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4 b h k6 I[?! T h bdk dt W P>PN O k6 5 * 6[ C? h k 5k6! h k [??! dt f!? a`8! bcid PQA [' ` /R8 < [ ` [h C f ^>? P N A5 <k?qr U W ` T^RC? R< ]d,[! #"$" Q A5 N [ *! A! * f W P>PN &%R A >W!,[ QR ' k R< bc c5 h W (%R A V k RC! T h bdka['q PQA! Dk* Dk 8 h! Dk]g dk WW *) +% Y6X-,[ h! Dk]g6e PQA! Dk* Dk / #"6" WCRTbcDA5 (% k bd W! P A[KC RC I a[ T! A! * c5 [ T! k6 A! * h bdk8! T! $h c7c [ *! A UW!Dk är8! h! Dk XL bc* bc R8 #) 7% Y6X$,[ h! Dk Xhe PQA! Dk* Dk W CRTbcDA5 8 U 2$-+54, 7:289?WV8X4-R=>C>Y+XGC>AG54, -"J`,3aC>A\71-+7:-RZ[,C]0F@BA_H 7:A8C>,=:9 -RA8@B=^5 Z[,X87BA\5$,=]0S B

5 5 #"6" Q >`8 A5! * h bdt o[ *! A! T! c5 W W P>P ]`2 bc WA[5Q W W `8 dk WW h! DkoB ) +% Y6X6,[ h! DkoBhe PQA! Dk* Dk W `2 A5 5"$ Q % k bd W! PU`8 A5? h 2`2 Af 6[gC )RC > *! [ T! h A W!Dk kh! T? R [< c5 W<f W P>PN.[ *! W bc k,[dvj h! DkQF R8 * bc W W! P `2 A5! bc ) 7% Y6X6,[ h! Dk F e PQA! Dk* Dk `2 Af F

6 b 5 Qk6 VPV c a\ gk6 bc d L [ T! k* Dk W! /ADA *k P> C P A [ T! k* Dk P gk6 bc 5c Id P>PNA[ C c* h 4` k T! Dk l k* Dk?! dt! S c5 P QR! bc f bc/ R [ P O A f bci P \a A! dka[dvjp [ k dr< ah GaADA h P BP `2 A, W A5 8dk! N dk A[ h! Dk Zhe\ A 5 PQA! Dk* Dk :+ 2:3 L O dt X Wbc k6 W PV c [! * bc Wh bc o[??! dt k PQA! Dk* Dk R P V A?: 9 WbcT? P> k RC,[kQ < bc lf k* Dk?! dk! h bc c c ` h *! Dk?: Ra [! B c bct W!] A ` R< >' Wf bc Dk k,[ P `2 A f A5 h! Dk g?c 6C ` * W! W bc?& R< P,[!9! P R8 $h %R? *! Vh! bc! f Dk k f,[!9! P h c 2R< bc*! Dk> W P -h P PQA! pass h f k* Dk stopp! h 2R< bc*! Dk W P> P PQA! Dk* Dk [ O P> δ 1 δ 8 bca % /A c h! Dk]g Ih,`C bc *,[9Q 2 Q! bch f bc WbdT bc/ YB lf k* Dk bc f pass 0 f! O Af bc [?Q f C *>P \ ADAf pass f stopp f stopp δ P Q! bc bc]! d\ ADA P O A f bc,[5qr! bc]h W 1 δ R8 gk 5 * O PQA 2 \B A h O bc P B ` h! Dk g bc 4 k W A[h\a W.C?: d PQA! Dk* Dk P> h bc ` h T! Dk /ADA h P> R< ) 7% Y6X$,[ Z

7 Q S O A `C! R< ch^[? h B?&bc * 6[ 0 /ADA h P> I PV 4?& 4[! dk!?c7r<c? P QR! bc bc$! bcb P bc )L dt &g*,[ h! Dk.She `C! R< A5 [ k `2 Wh Dk6! Dk R8 >c W! h l /k?! dt Ik W *) dk,e A 2 = A k 2n Ω 2n h %R bc A5 *! Dk dt! Dk A 0 Ω = f!? %? T! P> [ *!! Ih ` h! Dk f g W [? A[ QR! bca k6 bcah blf /k?! dk,e A[P ] = i (1 + a ip + b i P 2 ) C, k6 bc? k `2 W ct! Dk d? h k VW * [# [? P> k ` h *! Dk,e a i = 2 cos( [# [? P>! Dk ` h *! Dk b i = 1,e A 0 (2i 1)π ) 2n a i! b i e a 1 = 1 b 1 = 0 (i 1)π a i = 2 cos( ) n b i = 1 S

8 d "62435 "$2:3 >'bc A bc 0 bc h! bc> R< k S!?5 P Q! bch f bc! PV?: S!? P O A f bc! j['qr 7 k ` h *! Dk lf k* Dk?! dk C f 9 \ k 5 Wh k, P `C! R< c *)L dk X,[ h! Dk A5 [# B`2 bc 0 x 1 f W A h >bc ' Tk PV h k T Ra bc* 6 k bc59?,[s[# RCbc /h I>'bc ' n Tk PV x 1 PV,[ 8`2 Wh Dk6! Dk?O A5 P>PN P B,e A 2 ka 2 0 = 1 + ɛ 2 Tn(P 2 ) P> k R2c ` 5? x = 0 k6 A 2 = A \B V c k6d A[ 2 0 ɛ `C Dk6 P ]h l /k?! dt I ]h [ P A[P ] = i (1 + a ip + b i P 2 ) k a bc? h k VW *!? k `2 W ct! Dk,e a i b [# [? P> k ` h *! Dk,e i A 0 (2i 1)π á i = 2 b i sinh γ cos 2n 1 bi = cosh 2 γ cos 2 (2i 1)π 2n [% a[' 8P>! Dk ` h *! Dk,e á 1 = 1 sinh γ b1 = 0 á i = 2 b i sinh γ cos bi = 1 (i 1)π n cosh 2 γ cos 2 (i 1)π n i

9 "62435 "$2:3 S S O A >'bc VV&^[? R B * >'bc A[ Rc5 PV N?:!? P QR! bc f bc! R< k!? P bc,[ h! DkIXhe A5 5 _! ^[? ` T! bc A bc [' k 5 T` W W RC k S!? (R<c C P QR! bc! bc P O Af bc,[ O R8 P> C?: > A bc [! dt A A[?Q W I[! dk c d PQA! d6 R<? c *R! * ) dk &B-,, 5bcDk W dr<,[ h! Dko_he _! A5 X

10 4 5 h k6 [? Cd P 2 S f! Dk PV c T W2h k6 bwkh O k6 5! U, Th W P>PN O k6 5 * dt >! T h bdka[ QR \ A! Dk h W >h k6 [? >d / T^RC W!? `2 / dt bc O bc5! Dk A k [ P O /? ^R2 V PQA PV T RC ) dk,[ 5 * 0 P B 2 PQA! C WA [? 0 ct?p V >h k6 [? P> P k6 W VJPQA! A Wbc* c5,[ QR l /k?! dk W [)V:\8^[? & ` h T! Dk c5 h?: k b[ PIe H(z) = b 0 + b 1 z b N z N /bc [? V!?#k6! > bc* & dk! O! dk! > P>PV V! If W W A %R!DADA!? A *) 7% Y6X6,[ _

11 Y 3 "2O %7"2-/LN" %(f< 0" P>PN<P V *! I[ T! k* Dk k k f o[??! dk! QRC h 0!? h T 5[! ^>,? P R c5 0P f d6 * bc h QVJPQA! T^R< W W!Dk c7 k 2>5V O Tk PVA[,QR W QVJPQA! *]R< h[n] R! h bc Dk! $P> h! bc N[V]\ ^[' bc* A 5 A[ P h W W 0 P W! 6 P! h QVJPQA! *]R<!?2 W P>PN `2 bc f Wbc dt RC,[ QR WaR8 bc*` P P $h VPQA! T^RC h[n] P> ' RC k 4[ PV c T PN! A W A[ w[n] h[n] = h ideal [n]w[n] QR?&bc [ PV c T C 5\ Wbc* Wbd1?: A[ QR W? c ' W kbc W!! T 5 O bc1r8 Dk6! Dk P [ *! k* Dk [fqr W W 0c5 PV ]f A bch 'P! bc> % bc W 0c5 PV! d6 Nh! bc c T A5 [5 PV c T k P> sr<,[tq Cc? 0PV 0 R< \] c k 2 CRP>P> Dk^[ A[ W hamm [n] = { c [0, , 046 cos( 2πn) N+1 )] : n N 2 0 : sonst h! DkBgAYheDCRP>P> Dkg?:! Dk V h! Dk gay!? Wbc* O h B A VPQA! T^RC ` h [?! dk! h bc bc VPQA! *^RC W [V]\ ^[' C k W A[!? d O > A.[ T! k* Dk O Af dk! P W C P>P> Dkg?! bc bc PQA! Dk* DkB! A['QR VJPQA! T^R<! P Y Z + * 6 N 5bc] Wbc*R bc 9! P Wbc*A?&c k W! P bc ) +% Y6X6,[ U 2$-+54,"F@AJ`,Z\7:90RC^7BA;bX4C^,.=<C>A H"C>9 -R,.0R, 8@B24-RC>,.-DM ;<-+7:A89?(@:-R3c7:0RC>@A Z[,X87BA854,=]0S gay

12 [ Y &+ H%( L M! L "1%(fK< 0)" ga P [V:\8^[? W W27RCc f P QR! bc R8! bc P O Af bc k6 bc P 6 k G k6d A[ 0 *^R8!?:?&c.?O Cbc ga P E I[' f! ct<!?, \a P -k'k i>bc h P,[ 0 A T 0! dt h 0 P>! Dk$ k R8 bc* [5 c?! dk E(ω) h V! N P S!?2! $ N k PV W[5 P W!Dk c7 k [! dt c k cta[qr k VW * b['?! dk $ W!h R2c ' `2 /k] [ c?! dk P> P> R8 [ VJP [5 W A5 ^[? h W!]/ADA h P> [! dt dk k f ah! bc,e { 1 : 0 ω ωp H 1 (ω) = 0 : ω s ω π Q k6 W QR! bc f bc! 0 ω ω O A bc,[ p QR k R8 bc* [ c?! dk dr8 If W P>P h! bc,e ω s ω π E(ω) = W (ω)[ĥ1(ω) H 1 (ω)] Q k6 5? h % R8 bct! Dk?! dt W (ω) e { 1 W (ω) = K 0 ω ω p 1 ω s ω π '! () ^1\ ^ 6,R8 Bf >bc h P \] PN! P W k6 6! [5 c $Q! bc 8! O Af bc E(ω) bc*` RC E(ω) P> W S A bc 1RC Wbc!?#RC A[ O P> R8 N 0 +2 P> N\B-h P R< E max E(ω) f! 0 ω ω p ω s ω π k 5! Bf A bc `?& Bh W P P> W [ *! A $ h N D?O h k6 Ae E(ω i ) = E(ω i 1 ) = ±E max i = 1, 2,..., N P> ω 0 < ω 1 < ω 2 <... < ω N0 +1 [# h bl /k?! dk RCf?O Q(ω)!! Dk H(e jω ) = e jω(n 1)/2 Q(ω)P (ω) P (ω) k6 Ae Q(ω) = 1 k P (ω) = N 1 2 n=0 a(n) cos(nω) Q(ω) = cos( ω 2 ) g6g

13 N 2 1 P (ω) = b(n) cos(nω) n=0 k6 ` 5 h Nf % A O! P>PV N 0 P (ω) A[ Q S c5 H 1 (ω) P àa PN! P R8 P> ` T P?O E(ω i ) = δ k W W R8 [ 2 0 ω i ω p H 1 (ω i ) = Ĥ1(ω i ) E(ω i) W (ω i ) aw! S?:bc*! Dk H 1 (ω i ) = 1 Kδ 2 Dk P>PV dr8 K = δ 1 δ 2 `?: k6ae H 1 (ω i ) = 1 δ 1 [# V d T` < ω i+1 > ω i 0 ω i+1 ω P>W!?: k 6e p E(ω i+1 ) = δ 2 V! H 1 (ω i+1 ) = 1 + δ 1 Q! ^T$A bc [ <!? 5?: k d[ T! A ω i ω P QR! bc i+1 f bc S A bc R< Wbc ` 5R8 bct bc E(ω) H 1 (ω) Ih W P `C bca R8 bc!!? A[ 1 δ δ! R< /h!? 5?: k h P RC 2 E(ω) P O A bc k d A bc A[2, k6 P /c W! h W R8 ω s ω π W!Dk6! [ 5> c5 P,e H 1 (ω i ) = E(ω i ) = δ 2 V bci P H 1 (ω i+1 ) = E(ω i+1 ) = δ 2 O A f bc R8 bc! H 1 (ω) δ 2 δ 2 ) ^ \ ^ $,[ h! Dk]g6g6e PQA! Dk* Dk h! bc 2R< h! Dk aàa P PV c T gax

14 5. O * R & N PQA! WA [? ctp a [? P> dt!! dt! A[ 0 ]bc a[? ` h! Dk c5 h?: k [5 PIe H(z) = b 0 + b 1 z b N z N 1 + a 1 z a N z L 8bc >f d6 *` 6d6 V Vbc i[?!?#k6! V N 0 dt! O! dt! ' T! brc 6?P> ' bct A %R!DADA!? A!? ) 7% Y6X-,[ Y &+ Q#! "1MD-/" -9MO78^<-/LNM %(#< Q c5! k Dk RC S?&c W! P 0 *]R8!? VV]\ ^[' h & A >,? P [ O <h * W!! ' P 5gk Q[??! dk h k6 [??! dt c W!! R8! >, /A W k N VJ * k6 Dk A[ P?: DkV ctae dy a (t) dt = w a (t) dt a *! I O I T k6 R<? k6ae nt dy a (t) nt dt = y a (nt ) y a [(n 1)T ] = w a (t)dt (n 1)T dt (n 1)T Q VJ * k6 d I 5bca > /Af W k /ADA h P> arc R<!? k6ae y a (nt ) y a [(n 1)T ] = ( T 2 ){w a(nt ) + w a [(n 1)T ]} P h h dt &!f N Kk RCc * c5 P,e ` y(n) y(n 1) = ( T [w(n) + w(n 1)] 2 L *! Bh ^>,?: P> h W %R bc*! Dk k Wbc 5> T c*ae 2 1 z 1 Y (z) = W (z) T 1 + z 1 L $h W P> NW!Dk c7 k 5gk `2 W ct! Dk sy a (s) = W a (s) k6 bc B ct P 9 Rh >,?: P $ 5gk W! h k6?h! bc h N WA N R8 dt5r8 e s Y (s) 2 T 1 z 1 Y (z) 1 + z 1 W a (s) W (z) g7b

15 QR A.>,?: P / *A bc* ] O!! 5gk l /k6! Dk?! dt ]h! bc,e s = 2 1 z 1 T 1 + z 1 S! W! Dko?O $bc [? *]R8!?8 R 5gk Q[??! dk /RC bc 8h! bcqk W k6 aada h P PV c k P> R< >P! A[ ) ^1\ ^ 6,0QR N ^>? P $ R8 dk` 9 h B ^C f h! Dk]gAXhe h! Dk + 0 N!?0h + 0 k ' 2 > VJ 8 W 0 c dk W?O c R<,[*Q W)? c R8 W! P W!I P dvv]\ ^[' A[ g F

16 a Q I P h k6 \ Wbc *! *! PV bc! c h! bc h bc jc `C 2 k W 5RC d k bc \ k f Dk 5! k6d 9h IW! [ c d?o c,[ 5 "6! /d f Dk a (B + 1) T! 2B+1! c L A h RC,[ Q P> W! c Tbc k W R< d P Nk! k BDk Wbc R<,[ QR W àaf f A bc P j!5 *! Dk! d h! bc ADA? P> k j!5 T! Dk d k 5 T >bc5 dk.r<,[2q d6 h T!5 *! c *! V c PV,h > $c7c N>, ADA!?: x TA bc ) 7% Y6X,[ Q h! Dk]g7Bhe5 \! h! Dk d N bc d. 5 Q Nh \ W! bw WCh k6 o\ Wbc ' W kbc f k6 W2 * <! bc h! c! Dk k6 WA[*Q [ c,[ Wik6 P> P 9! >P -k P f 5 a! c &[8 h W < bc*r8 9k6? W? R< k! * bc Wh bc \ gk6 bc d I!?0W! A/k6,[ 0 *]R< W R8 Ih W! c, k c5 DRC P ] O k6! Dk$ A bc ` 0k >h! c R8 /Dk Wbc! R< Wbc >W! P ' R< P h P RC ` src P lf!?rf A bc A[ QR WQR8 h! bc h! Dkag FV bc* bc ) 7% Y6X,[ h! Dk]g F e\! h! Dk <! l!? d gaz

17 1 Q S k c ]W! \ A! Dk W QR k6 (! P?& RA dt 6e g6[ `2! C RC QR I [ /k [ <h a bc V ` [? bcta k V bc* Wbc*A RC V &[E`2 bc* Wbc*A W?P gk6 bc>h Q R8 bc W! Af bc,[hq 5bc RC h! bc I[? k c P! Rf A! af?`2 W?sk6 /A c bc k W A[ QR W? d V W C RC 6[)VJP bcta `2 d /k k c c?:! Dk$ h ocr R2.k W R<,[VJ `2 bct d P>PV c O k6 5 A A W ka W QR k6 ( ^V _ Ck C bc b\ Wbc 5! ) +% Y6X,[ Xh[c T Dk\ R!? T! Q [ *! k* Dk W Q k6 ( A Th bc ` kp h f Dkg?: c P> W R<,[ Q c s PN! * Dk\ Vc7c? *! T QR! bc f bc! T h bd,[hqr W c5?: k Dk6 bc*! Dk W! [ k 6e f stopaaf < f s f stop f h O Af? *! Wa * Dk\ ` h G f stopaaf f? T!! h O Af? T! N W QR k6 ( A[ s f! k stop %R bc*! Dk?: k65? h 6e f s > f stop + f stopaaf ap dt R2c ` 9 k6 h?! Dk ]h b[? d f c W T s Dk\ ` <! RC P k6 NR2c ` < k6 h f?:! Dk h CR R2 6[ QR W W QR P>P s d W QR h `? N! T bc Wh bc? T! A? 0 Dk* Dk! I! k* Dk> W W` P> ) 7% Y6X$,[ g7s

18 Bh[a c [? C h A * [ /k [ RC bc O! dk! k RCc brc 6[ Q c 6` P V 2 dt! 8k 2 bct & dt! O! dt! f 7 k 6 Q[V:\8^[?? T^R8 bd CT Tbc> VV]\ ^[' f 7 k6a[dvj?: k >' h P k bc a * bc W W! P>PV 7% Y6X$,e h! DkagAZhe 0 k bc59? 4[V:\! VV:\ [? F [ O Af W ( d6 T O Af W ( d6 $ c* P $ Q W >' Abc P P> QR! bc R! O Af bc (R8 W! k \ Ah Af P Q! bc! O Af bc,[t PVk Dk Ch 8W! k I> A 5! P k6 5bc c Rh ` h *! Dk! a P>W!?: k! bc \ Wbc 5!?#R2 #) +% Y6X6,[ Zh[ [? *^R8!? V ] 0 *]R8!? A c5 R8 a ' R< k [? P> ac?: W >' bc P! I A5 ADA *k P Wbc *) 7% Y6X,[ Sh[! dt! QR bc! dk!?o [V]\ ^[? N h bc* dk! QR dt?: P! dt! ` 6! bc > (! dk! sk 5 *A[6QR <c5! k! dt!? VV]\ ^[? f,h d6 h! dt! 'P>,`< kbd Xh[ ` h *! Dk7) 7% Y6X6,[ ik[ O PN! VPQA PV T ]! * Rbc 2`2 W P>PN! Dk>? T! Q k [? d k VW *! ['! dt! d6 0QR k6 ( 0P> )C?:! * bc Wh bc O /? ]RC RC dta!dk $ P!! a! bci A RC ) +% Y6X,[ gji

19 o. ) 7% Y6X-,!"# %$'&)(*!+,-. Q _ c,[ % k 5X6Y6Y6X ) C %, / !"# : ; 2<!## C [% Q,g7 /F ) ^ [ \ [, A A A A ^1\ ^ $,!"# ^ c g7 hg )L dk =% 9CBD/,"EGF>-;HI?'JK7 9 c*a,emll 6[ R8 dk Wh h[ k#l9r8 dt 6LQR k6 WN [? )L dk &g, =% 9CBD/,"EGF>-;HI?'JK7 9 c*a,emll 6[ R8 dk Wh h[ k#l9r8 dt 6L`C! R< c )L dk X-, =% 9CBD/,"EGF>-;HI?'JK7 9 c*a,emll 6[ R8 dk Wh h[ k#l9r8 dt 6L/>'bc )L dk &B-, =% 9CBD/,"EGF>-;HI?'JK7 9 c*a,emll 6[ R8 dk Wh h[ k#l9r8 dt 6L8 0 A bc N [! 5dT g7x