì Fourier Re{X(e j )}. (a) X(e j ) + n xn]e jn + n xn] 6. (7.7.65)

4 K. Kotrìpouloc: S mata-sust mata (b) ParathroÔme ìti h akoloujða eðnai summetrik perð to. ToÔto shmaðnei ìti X(e j ) + n e j + xn]e jn nm+ m { e j n m m + m xm +]e jm xm +]e j(m+) xm +]e jm + x] e j + { + e j x n] e jn ++ n + n + m xn +]e jn } + { e j x n] e jn + xn +]e jn}] (xn+]x n]) n + e j xn +](e jn + e jn ) e j { n + n xn +]cosn } } {{ } A() xm +]e jm } (7.7.66) ìpou A() pragmatik sun rthsh ra mhdenik c f shc. Sunep c X(e j ), <. (g) X(e j )d X(e j )e jn d x] 4. (7.7.67) n (d) Re{X(e j )} Re{A() e j } A() cos A() {e j + e j I } yn] {αn +]+αn ]}. (7.7.68) All X(e j )e j A() xn] αn ]. (7.7.69) opìte αn +]αn }{{ } +4] xn +4]. (7.7.7)

K. Kotrìpouloc: S mata-sust mata 5 Epomènwc yn] {xn]+xn +4]}. (7.7.7) Ta s mata xn +4] kai yn] sqedi zontai sto Sq ma 7.7.6. xn +4] n - - -8-6 -4-4 6 8 yn].5.5 n -.5 - -8-6 -4-4 6 8 Sq ma 7.7.6: S mata xn +4]kai yn]. 7.7.8. 'Estw xn] kai X() anaparistoôn mia akoloujða kai to metasqhmatismì Fourier thc, antistoðqwc. ProsdiorÐste me ìrouc tou X() touc metasqhmatismoôc twn y s n], y d n] kai y e n]. Se k je perðptwsh na sqedi sete touc Y () gia to metasqhmatismì X() tou Sq matoc 7.7.7. (a) Deigmatol pthc (sampler): xn] n rtioc y s n] n perittìc. (7.7.7) (b) Sumpiest c (compressor): y d n] xn]. (7.7.73) (g) Aposumpiest c (expander): xn/] n rtioc y e n] n perittìc. (7.7.74)

6 K. Kotrìpouloc: S mata-sust mata X(e j ) Sq ma 7.7.7: X(e j ). Upìdeixh: DÐnetai ìti: y s n] {xn]+( )n xn]} kai e j. (7.7.75) (a) Apì thn (7.7.75)kai to Sq ma 7.7.7 prokôptei ìti y s n] {xn]+ejn xn]} Y s () {X() + X( + )}. (7.7.76) Qrhsimopoi ntac to Sq ma 7.7.7, o metasqhmatismìc Fourier D.Q. Y s () sqedi zetai sto Sq ma 7.7.8. X() 3 3 X(+) Y s() Sq ma 7.7.8: Upologismìc Y s (). (b) Apì ton orismì thc sumpiesmènhc akoloujðac y d n] xn] (7.7.77)

K. Kotrìpouloc: S mata-sust mata 7 X() 3 3 X( ) X( +) 4 4 Sq ma 7.7.9: Upologismìc Y d (). o metasqhmatismìc Fourier D.Q. thc sumpiesmènhc akoloujðac dðnetai apì thn Y d () mn + n + m y d n] e jn + n m rtioc X( ] )+X( + ) xn] }{{} e jn xm] e j m F{y s n]} Y s ( ). (7.7.78) Sto Sq ma 7.7.9 sqedi zontai oi metasqhmatismoð X( ) kai X( + ). (g) Epeid y e n] x () n] Y e () X() (7.7.79) o metasqhmatismìc Fourier D.Q. thc aposumpiesmènhc akoloujðac eðnai ìpwc sto Sq ma 7.7.. Y e() 3 4 3 4 Sq ma 7.7.: Upologismìc Y e (). 7.7.9. Gia to sôsthma tou Sq matoc 7.7. prosdiorðste thn èxodo, ìtan h eðsodoc eðnai δn] kai ìtan H(e j ) eðnai èna idanikì katwdiabatì fðltro, dhlad : < H(e j ) (7.7.8) < <.

8 K. Kotrìpouloc: S mata-sust mata xn] H(e j ) wn] ( ) n + yn] Sq ma 7.7.: SÔsthma tou Probl matoc 7.7.9. An xn] δn], tìte X() kai W () H(e j )H(e j ). (7.7.8) All ìpou ( ) n wn] e jn wn] yn] yn] wn]+( ) n wn] (7.7.8) W ( ) H(e j( ) ) (7.7.83) Y () H(e j )+H(e j( ) ). (7.7.84) O metasqhmatismìc Fourier H(e j( ) ) èqei th morf tou Sq matoc 7.7., dhlad < H(e j( ) ) alloô. (7.7.85) Epomènwc Y (), ) (7.7.86) kai kat sunèpeia yn] δn]. (7.7.87)

K. Kotrìpouloc: S mata-sust mata 9 H(e j ) H(e j( ) ) 3 3 Sq ma 7.7.: Upologismìc H(e j( ) ).