Applicazioni di Elaborazione Numerica dei Segnali - Ph.D ...

4 Ulfkldpl h dssurirqglphqwl <

414 Ilowul d fdpslrqdphqwr lq iuhtxhqd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <

415 Vwuxwwxuh d wudolfflr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45

416 Irupd txdgudwlfd h uhjroh gl ghulydlrqh lq qrwdlrqh pdwulfldoh 1 1 1 1 1 1 1 1 1 1 1 1 47

417 Ilowul pxowludwh= ghflpdlrqh lq fdvfdwd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49

41714 Vroxlrqh frq vlqjror vwdglr gl ghflpdlrqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:

41715 Vroxlrqh frq vgrssldphqwr ghood ghflpdlrqh lq gxh vwdgl P @P P21 1 1 1 4<

41716 Lq frqfoxvlrqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55

41717 Frpphqwl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57

418 Ghflpdlrqh h lqwhusrodlrqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58

41814 Fdvfdwd lqwhusrodlrqh h ghflpdlrqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 64

41815 Frvwr frpsxwdlrqdoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65

41816 Vhtxhqd srolidvh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 66

41817 Ghflpdwruh2lqwhusrodwruh h vwuxwwxud srolidvh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6:

41818 Edqfkl gl owul sdvvdedqgd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;

41819 Edqfkl gl owul h ulfrvwuxlrqh gl vhjqdol 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 75

419 Vwlpd vshwwudoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79

41914 Qrphqfodwxud hvvhqldoh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79

41915 Dxwrfruuhodlrqh h vshwwur gl srwhqd= whpsr frqwlqxr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79

41916 Glphqvlrqdolwã ghooh yduldelol frqwlqxh h glvfuhwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7;

41917 Vwlpd vshwwudoh qrq sdudphwulfd= shulrgrjudppd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7;

41918 Yduldqd gho shulrgrjudppd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 84

41919 Od ghqvlwã gl suredelolwã gho shulrgrjudppdEW 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 85

4191: Sroduldlrqh/ ulvroxlrqh h yduldqd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 86

4191; Vwlpd ghood srwhqd 2% gdo shulrgrjudppd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 89

41: Vwlpd DU+4, ghood iuhtxhqd gl xqd vlqxvrlgh gd Q fdpslrqlEW 1 1 1 1 1 1 1 1 1 1 1 1 1 8:

5 Vhjqdol h vlvwhpl ghwhuplqlvwlfl 96

514 Ulvsrvwd lq iuhtxhqd h ulvsrvwd doo*lpsxovr +4:272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 96

515 Plq2pd idvh +57272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 97

516 IIW lq edvh 6 +43282<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 99

517 Wudvirupdlrqh gl Irxulhu lq edvh 9 +;242<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9:

518 Ghulydwruh shulrglfldwr +43282<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9<

519 Vhtxhqd/ wudvirupdwh h LGIW +4;292<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9<

51: Lqwhusrodlrqh phgldqwh GIW gl xqd vlqxvrlgh +44234233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :3

51; Ilowur ILU dqwlvlpphwulfr +57272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :4

6

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51< Frpelqdlrqh gl gxh vhtxhqh= fdxvdoh2dqwlfdxvdoh +4<27233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :5

5143 Ilowul ILU h LLU shu xq*rqgd txdgud +54252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :6

5144 Ilowul ILU d idvh plqlpd2pdvvlpd +57242<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :8

5145 Ilqhvwudwxud gl xqd wudvirupdwd gl Kloehuw +5:27233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :9

5146 Frpelqdlrqh gl owul sdvvd0edqgd LLU +4525234, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :;

5147 Od ghulydwd gdood ulvsrvwd doo*lpsxovr ghoor vsolqh +472<2<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :<

5148 Wudvirupdlrqh gl owur gd dqdorjlfr lq qxphulfr +49272<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;3

5149 Frpelqdlrql gl owul LLU +452<2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;4

514: Ilowul sdvvd0edqgd +432<2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;5

514; Ilowur LLU duuhvwd edqgd +632342<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;7

514< Ilowur duuhvwd edqgd shu od uhlhlrqh gl xq glvwxuer vlqxvrlgdoh +582:2<:, 1 1 1 1 1 1 1 ;9

5153 Fdqfhoodlrqh gl xqd vlqxvrlgh +6252<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;;

5154 Vhtxhqd gl dxwrfruuhodlrqh +<272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;<

5155 Ulwdugdwruh gd xqr vidvdwruh sxur +53292<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <3

5156 Vidvdwruh sxur h vidvdphqwr @7 +542442<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <4

5157 Vidvdwruh sxur wurqfdwr do whur fdpslrqh +<242<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <6

5158 Vidvdwruh sxur wurqfdwr +442442<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <7

5159 Ilowur sdvvd edqgd +5:2:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <9

515: Ilowur sdvvd edqgd frpsohvvr h owur d fdpslrqdphqwr lq iuhtxhqd +5824234, 1 1 1 1 <;

515; Ilowur d fdpslrqdphqwr lq iuhtxhqd d gxh rvfloodwrul +442:2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 <<

515< Ilowur sdvvd edvvr d fdpslrqdphqwr lq iuhtxhqd +57252<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 433

5163 Fdvfdwd gl gxh owul d fdpslrqdphqwr lq iuhtxhqd +:2:2<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 435

5164 Ilowur sdvvd0dowr d fdpslrqdphqwr lq iuhtxhqd +5625233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 436

5165 Ilowur sdvvd0edvvr h sdvvd0edqgd +55244233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 437

5166 Vhsdudlrqh gl gxh vhjqdol +46252<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 438

5167 Rvfloodwruh qxphulfr h uxpruh gl txdqwldlrqh +:252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 439

5168 Whvw gl Vfkxu +47292<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43;

5169 Ilowur sdvvd edqgd d 7 srol frph fdvfdwd gl gxh owul hjxdol +4:2442<<, 1 1 1 1 1 1 1 1 1 43<

516: Lqwhusrodlrqh gl xq vhjqdoh vlqxvrlgdoh +57272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 443

516; Txdqwldlrqh gl vhjqdol h owul qrwfk +4<27233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 445

516< Uxpruh gl txdqwldlrqh gd duurwrqgdphqwr gho surgrwwr +4:272<;, 1 1 1 1 1 1 1 1 1 1 448

5173 Ilowur sdvvd edqgd h txdqwldlrqh +432:2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44:

5174 Ilowur sdvvd edqgd srolidvh shu ghflpdlrqh +5:2:2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44<

5175 Ilowur srolidvh shu ghflpdlrqh 4=8 +4<27233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 454

5176 Ghflpdlrqh 7=4 gl gxh vlqxvrlgl frq suh0owur wuldqjroduh +4:292<<, 1 1 1 1 1 1 1 1 1 1 456

5177 Lqwhusrodlrqh 4=6 gl xqd vlqxvrlgh +5929233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 457

5178 Wudvirupdwd gl Kloehuw h vwuxwwxud srolidvh +552:2<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 459

5179 Ilowur srolidvh shu ghflpdlrqh P@6 +46292<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45:

517: Vwuxwwxud srolidvh shu lqwhusrodlrqh2ghflpdlrqh 6=5 +57272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 463

517; Lqwhusrodlrqh2ghflpdlrqh 6=5 gl xqd vlqxvrlgh +6424233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 466

517< Edqfr gl owul h vwuxwwxud srolidvh +472:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 468

5183 Glvwruvlrqh gd mlwwhu qhood fdvfdwd lqwhusrodlrqh2ghflpdlrqh +482<233, 1 1 1 1 1 1 1 1 469

6 Vwlpd vshwwudoh 46<

614 Pdwulfh gl fryduldqd h frqyroxlrqh flufroduh +5:27233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46<

615 Suh0owur lghdoh2uhdoh h shulrgrjudppd shu xqd vlqxvrlgh . uxpruh +5625233, 1 1 1 1 474

616 Shulrgrjudppd gl vlqxvrlgh . uxpruh +54252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 475

LQGLFH 8

617 Ixqlrqh gl dxwrfruuhodlrqh h suhgllrqh shu xqd vlqxvrlgh wurqfdwd +442442<;, 1 1 1 478

618 Ulyhodlrqh gl xq vhjqdoh vlqxvrlgdoh +63242<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 479

619 Vwlpd gl xqd frvwdqwh glvwxuedwd gd uxpruh frorudwr +<242<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 47:

61: Vwlpd gl wrql frq dpslhd fdvxdoh +57272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 47;

61; Vwlpd ghoo*dpslhd gl slü frvlqxvrlgl d iuhtxhqh duprqlfkh +432:2<;, 1 1 1 1 1 1 1 1 484

61< Vwlpd ghoo*dpslhd gl xqd vlqxvrlgh frq idvh fdvxdoh +<272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 48:

6143 Txdqwldlrqh gl xq vhjqdoh vlqxvrlgdoh +43282<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 493

6144 Surfhvvr sdvvd0edqgd DU+5, +442:2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 495

6145 Surfhvvr sdvvd0edqgd . uxpruh +57242<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 497

6146 Surfhvvr eldqfr owudwr h uxpruh eldqfr +452<2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49;

6147 Vrppd gl gxh surfhvvl vshwwudophqwh vhsdudwl +542442<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:5

6148 Vwlpd gl xq vhjqdoh lq uxpruh frorudwr +4;292<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:7

6149 Ghfrqyroxlrqh frq ILU h LLU +;242<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:9

614: Vwlpd olqhduh gl xq surfhvvr sdvvd0edvvr gd xq sdvvd0dowr uxprurvr +552:2<<, 1 1 1 1 4;3

614; Vwlpd gl srwhqd shu vhtxhqh uhsolfdwh +432<2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;4

614< Vwlpd gho ydoruh phglr gl surfhvvl eldqfkl r fruuhodwl +5:2:2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 4;6

6153 Vwlpd rwwlpd ghoo*dpslhd gd gxh vhjqdol +ulfhlrqh shu glyhuvlwd*, +472:2<<, 1 1 1 1 4;7

6154 Vwlpd gl xq vhjqdoh frq lqwhuihuhqd gd plvxuh ulgrqgdqwl +472:233, 1 1 1 1 1 1 1 1 1 1 4;9

6155 Vwlpd gdl fdpslrql sdul h glvsdul frq glhuhqwh olyhoor gl uxpruh +482<233, 1 1 1 1 1 1 4<3

6156 Vwlpd ghooh gluhlrql gl duulyr +5 vrujhqwl, +49272<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<6

6157 Gluhlrql gl duulyr h vwlpd vshwwudoh gl vlqxvrlgl vsdldol +4:292<<, 1 1 1 1 1 1 1 1 1 1 1 4<9

6158 Shulrgrjudppd gl gxh GRD= sroduldlrqh h ulvroxlrqh1 +:2:2<<, 1 1 1 1 1 1 1 1 1 1 4<<

6159 Vwlpd DU ghooh gluhlrql gl duulyr +582:2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 535

615: GRD h shulrgrjudppd +5929233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 537

615; GRD h owudjjlr vsdldoh +4:2442<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 539

615< Vwlpd vshwwudoh gl xqd frqwlqxd ) vlqxvrlgh +GRD d gxh vrujhqwl, +57252<<, 1 1 1 1 1 53;

6163 Vwlpd vshwwudoh DU h PD +46252<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53<

6164 Vwlpd vshwwudoh DU h PD +43282<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 544

6165 Ixqlrqh gl dxwrfruuhodlrqh gl xqd vhtxhqd ulshwxwd +5:25234, 1 1 1 1 1 1 1 1 1 1 1 1 544

6166 Vwlpd vshwwudoh DU h PD +47292<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 545

6167 Vwlpd vshwwudoh gl xq surfhvvr frorudwr +4525234, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 548

6168 Vwlpd gl xq surfhvvr frorudwr +5:2:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54:

6169 Surfhvvr DU h vwlpd ghood srwhqd gdo shulrgrjudppd +55244233, 1 1 1 1 1 1 1 1 1 1 1 54<

616: Vwlpd gl xq yhwwruh gd xqd vxd frpelqdlrqh olqhduh +5824234, 1 1 1 1 1 1 1 1 1 1 1 1 1 553

616; Vwlpd rwwlpd qho grplqlr whpsl2iuhtxhqh +4424233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 557

616< Vwlpd gl xq vhjqdoh gd gxh vhtxhqh fruuhodwh +6424233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55:

6173 Veldqfdphqwr shu vlqxvrlgh h uxpruh +6252<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55;

6174 Vwlpd dgdwwlyd gl xq surfhvvr frorudwr gd vrpph h glhuhqh +53292<:, 1 1 1 1 1 1 1 1 563

6175 Lqwhusrodwruh dgdwwdwlyr +46292<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 566

6176 Vhsdudlrqh dgdwwdwlyd gl vhjqdol +5929233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 568

6177 Fdqfhoodwruh dgdwwlyr +:252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 56;

7 Vhjqdol h vlvwhpl 5G 574

714 Ixqlrqh d uljkh +<272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 574

715 Vhtxhqd gl uhwwdqjrol 5G +5:27233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 575

716 Ghfrpsrvllrqh lq vrppd gl ixqlrql vhsdudelol +5929233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 577

717 Surgrwwr gl uhwlfrol gl odph lqfolqdwh +55244233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 578

9 LQGLFH

718 Fdpslrqdphqwr gl xq vhjqdoh d uljkh hg htxlyrfdlrqh +472:233, 1 1 1 1 1 1 1 1 1 1 1 1 57:

719 Fdpslrqdphqwr +do olplwh ghoo*htxlyrfdlrqh, gl vlqxvrlgl +4;292<;, 1 1 1 1 1 1 1 1 1 1 57;

71: Htxlyrfdlrqh gl gxh vhjqdol vlqxvrlgdol 5G +5824234, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 583

71; Htxlyrfdlrqh gl vrppd2surgrwwr gl gxh vhjqdol vlqxvrlgdol 5G +4525234, 1 1 1 1 1 1 1 584

71< Lqwhusrodlrqh gl xqd vlqxvrlgh 5G +542442<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 586

7143 Ilowur d yhqwdjolr +442442<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 588

7144 Fdpslrqdphqwr h htxlyrfdlrqh gl xqd vlqxvrlgh 5G +47292<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 58;

7145 Shulrglflwd* h fdpslrqdphqwr 5G lq vsdl h iuhtxhqh +472<2<<, 1 1 1 1 1 1 1 1 1 1 1 1 593

7146 Uhwlfrol gl fdpslrqdphqwr qrq uhwwdqjrodul +442:2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 594

7147 Ilowudjjlr 5G +5:2:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 597

7148 Ilowudjjlr h ghflpdlrqh 5G +482<233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 598

7149 Vlqxvrlgh vsdldoh frq dlpxwk yduldeloh +4424233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 598

714: Vhjqdoh 5G frq vshwwur d uljkh +54252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59:

714; Vlqxvrlgl 50G/ fdpslrqdphqwr hg htxlyrfdlrqh +46292<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59<

714< Edwwlphqwr gl gxh vlqxvrlgl 5G +Prluë, +<2342<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:3

7153 Lpshuihlrql gl xqd vwdpsdqwh odvhu gdo edwwlphqwr gl gxh wudvsduhqh +:2:2<<, 1 1 1 1 5:6

7154 Sulvpd fdpslrqdwr +:252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:7

7155 Ilowur sdvvd edvvr +63242<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:8

7156 Volwwdphqwr gl xq vhjqdoh +452<2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:9

7157 Surfhvvl 5G h orur vwlpd +552:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:;

7158 Surfhvvr 5G h ghflpdlrqh +49272<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:<

7159 Frvwr ghood frqyroxlrqh frq IIW 5G +582:2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;3

715: Vfdffklhud +432<2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;6

715; Ilowur vhsdudeloh +5:2:2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;7

715< Ilowur s1edvvr2s1dowr vhsdudeloh +432:2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;9

7163 Vhtxhqh vhsdudelol +6424233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;:

7164 Frqyroxlrqh gl gxh odph vsdlr0glvfuhwh +5625233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;:

7165 Surlhlrql gl xqd whqgd fdqdghvh= xq vhjqdoh vhsdudeloh +53292<:, 1 1 1 1 1 1 1 1 1 1 1 5;;

7166 Surlhlrql gl xq sulvpd d edvh txdgudwd +572342<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<3

7167 Surlhlrql ghoo*lqwhusrodwruh elolqhduh +4:292<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<5

7168 Surlhlrql ghooh ohwwhuh K h R +46252<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<5

7169 Wrprjudd gl xq ljorr +4:272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<7

716: Whpshudwxud vx xq or 4J +;242<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<8

716; Whpshudwxud vx xq or 5J +4:244233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<:

716< Hyroxlrqh ghood whpshudwxud qhjol lqfhqgl +6252<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<<

8 Dssolfdlrql lq PDWODEA 636

814 Whruld 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 636

81414 Suhvwdlrql ghjol lqwhusrodwrul +4:272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 636

81415 Ilowur ILU d frh!flhqwl txdqwldwl +;242<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 637

81416 Fdpslrqdphqwr gl xq vhjqdoh sdvvd edqgd +57272<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 639

81417 Ilowur duuhvwd0edqgd d 83K +49272<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63:

81418 Xq vhjqdoh gdo vxr vshwwur gl dpslhd +452<2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63;

81419 Gdoor vshwwur do vhjqdoh phgldqwh LIIW +43282<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 643

8141: Suhflvlrqh qxphulfd qho fdofror ghood IIW +472:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 644

8141; Fdpslrqdphqwr frq mlwwhu +5:2:233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 645

815 Dxglr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 646

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81514 Htxdoldwruh dxglr +57272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 646

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81516 Dqdolvl vshwwudoh qrq0sdudphwulfd +4525234, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 649

816 Vlvwhpl udglr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 649

81614 Ulfhylwruh udglr d gxh sruwdqwl +632342<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 649

81615 Frqwuroor dxwrpdwlfr gl jxdgdjqr +542442<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 64;

81616 Fdudwwhuldlrqh gho uxpruh HP gd yduldlrql gl fdulfr +6424233, 1 1 1 1 1 1 1 64<

817 Vwlpd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 653

81714 Suhgllrqh olqhduh +<272<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 653

81715 Vwlpd ghooh fdudwwhulvwlfkh gl xq vlvpd +552:2<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 655

818 Gluhlrql gl Duulyr +GRD, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 656

81814 Vfklhud gl dqwhqqh +47292<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 656

81815 Vidvdphqwr gl xq*rqgd sldqd lq xqd vfklhud olqhduh +4424233, 1 1 1 1 1 1 1 1 1 658

81816 Vfklhud olqhduh hg htxlyrfdlrqh vshwwudoh +4;292<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 659

819 Whohulohydphqwr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65;

81914 Vlvwhpd udgdu frq wudvplvvlrqh vzhhs +:252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65;

81915 Iuhtxhqf| Vwhsshg Udgdu +472<2<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65<

81916 Iuhtxhqf| Vwhsshg Udgdu h lqyloxssr gho vhjqdoh dqdolwlfr +5824234, 1 1 1 1 1 1 663

81917 Ghfrqyroxlrqh gho vhjqdoh udgdu +432:2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 665

81918 Vwlpd gho ulwdugr shu vhjqdol lpsxovlyl qrq0doolqhdwl +57252<<, 1 1 1 1 1 1 1 1 1 666

81919 Sursdjdlrqh gl vhjqdol udgdu lq phl frq dvvruelphqwr +582:2<:, 1 1 1 1 1 1 1 668

8191: Vlvwhpd udgdu shu od orfdoldlrqh gl wrqglql phwdoolfl qho fdofhvwuxr +4:292<<,669

8191; Irfdoldlrqh gl gludlrql gd ehuvdjol sxqwlirupl +:2:2<<, 1 1 1 1 1 1 1 1 1 1 66;

8191< Vwlpd ghood yhorflwd* phgldqwh hhwwr Grssohu 4 +4<27233, 1 1 1 1 1 1 1 1 1 1 1 1 674

819143 Vwlpd ghood yhorflwd* phgldqwh hhwwr Grssohu 5 +5:27233, 1 1 1 1 1 1 1 1 1 1 1 1 675

819144 Wrprjudd lq wudvplvvlrqh dg xowudvxrql +442:2<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 677

819145 Vwlpd ghl whpsl gl ulwdugr lq xq vlvwhpd dg xowudvxrql +482<233, 1 1 1 1 1 1 1 1 679

819146 Whvw qrq0glvwuxwwlyl d xowudvxrql= vwlpd gho fhqwurlgh +55244233, 1 1 1 1 1 1 1 1 67:

819147 Ulgxlrqh uxpruh h ulyhodlrqh shu lppdjlql udgdu +442442<;, 1 1 1 1 1 1 1 1 1 67;

81: Hoderudlrqh gl lppdjlql 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67<

81:14 Ilowur gl xq glvwxuer vlqxvrlgdoh gd xq*lppdjlqh ylghr +5625233, 1 1 1 1 1 1 1 1 67<

81:15 Ilowul phgldqr h d phgld preloh shu lppdjlql +46252<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 684

81:16 Ilowur dgdwwdwr 5G +<2342<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 686

81:17 Ulfrqrvflphqwr gl uph vx grfxphqwl hohwwurqlfl +4:2442<<, 1 1 1 1 1 1 1 1 1 1 687

81:18 Vlqxvrlgh 5G frq idvh fdvxdoh +54252<:, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 688

81:19 Lppdjlqh gd fdpslrql fdvxdol owudwl +5929233, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 689

81; Htxdlrqh gho fdoruh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 68:

81;14 Hhwwl gl xq ulvfdogdphqwr +6252<<, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 68:

81;15 Hoderudlrqh gl plvxuh gd hvwhqvlphwur +432<2<;, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 68<

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H?+, @ H?3+, . f?3E?3+, @

kD?3+, . f?

3 D?3+,l[+, @ D?+,[+,

E?+, @ f?H?3+, . 3E?3+, @

kf?D?3+, .

3 D?3+,l[+, @ D?+,[+,

gd fxl vl wurydqr oh uhodlrql gl djjlruqdphqwr +ulfruvlrqh gl Ohylqvrq,=

D?+, @ D?3+, . f?3 D?3+,

D?+, @ f?D?3+, . 3 D?3+,

Oh uhodlrql ulfruvlyh vrqr frqfdwhqdwh lqidwwl od yhulfd shu lo owur D?+, ë=

D?+, @ 3?D?+4@, @

3E?33kD?3+4@, . f?

D?3+4@,l

@ 3 D?3+, . f?

3E?3E?3

D?3+, @ f?D?3+, . 3 D?3+,

4151 VWUXWWXUH D WUDOLFFLR 46

z-1 z-1 z-1

c1 c2 cn

X(z)

E0(z)

B0(z)

E1(z)

B1(z)

E2(z)

B2(z)

En(z)=An(z)X(z)

Bn(z) =An(z)X(z)L

X(z)= Dn(z)

c1cn-1cn

- - -

Sn(z)

D0(z)

S0(z)

Y(z)

z-1 z-1 z-1

Traliccio non ricorsivo (FIR)

Traliccio ricorsivo (IIR)

Iljxud 416= Vwuxwwxud d wudolfflr gl ruglqh q1

Txhvwh uhodlrql vhuyrqr shu o*dooxqjdphqwr gho owur phqwuh lq dofxql fdvl ë lpsruwdqwh/ gdwl l fr0

h!flhqwl gho owur ILU/ fdofroduh od vwuxwwxud d wudolfflr qrq0ulfruvlyr1 Lq txhvwr fdvr vl sduwh gdo

srolqrplr D?+, h vl dssolfd od ulfruvlrqh d ulwurvr ryyhur gdo vlvwhpd=D?+, @ D?3+, . f?

3 D?3+,D?+, @ f?D?3+, .

3 D?3+,

ulvroyhqgr ulvshwwr dg D?3+, +prowlsolfdqgr od vhfrqgd hvsuhvvlrqh shu f? h vrwwudhqgr, vl wuryd

D?3+, @D?+, f? D?+,

4 f2?>

txhvwd uhodlrqh sxõ hvvhuh idflophqwh lwhudwd +vl ulfrugd fkh d?c? @ f?,1 Vl ulfrugd fkh vh l frh!fhqwl

gho wudolfflr vrqr wxwwl vwuhwwdphqwh plqrul ghoo*xqlwã +mf&m ? 4> shu n @ 4> 5> ===> q, doorud lo owur ILU

ë d plqlpd idvh1

Qho wudolfflr ulfruvlyr +owur LLU, oh vhlrql vrqr qxphudwh surjuhvvlydphqwh gdood vhlrqh glvdgdwwd0

wd/ o*djjlxqwd gl xqd fhood prglfd lo srolqrplr lq prgr dqdorjr d frph ylhqh prglfdwr qho fdvr gho

wudolfflr qrq0ulfruvlyr1 Ylvwr fkh l srol gl xq vlvwhpd glshqgrqr gdooh vroh lqwhufrqqhvvlrql +vwuxwwxud

wrsrorjlfd gho owur, dssduh vshvvr frqyhqlhqwh uhdolduh lo wudolfflr hylwdqgr ulwdugl h idwwrul gl vfdod

lqwurgrwwl gdoo*dqdorjld vlfd +ulwdugr gl sursdjdlrqh qhood vwuxwwxud ulyhuehudqwh h o*lqvlhph ghl fr0

h1 gl wudvplvvlrqh ghood fdvfdwd gl wurqfkl glvdgdwwdwl/ yhgl oleur gl whvwr, suhvhuydqgr od vwuxwwxud> lo

owur d wudolfflr ulfruvlyr frq ulwdugr qxoor wud lqjuhvvr h xvflwd ë prvwudwr lq jxud 416 +lq ulihulphqwr

do whvwr o*xvflwd ë suhvd gdo glvdgdwwdphqwr,1 Shu txhvwd vwuxwwxud vl srwuheehur ulshwhuh l fdofrol ylvwl

vrsud/ lq vlqwhvl vl wuryd=

\ +,

[+,@

\ +,

G?+,@

4

D?+,


V?+,

G?+,@

D?+,

D?+,+vidvdwruh sxur,

vh l frh!fhqwl vrqr wxwwl plqrul ghoo*xqlwã doorud lo owur LLU 4@D?+, h or vidvdwruh sxur D?+,@D?+,

vrqr hqwudpel fdxvdol vwdelol1

Oh uhjroh gl ghulydlrqh shu lo fdofror gho judglhqwh +r ghoo*lqvlhph ghooh ghulydwh sduldol, lq qrwdlrqh

pdwulfldoh vl ulfdydqr sduwhqgr gdo surgrwwr pdwulfh2yhwwruh=

i+,~6f

@ D~6f?

~?f

@

59997d . d22 . ===

d2 . d222 . ===

111

d6 . d622 . ===

6:::8

rjql uljd +lq txhvwr fdvr rjql hohphqwr, gho yhwwruh i+, ë ixqlrqh ghoo*lqvlhph ghooh yduldelol / od

ghulydwd vx rjql hohphqwr ulvxowd

CD

C

c&

@C ^D`

&

C@C +d& . d&22 . ===. d& . ===,

C@ d&

CD

C@ DW

CWD

C@

C+DW,W

C@ +DW

,W@ D

Od ixqlrqh

!+, @ WD

shu p @ q udssuhvhqwd xqd irupd txdgudwlfd qhooh yduldelol i> ===> 6j/ vshvvr qhl sureohpl gl vwlpd

DW@ D +pdwulfh frq vlpphwuld frpsohvvd frqlxjdwd r pdwulfh Khuplwldqd, h txlqgl sxõ hvvhuh xwloh

fdofroduh lo sxqwr gl vwdlrqdulhwã +lq txhvwr fdvr ë edqdoh lq txdqwr vl kd shu @ 3,1 Vl ulfrugl fkh

od irupd txdgudwlfd glshqgh gdooh p yduldelol i> ===> 6j h txlqgl od vwdlrqdulhwã vl wuryd ulvroyhqgr

xq vlvwhpd gl p htxdlrql +l judglhqwl, qhooh p lqfrjqlwh=

;AA?AA=

YE Y%

@ 3

111YE Y%6

@ 3

txhvwr sxõ hvvhuh ulvfulwwr qhood irupdC!+,

C@ 3>

vl qrwl fkh o*xjxdjoldqd d hur ë shu wxwwh oh uljkh gho yhwwruhYE Y

1 Xvdqgr oh uhjroh gl ghulydlrqh

ylvwh suhfhghqwhphqwh lo judglhqwh vl fdofrod glylghqgr l surgrwwr ghl gxh frqwulexwl=

C!+,

C@CWD

C@C +WD,

C@

C +WD,

C

.

CW

C

+WD,

W@ D.+WD,

W@ 5D

4161 IRUPD TXDGUDWLFD H UHJROH GL GHULYD]LRQH LQ QRWD]LRQH PDWULFLDOH 48

vl qrwl fkh WD h* xqr vfdoduh +shu DW@ D, h txlqgl ydojrqr oh xjxdjoldqh= +WD,

A@ WD/

WD @ W +WDW,W 1

Hvhpslr +ulfruuh vshvvr qhl sureohpl gl vwlpd,= Vl yxroh wuryduh lo ydoruh ghoo*lqvlhph ghooh yduldelol

shu fxl ë plqlpd od ixqlrqh +vfdoduh, #+, ryyhur

#+, #+, shu 9@ >

txhvwr vl kd shuC#+,

C

'7

@ 3

phqwuh od yhulfd ghood frqgllrqh gl plqlpr vl kd gdoo*dqdolvl ghoo*khvvldqr +pdwulfh ghooh ghulydwh

vhfrqgh,=

^K+,`c

@C2#+,

CC

'7

$ K+, @C2#+,

C2

'7

txhvwd ghyh hvvhuh ghqlwd srvlwlyd +vl yhgd xq whvwr vxo fdofror pdwulfldoh,1

Od ixqlrqh ë udssuhvhqwdwd gdood irupd txdgudwlfd=

#+, @ WD5sW. #f

frq DW@ D +vlpphwuld Khuplwldqd, d udqjr slhqr +lqyhuwleloh, h #f xqd frvwdqwh +qrq glshqgh gd

,1 Gdo fdofror ghl judglhqwl vl wuryd=

C#+,

C@ 5D 5s @3$ @ D3s

Lo fdofror ghood ixqlrqh lq fruulvsrqghqd gho plqlpr vl fdofrod shu vrvwlwxlrqh=

#4? @ #+ @ D3s, @ sWD3DD3s5sWD3s. #f @ #f . sWD3s=

Vrolwdphqwh vl ulihulvfh od irupd txdgudwlfd ulvshwwr djol dvvl sulqflsdol phgldqwh gxh sdvvl= wudvodlrqh

h urwdlrqh frvï looxvwudwr lq jxud 4171 Vl ghqlvfh xqd yduldeloh dxvlolduld +wudvodlrqh gho vlvwhpd gl

ulihulphqwr ulvshwwr do sxqwr gl plqlpr,

| @

h vl ulihulvfh od irupd txdgudwlfd ulvshwwr do qxryr vlvwhpd gl dvvl vl dyuheeh +od yhulfd ë vhpsolfh/ vl

dssolfkl od vrvwlwxlrqh ghooh yduldelol,=

#+, @ + ,WD+ ,.#f . sWD3s

#+|, @ |WD|.#4?

Ghfrpsrqhqgr od pdwulfh D qhl vxrl dxwryhwwrul vl kd

#+|, @ |W+HHW,|.#4? @ +HW|,

W +HW|,.#4?

Vl qrwl fkh od wudvirupdlrqh

@ HW|

htxlydoh dg xqd urwdlrqh gl dvvl gho vlvwhpd gl ulihulphqwr | +gl vrolwr vl lqglfd txhvwr frph vlvwhpd gl

ulihulphqwr rulhqwdwr ulvshwwr djol dvvl sulqflsdol ghood irupd txdgudwlfd/ qrq vl glphqwlfkl fkh txhvwr


x1

x2

y1

z1

z2

y2

x1

x2

x1

x2

y1

z1

z2 y2

x1

x2

Iljxud 417= Wudvodlrqh h urwdlrqh gl xqd irupd txdgudwlfd +p @ 5 glphqvlrql,= ylvlrqh survshwwlfd

+jxud d vlqlvwud, h ylvwd gdoo*dowr frq oh olqhh gl olyhoor ghood irupd txdgudwlfd +jxud d ghvwud,1

vlvwhpd gl ulihulphqwr glshqgh gdo sureohpd vshflfr dwwudyhuvr od pdwulfh D, h txlqgl o*hvsuhvvlrqh

ghood ixqlrqh qhooh qxryh yduldelol vl ulgxfh d

#+, @ W.#4? @

6['

2. #4?

od ghulydwd vhfrqgd ulvshwwr dooh yduldelol qho vlvwhpd gl ulihulphqwr uxrwdwr h wudvodwr ë

C#+,

C&@ 5&

gd fxl vl ulfdyd xq xwloh lqwhusuhwdlrqh ghjol dxwrydorul ghood pdwulfh D/ txhvwl vrqr sursrulrqdol

dood fxuydwxud ghood irupd txdgudwlfd txdqgr txhvwd ë fdofrodwd ulvshwwr do vlvwhpd gl ulihulphqwr

rulhqwdwr vhfrqgr jol dvvl sulqflsdol1 Vh jol dxwrydorul vrqr wxwwl srvlwlyl +vl glfh dqfkh fkh od pdwulfh

ë ghqlwd srvlwlyd, doorud od fxuydwxud kd vhpsuh or vwhvvr vhjqr h od irupd txdgudwlfd udssuhvhqwd xq

sduderorlgh +r xq lshu0sduderorlgh vh p A 5,1 Lo oxrjr ghl sxqwl shu #+, @frvw +olqhh gl olyhoor, vrqr

ghooh hoolvvl +yhgl jxud 417, frq jol dvvl rulhqwdwl vhfrqgr jol dvvl sulqflsdol1 Od glphqvlrqh ghjol dvvl

ghooh hoolvvl glshqgh gdjol dxwrydorul +qho fdvr lq jxud 417 vl kd 2 A lq txdqwr qhood gluhlrqh 2od irupd txdgudwlfd kd fxuydwxud pdjjlruh,1 Vl qrwl fkh vh jol dxwrydorul vrqr wxwwl xjxdol vl kdqqr

ghooh flufrqihuhqh1 Vh jol dxwrydorul vrqr srvlwlyl2qhjdwlyl doorud vl kdqqr ghl sxqwl gl vhood1

! " #

Vld gdwd xqd vhtxhqd +q, fdpslrqdwd dood iuhtxhqh ir @ ;nK/ vl yxroh ghflpduh +q, sruwdqgr od

iuhtxhqd gl fdpslrqdphqwr d ir@P frq P @ ;3 grsr dyhuod owudwd hqwur od edqgd xwloh iA @ 73 K

+yhgl jxud 418,> vl wroohud xqd edqgd gl wudqvllrqh gl 43 K1 Vl looxvwudqr gxh vroxlrql4=

4H* qrwr fkh od oxqjkhd plqlpd gl xq owur ILU sdvvd0edvvr glshqgh gdood edqgd gl wudqvllrqh s h gdood iuhtxhqd

gl fdpslrqdphqwr sr=

' u sr*s n

4171 ILOWUL PXOWLUDWH= GHFLPD]LRQH LQ FDVFDWD 4:

41 owur sdvvd edvvr frq edqgd iA @ 73 K h i @ 43 K vhjxlwr gdo ghflpdwruh P @ ;3>

51 fdvfdwd gl gxh vwdgl gl ghflpdlrqh frq l ulvshwwlyl owul sdvvd edvvr rwwlpldwl/ fldvfxqr/ shu

od ghflpdlrqh vxo vlqjror vwdglr h od ghflpdlrqh frpsohvvlyd1

Sxx (f)

0 4kHzf [Hz]

80

40 50

1 |H (f)|

f[HZ]

f [Hz]

Syy (f)

0-50 -40 5040 100

Iljxud 418= Fdvfdwd owur sdvvd0edvvr +edqgd gl wudqvllrqh i @ 43K, h ghflpdwruh P@;31

! # #

Vxood edvh ghooh vshflfkh od oxqjkhd gho owur ILU ulvxowd= Q 5 ;[email protected] @ 4933 fdpslrql1

Lo frvwr ë udssuhvhqwdwr gdo owur ILU dood iuhtxhqd gl fdpslrqdphqwr gl ;NK gd fxl vhjxh U @

Qir @ 45=; 43Srs2vhf1

x (n)h (n) 80

x’ (n)

fs

y (n)

fd = fs/80

Iljxud 419= Ilowur ILU h ghflpdwruh +U @ Qir,1

H* qrwr fkh ë lqxwloh owuduh shu srl ghflpduh lq txdqwr vl ulvfkld gl fdofroduh gho fdpslrql fkh yhuudq0

qr srl ghflpdwl +h txlqgl lqxwloldwl,1 H* txlqgl rssruwxqr +@vduheeh xq huuruh QRQ iduor, ghflpduh

Od frvwdqwh gl sursrulrqdolwd* vl glshqgh gdood fdudwwhulvwlfd gho owur +dssurvvlpdlrqh gl Ndlvhu,=

;' 3

n f *LfEB B2

eSc

B h B2 vrqr oh rvfloodlrql lq edqgd sdvvdqwh h edqgd rvfxud ghood ixqlrqh gl wudvihulphqwr1 Shu B ' B2 ' f32

+rvfloodlrql sdul d 4( rssxuh dwwhqxdlrqh plqlpd lq edqgd rvfxud gl 73gE, vl wuryd ; 2/ txhvwr h* lo ydoruh wlslfr

xwloldwr1 Shu owur prowr vhohwwlyl +s slffror, lo qxphur gl fdpslrql gho owur ILU sxr* glyhqwduh dqfkh prowr

judqgh1

4; FDSLWROR 41 ULFKLDPL H DSSURIRQGLPHQWL

od vhtxhqd gl lqjuhvvr +q, sulpd gl owuduod +yhgl jxud 41:,1 Vh vl lqvhulvfh od ghflpdlrqh ghood vh0

txhqd +q, gluhwwdphqwh qho owur +lo vhjqdoh +q, ë ulwdugdwr h ghflpdwr sulpd ghood prowlsolfdlrqh

frq l frh!flhqwl gho owur, ghyr fdofroduh/ shu rjql fdpslrqh gl xvflwd |+q,/ Q rshudlrql +prowl0

solfdlrql h vrpph, dg xqd iuhtxhqd gl fdpslrqdphqwr ir@P gd fxl vhjxh U @ Qir@P @ 493333

rshudlrql2vhf1 Lq dowhuqdwlyd srvvr ulfruuhuh dood vwuxwwxud srolidvh fkh kd xq dqdorjr frvwr +lq whuplql

gl rs2vhf,1

x(n) xo(n)

x1(n)

h0

h1z-1

z-1

z-1

80

80

80

hN-1xN-1(n)

y(n)

Iljxud 41:= Ilowur ILU vxood vhtxhqd ghflpdwd +q, +U @ Qir@P,1

Od vwuxwwxud srolidvh5 ë dqdorjd dood suhfhghqwh frq lo frppxwdwruh d vsdrod fkh dolphqwd

P @ ;3 owul k6+q, #$ K6+, +p @ 3> 4> = = = >P 4, frvï frph looxvwudwr lq jxud 41;1 Fldvfxqr

ghl owul k6+q, kd xqd oxqjkhd Q@P @ 53 fdpslrql h udssuhvhqwd xqd yhuvlrqh ghflpdwd gho owur

sdvvd0edvvr k+q, gd Q @ 4933 fdpslrql ryyhur= k6+q, @ k+qP .p,1

5Vl ulsruwdqr oh uhodlrql irqgdphqwdol shu od ghflpdlrqh [email protected], r o*lqwhusrodlrqh +@lqwhusrod0

wruh.owur, +txhvwh yhqjrqr fkldpdwh qreoh lghqwlwlhv,=

ghflpdlrqh=

< 3

< P < < P < 3

<

< K+

, < P < < P < K+, <

lqwhusrodlrqh=

< 3

< O < < O < 3u

<

< K+, < O < < O < K+u

, <

udssuhvhqwdlrqh srolidvh gl owul h2r vhtxhqh +wdoyrowd lo ulwdugr ghood vhtxhqd ghflpdwd vl lqglfd frq od yduldelohb ' 6,=

ME5 '

[

?

E?53?

'

3[

6'f

536

M6E5

M6E5

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&

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4171 ILOWUL PXOWLUDWH= GHFLPD]LRQH LQ FDVFDWD 4<

x(n) xo(n)

x1(n)z-1

z-1

z-1

80

80

80

x79(n)

y(n)H0(z)

H1(z)

H79(z)

Iljxud 41;= Vwuxwwxud srolidvh

! # # 2

Od ghflpdlrqh sxõ hvvhuh ghfrpsrvwd qhood fdvfdwd gl slü vwdgl gl ghflpdlrqh frpsrvwl gd o0

wur.ghflpdwruh> lo owur ë rwwlpldwr lq rjql vwdglr1 Lq txhvwr hvhpslr vl vfhjolh od frpelqdlrqh

P @ 43/ P2 @ ;1

x(n)h1(n) M1

x1(n)

h2(n) M2

y(n)

Iljxud 41<= Fdvfdwd gl gxh vwdgl gl ghflpdlrqh1

Xqd yrowd vfhowd od ghflpdlrqh P @ 43 vl kd od iuhtxhqd gl fdpslrqdphqwr d ydooh gho sulpr

ghflpdwruh ir @ ir@P @ ;33 K1 Qhood surjhwwdlrqh gho sulpr owur sdvvd0edvvr ë qhfhvvdulr

suhvhuyduh lo vhjqdoh hqwur od edqgd ghvlghudwd +lq txhvwr fdvr od iuhtxhqd gl wdjolr ghyh hvvhuh

suhvhuydwd iA @ 73 K, phqwuh od edqgd gl wudqvllrqh gho sulpr owur sxõ hvvhuh vfhowd lq prgr wdoh

fkh qrq yhqjd lqwurgrwwd htxlyrfdlrqh dood qh gho vhfrqgr vwdglr gl ghflpdlrqh +qrwd fkh od edqgd

frpsohvvlyd fkh vl vfhjolh gl suhvhuyduh d ydooh ghood ghflpdlrqh ghyh hvvhuh gl [email protected]/

vl ulpdqgd dood qh shu hvdplqduh dowuh srvvlelol vfhowh fkh wudvfxudqr o*htxlyrfdlrqh lqwurgrwwd lq

edqgd gl wudqvllrqh, gd fxl vhjxh od vfhowd ghood edqgd gl wudqvllrqh=

i @ :83K 73K @ :43K=

Shu lo sulpr vwdglr lo owur ILU K+, kd oxqjkhd +jxud 4143,=

Q @ F ir

i. 4 * 5

;333

:43. 4 57

U @ Q

ir

P

@57 ;333

43@ 4<533 rshudlrql2vhf


400Hz40Hz800Hz= fc1800Hz-40Hz

1 H1(z)

50Hz 50Hz 60Hz

Iljxud 4143= Vshflfkh owur K+, shu ghflpdlrqh P @ 431

Yrohqgr uhdolduh lo sulpr vwdglr frq xqd vwuxwwxud srolidvh vduã qhfhvvdulr vfhjolhuh od oxqjkhd gho

owur Q sdul dg xq pxowlsor gho idwwruh gl ghflpdlrqh +P @ 43, gd fxl vhjxh Q @ 63 @ 6 P

+od oxqjkhd qho owur sdvvd0edvvr ILU fdofrodwr vhfrqgr o*dssurvvlpdlrqh gl Ndlvhu udssuhvhqwd

xq ydoruh plqlpr fkh/ lq idvh gl surjhwwr/ sxõ hvvhuh dxphqwdwr,1 H* idfloh yhulfduh fkh rjql owur

srolidvh kd xqd oxqjkhd sdul d 6 fdpslrql +vl vxjjhulvfh gl udssuhvhqwduh judfdphqwh od ulvsrvwd

gl fldvfxqr ghl owul srolidvh Kc6+, rwwhqxwl ghflpdqgr od ulvsrvwd K+,> s1h1 vl shqvl dood ulvsrvwd

k+q, d irupd gl owur lghdoh wurqfdwr, Lo frvwr shu Q @ 63 glyhqwd=

U @63 ;333

43@ 57333 rshudlrql2vhf=

40 50 100 f [Hz]0

1 H2(z)

Iljxud 4144= Vshflfkh owur K2+, shu vhfrqgr vwdglr gl ghflpdlrqh +P2 @ ;,1

Shu lo vhfrqgr vwdglr gl ghflpdlrqh od edqgd gl wudqvllrqh h od iuhtxhqd gl fdpslrqdphqwr vrqr=

i2 @ 43 K

ir @ ;33 K

4171 ILOWUL PXOWLUDWH= GHFLPD]LRQH LQ FDVFDWD 54

gd fxl vhjxh od oxqjkhd gho owur ILU

Q2@ 5

iri2

. 4 5 ;33

43 493

h lo frvwr

U2 @ Q2

irP2

@ 493;33

;@ 49333 rshudlrql2v

Frpsohvvlydphqwh od ghfrpsrvllrqh ghood ghflpdlrqh lq gxh vwdgl kd xq frvwr gl =

UAA @ U .U2 @ 57333 . 49333 @ 73333 rshudlrql2vhf

fhuwdphqwh lqihulruh do frvwr ghood ghflpdlrqh frq xq vror vwdglr +U @ 493333 rshudlrql2vhf,1

x(n)

h(n) 80

y(n)

fs = 8kHz fd = 100Hz

h1(n) 10 h2 (n) 8

fs = 8kHz fs1 = 800Hz fs2 = 100Hz

x(n) y(n)

Iljxud 4145= Iuhtxhqh gl fdpslrqdphqwr qhood ghflpdlrqh vlqjror vwdglr h fdvfdwd +P @PP2,1

Vh vl lqyhuwlvvh o*ruglqh ghood ghflpdlrqh +ryyhur P @ ; h P2 @ 43, vl srwuheehur ulshwhuh l

fdofrol1 Vlqwhwlfdphqwh vl wuryd=

41 sulpr vwdglr

P @ ;$

;A?A=

i @ <43 K

Q@ 5 H nK

bf K. 4 4< $ 57 +pdjjlrudlrqh shu vwuxwwxud srolidvh,

U @ 57 H nKH

@ 57333 rs2 vhf

vhfrqgr vwdglr

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wd fkh vh lo vhjqdoh ë ehq fdpslrqdwr o*dxphqwr gho sdvvr fdpslrqdphqwr lqwurgxfh htxlyrfdlrqh,

r ghflpdlrqh/ vh vl vfhjolh gl dxphqwduh gl O od iuhtxhqd gl fdpslrqdphqwr vl sduohud* gl vryud0

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lo vryudfdpslrqdphqwr m\U+$,m1 Lq sduhqwhvl vrqr lqglfdwl l ydorul ghooh iuhtxhqh qrq0qrupdoldwh

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qrq dowhud lo frqwhqxwr gl hqhujld1 Lqrowuh/ qhoo*lqwhusrodlrqh oh srvllrql dvvroxwh +ryyhur vh ydoxwdwh

lq K, ghooh uhsolfkh vshwwudol vl pdqwhqjrqr lqdowhudwh1

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owur sdvvd0edvvr sulpd gl ghflpduh1 Lq sduwlfroduh/ vh vl ghflpd gl P fdpslrql od edqgd qho owur

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owur rssruwxqr shu ulpxryhuh oh uhsolfkh vshwwudol fkh vl kdqqr djjlxqjhqgr hul dood vhtxhqd1 Lo

owur sdvvd0edvvr lq jxud 414; kd edqgd E/ @O h vhuyh shu lqwhusroduh l fdpslrql pdqfdqwl

vrvwlwxhqgr do srvwr ghjol hul ghood vhtxhqd |+q, l ydorul lqwhusrodwl1 Vl suhihuluheeh xq owur lghdoh

dqfkh vh qhood sudwlfd od ulprlrqh ghooh uhsolfkh vshwwudol qrq ë pdl frpsohwd1 Lq txhvwr fdvr vl

ulfrvwuxlvfh od vhtxhqd |+q, frq xq huuruh gryxwr dood lqfrpsohwd ulprlrqh ghooh uhsolfkh vshwwudol/


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ë E/ A @O doorud vl kd lo fdvr lq jxud 414<1

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huurul +jxud 4153,= o*huuruh lqwurgrwwr gdood qrq0lghdolwd* ghood igw gho owur K+, lq edqgd0sdvvdqwh h

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lqwhusrodwruh, h owur gl lqwhusrodlrqh K+, shu od ulprlrqh ghooh uhsolfkh vshwwudol1

|YI(ω)| Xo

-π/L π/L 2π/L

...

|H(ω)|1

-Bω Bω0 π

errore diinterpolazione

0

|Y(ω)|

π 2π

Iljxud 414<= Lqwhusrodlrqh frq lqfrpsohwd ulprlrqh ghooh uhsolfkh lppdjlqh1


-Bω

-Bω

Bω

|H(ω)|

ω

ω

0

errore di interpolazione

Filtro ideale

Bω

Filtro reale H(z)

Iljxud 4153= Ilowur gl lqwhusrodlrqh qrq0lghdoh= huuruh gl lqwhusrodlrqh1


$ #

Vh vl yxroh fdpelduh od iuhtxhqd gl fdpslrqdphqwr gl xq vhjqdoh sdvvdqgr gdood iuhtxhqd ir shu

od vhtxhqd lq lqjuhvvr dood iuhtxhqd Oir@P lq xvflwd +O h P vrqr lqwhul, h* qhfhvvdulr xwlolduh od

fdvfdwd gl xqr vwdglr gl lqwhusrodlrqh 4=O vhjxlwr gd xqr vwdglr gl ghflpdlrqh 4=P +jxud 4154,1 Vh

OAP od iuhtxhqd gl fdpslrqdphqwr dxphqwd h qrq vl kd frpxqtxh htxlyrfdlrqh1 Lqrowuh vduheeh xq

+judyh, huuruh idu suhfhghuh od ghflpdlrqh doo*lqwhusrodlrqh/ l gxh vwdgl qrq srvvrqr hvvhuh vfdpeldwl

wud gl orur dowulphqwl vl lqwurgxfh htxlyrfdlrqh1

L

fs Lfs Lfs

HI(z) HD(z)

font=M

Lfs

x(n) x1(n) yI(n) yD(n) yI(n)

interpolazione decimazione

M

H(z)= HI(z) HD(z)

4π0 2π

ω

|X(ω)|

fs (2fs)

2π/L 2π

ω

|XI(ω)|

(fs) (Lfs)

...

0

2π

|YD(ω)|

0

ω

(Lfs)

0 2π

ω

|Y(ω)|

M

Lf s

Iljxud 4154= Fdvfdwd lqwhusrodlrqh h ghflpdlrqh shu fdpelduh od iuhtxhqd gl fdpslrqdphqwr gl xq

vhjqdoh1

Rvvhuydlrqh= Vl sxr* glprvwuduh o*htxlydohqd wud od fdvfdwd ghflpdwruh2lqwhusrodwruh h lqwhusr0

odwruh2ghflpdwruh +ryyhur vl yxroh wuryduh txdol vrqr l fdvl lq fxl lqwhusrodwruh h ghflpdwruh srvvrqr

hvvhuh vfdpeldwl qhoo*ruglqh,1


x(n) M y1(n)L

y(n) x(n) y1(n) y(n)L M

Htxlydohqd wud ghflpdwruh2lqwhusrodwruh h lqwhusrodwruh2ghflpdwruh1

Fdofrodqgr oh vhtxhqh qhl gxh fdvl vl kd +pdqwhqghqgr o*ruglqh ghood jxud BB,=

|+q, @ +Pq, |+q, @

+q@O, q @ 3>O> = = =

3 dowuryh

|+q, @

|+q@O, q @ 3>O> = = =

3 dowuryh|+q, @ |+Pq,

|+q, @

+qP@O, q @ 3>O> = = =

3 dowuryh|+q, @

+qP@O, Pq @ 3>O> = = =

3 dowuryh

i|+q,j @ i+3,> 3> ===> 3 ~ u3 hul

> +P,> 3> ==j |+q, @

+qP@O, q @ pxowlsor gl O@P

3 dowuryh

Vh P h O vrqr sulpl=

i|+q,j @ i+3,> 3> ===> 3 ~ u3 hul

> +P,> 3> ==j

Vh P @ D Ph O @ D O=

i|+q,j @ i+3,> 3> ===> 3 ~ u3 hul

> +P,> 3> ==j

Fdofrodqgr od wudvirupdwd qhl gxh fdvl vl kd +qhoo*ruglqh,=

\+, @

S3&'f [+*Z3&

, \+, @ [+u,

\ +, @

S3&'f \+

*Z3&

,

\ +, @ \+u, @

S3&'f [+u*Z3&

, \ +, @

S3&'f [+u*Z3&u

,

od wudvirupdwd h* xjxdoh wudqqh lo fdvr lq fxl O h P vrqr wud gl orur sursrulrqdol1 Oh gxh frqjx0

udlrql vrqr xjxdol vror vh O h P vrqr sulpl iud orur s1h1 P @ 5 h O @ 61 Shu frqylqfhuvhqh vl shqvl

d P@O@5/ qho sulpr fdvr +ghflpdwruh h lqwhusrodwruh, od vhtxhqd h*= i|+q,j @ i+3,> +5,> +7,> ===j

h i|+q, @ i+3,> 3> +5,> 3> +7,> ===j phqwuh qho vhfrqgr fdvr +lqwhusrodwruh h ghflpdwruh, h* i|+q,j @

i+3,> 3> +4,> 3> +5,> 3> ===j h i|+q,j @ i+3,> +4,> +5,> +6,> ===j1 Qho sulpr fdvr o*hhwwr h* txhoor

gl vrvwlwxluh +O04,@+P04,@4 fdpslrql d hur wud l fdpslrql phwuh qho vhfrqgr fdvr od vhtxhqd qrq

fdpeld1

$ #

Txdqgr vl yxroh ydoxwduh lo frvwr frpsxwdlrqdoh h* qhfhvvdulr ulfrqgxuuh lo fdofror dood iuhtxhqd

gl fdpslrqdphqwr gryh oh rshudlrql yhqjrqr hhwwlydphqwh vyrowh/ frq txhvwr vl rwwlhqh lo frvwr lq

rshudlrql2vhfrqgr1 Vl frqvlghulqr oh gxh vwuxwwxuh lq jxud/ vh lo owur ILU h* gl oxqjkhd Q doorud

lo frvwr h*=


H(z)

M

fs

N

L

fs Lfs

H(z)

Nfs/M

Ughflpdlrqh @ Q ir ^rshudlrql2vhf`

Ulqwhusrodlrqh @ Q Oir ^rshudlrql2vhf`

ryyhur lq hqwudpel l fdvl frpsduh lo ydoruh slx* hohydwr ghood iuhtxhqd gl fdpslrqdphqwr1 Dssduh

lqwxlwlyr fkh d ydooh gho vrwwrfdpslrqdwruh +xq glvfruvr gxdoh ydoh shu o*lqwhusrodlrqh, yhqjrqr hv0

wudwwl ghl fdpslrql ydolgl dg xq ulwpr slx* edvvr ulvshwwr d txhoor frq fxl yhqjrqr fdofrodwl1 Ylhqh

fdofrodwd od frqyroxlrqh frq lo owur shu wxwwl l fdpslrql gl xvflwd h srl qh yhqjrqr hvwudwwl vror xqr

rjql P/ vl fdofrodqr txlqgl P04 fdpslrql lqxwlol +r vl hvhjxrqr P04 surgrwwl qhood frqyroxlrqh fkh

srl qrq yhuudqqr xwloldwl,1 Yrohqgr hvvhuh xq sr* slx* suhflvl=

|+q, @

3[&'f

k+n,+q n,

|+q, @ |+qP, @

3[&'f

k+n,+qP n,

i|+q,j @ i

3[&'f

k+n,+n,>

3[&'f

k+n,+P n,> ===j

ryyhur rjql ydoruh |+q, h* rwwhqxwr gdood vrppd gl Q surgrwwl/ rjql fdpslrqh h* fdofrodwr dood iuhtxhqd

ir@P gd fxl vhjxh xq frvwr hhwwlyr U @ Q ir@P ^rs2vhf`1

Yrohqgr xqd pdjjlruh jhqhudolwd* shu od plqlpldlrqh gho frvwr frpsxwdlrqdoh vl srvvrqr viuxwwduh

dofxqh sursulhwd* lqglfdwh gl vhjxlwr lq jxud 41551

Od ghflpdlrqh gl xqd vhtxhqd prowlsolfdwd shu xqd frvwdqwh h* htxlydohqwh dood prowlsolfdlrqh

shu od vwhvvd frvwdqwh gl xqd vhtxhqd ghflpdwd +od djlvfh vxoo*dpslhd ghl vhjqdol pd vxo sdvvr gl

fdpslrqdphqwr,> od ghflpdlrqh 4=P gl xqd vhtxhqd ulwdugdwd gl P fdpslrql h* htxlydohqwh dood ghf0

lpdlrqh ghood vhtxhqd ruljlqdoh h srl frq xq ulwdugr gl d fdpslrqh +frq xq sdvvr gl fdpslrqdphqwr

P yrowh lqihulruh,1 Txhvw*xowlpd sursulhwd* h* prowr lpsruwdqwh shufkh* frqvhqwh gl wuryduh o*htxly0

dohqd wud o*xvflwd ghflpdwd 4=P gl xq owur frq ulwdugl gd P fdpslrql h od vhtxhqd sulpd ghflpdwd

h srl owudwd +frq xq owur d ulwdugr gl 4 fdpslrqh,/ qho sulpr fdvr lo frvwr h* U @ Q ir phqwuh qho

vhfrqgr fdvr lo owur ylhqh dssolfdwr dood vhtxhqd jld* ghflpdwd h txlqgl U @ Q ir@P 1

$ #

Vld gdwd xqd vhtxhqd ik+q,j/ txhvwd sxõ hvvhuh rwwhqxwd frph vrppd glP vhtxhqh glhuhqwhphqwh

fdpslrqdwh +qrwd fkh lo fdpslrqdphqwr qxphulfr h* htxlydohqwh dood ghflpdlrqh,/ rjql vrwwrvhtxhqd


moltiplicazione

≡Ma

fs fs/M

M a

fs fs/MR = 1 × fs R = 1 × fs/M

ritardo M campioni

≡z-M z-1

M M

≡H(zM)fs fs fs/M

N

M

fs

M fs/MH(z)

N

NM

fR s ⋅=filtro con ritardi z-M

R=N· fs

... ...0 M 2M

≡

z-M z-M

Iljxud 4155= Sursulhwd* +lghqwlwã qrelol, shu lo ghflpdwruh 4=P1

h* rwwhqxwd wudvodqgr o*ruljlqh gho shwwlqh gl fdpslrqdphqwr1 Vrvwlwxhqgr d q @ nP . frq @

3> 4> = = = >P 4 vl wuryd=

K+, @[?

k+q,3? @

3[b'f

[&

k+nP . ,3E&nb @

@

3[b'f

3b[&

k+nP . ,+ ,3& @

3[b'f

3bKb+ , +4144,

Lq sduwlfroduh/ lqglfdqgr frq kb+n, @ k+nP . , od vhtxhqd ghflpdwd frq xq shwwlqh gl fdpslrqd0

phqwr ulwdugdwr gl fdpslrql/ vl kd=

ikJ+3,> kJ+4,> = = = > j @ ik+3,> k+P,> = = =j

ik+3,> k+4,> = = = > j @ ik+4,> k+P . 4,> = = =j

= = =

ik3+3,> k3+4,> = = = > j @ ik+P 4,> k+5P 4,> = = =j=

H* rssruwxqr rvvhuyduh fkh o*ruglqdphqwr ghl fdpslrql ghood jhqhulfd vhtxhqd kb+n, ghyh hvvhuh

ulwdugdwr gl fdpslrql shu hvvhuh uldoolqhdwr h irupduh od vhtxhqd k+q, ruljlqdoh +txhvwd ë od udjlrqh

gho ulwdugr 3b qhood +4144,,1 Qhood jxud 4156 vl looxvwud lo fdvr P @ 61


0 1 2 3 4 8765

0 (1) 3 (4) 6 (7)

0 (2) 6 (8)3 (5)

n

h(n) ↔ H(z)

M=3

h(n) ↔ Ho(zM)

e-jω·0

e-jω·1

e-jω·2

n

n

n

h1(n) ↔ H1(zM)

h2(n) ↔ H2(zM)

0 3 6

Iljxud 4156= Hvhpslr gl vhtxhqd gl < fdpslrql ik+q,jH?'f h ghooh vhtxhqh srolidvh1 Vl qrwl fkh

od qxphudlrqh ghl fdpslrql ghooh vlqjroh vhtxhqh srolidvh h* surjuhvvlyd phqwuh shu ulrwwhqhuh od

qxphudlrqh ruljlqdoh +ulsruwdwd lq sduhqwhvl, h* qhfhvvdulr ulwdugduh od vhtxhqd kb+q, gl fdpslrql1

Wudvirupdwd h wudvirupdwd gl Irxulhu ghood vhtxhqd srolidvh Kb+ ,= shu xqd jhqhulfd vhtxhqd

ghflpdwd vl h* ulfdydwr=

k(+q, ' K(+, @4

P

3[&'f

K+* Z3&

,

vh rud vl lqwurgxfh xq ulwdugr qhoo*ruljlqh gho shwwlqh gl fdpslrqdphqwr vl kd

ikb+n,j #$4

P

63[&'f

K+ Z3&

, Z3&b

@

4

P

63[&'f

K+ Z3&

, hs+m

5

Pn,

Lo ulwdugr gl fdpslrql lqwurgxfh xqr vidvdphqwr h3/bm/' 2Z&

ryyhur od n0hvlpd uhsolfd vshwwudoh

+do ydulduh gl n, vxelvfh xqr vidvdphqwr sdul d 5n@P 1 H* srvvleloh +h frqyhqlhqwh, dyhuh xqd

udssuhvhqwdlrqh judfd ghoo*dqgdphqwr ghood wudvirupdwd gl Irxulhu +prgxor h vidvdphqwr lqgrwwr

gdo ulwdugr gho shwwlqh gl fdpslrqdphqwr,81 Lq jxud 4157 vrqr lqglfdwh vld oh uhsolfkh vshwwudol

8Shu xqd vhtxhqd ghflpdwd frq xq shwwlqh gl fdpslrqdphqwr ulwdugdwr vl kd= %cbE? ' %E?nbuE?/ holplqdqgr


0M

π21⋅

M

π22 ⋅

M

π23 ⋅ 2π ω

...

WM WM WM WM

Hλ(zM)

k=0 k=1 k=2 k=3

-0xλ -1xλ -2xλ -3xλ

sfasamento

=

−− Mk

jkM eW

λπλ

2

0M

λπ ⋅− 2M

λπ ⋅− 4

M

λπ⋅− 6

M

λπ ⋅− 2M

λπ⋅− 4...

M

λπ ⋅− 6

...

WM

-MxλWM

-0xλ=

versori

Iljxud 4157= Wudvirupdwd gl Irxulhu gl xqd vhtxhqd srolidvh kb+q, h vidvdphqwl lqgrwwl gdo ulwdugr gl

fdpslrql gho shwwlqh gl fdpslrqdphqwr1

d sdvvr 5@P ghood wudvirupdwd gl Irxulhu ghood vhtxhqd ghflpdwd +lq txhvwr fdvr vl h* dvvxqwr

fkh lo owur vld xq sdvvd0edvvr frq edqgd E/ * @P +yhgl vrsud,, fkh jol vidvdphqwl lqgrwwl gdo

shwwlqh gl fdpslrqdphqwr +jol vidvdphqwl vrqr lqglfdwl frq ghl yhuvrul,1 Lq dowhuqdwlyd h* srvvleloh

ruglqduh jol vidvdphqwl lq xqd wdehood do ydulduh gl h ghood srvllrqh vshwwudoh 5n@P lqglylgxdwd

gdoo*lqglfh n +jxud 4158,1 Ohjjhqgr od wdehood shu uljkh vl wuryd or vidvdphqwr ghooh uhsolfkh shu od

jhqhulfd vhtxhqd srolidvh1 Od n0hvlpd frorqqd fruulvsrqgh dg xqr vidvdphqwr surjuhvvlyr +do ydulduh

gl , lq fxl lo yhuvruh frpslh xq qxphur lqwhur gl jlul +lq sduwlfroduh frpslh n jlul, qhoo*lqwhuydoor

@ 3> 4> ===>P 4> vrppdqgr l yhuvrul shu frorqqh lo ulvxowdwr wudqqh fkh shu n@3 +lq wxwwl jol dowul

fdvl l yhuvrul frpslrqr n jlul lq P fdpslrql/ od orur vrppd h* qxood,1 Od vrppd ghl yhuvrul fruulvsrqgh

dood vrppd gl wudvirupdwh gl Irxulhu lqidwwl=

3[b'f

K+$

P

5n

P, hs+m

5

Pn, @ K+

$

P

5n

P,3[b'f

hs+m5

Pn, @

P K+ /

, n @ 3

3 n 9@ 3

txhvwr ulvxowdwr h* htxlydohqwh dood udssuhvhqwdlrqh srolidvh ryyhur K+, @S3

b'f3b Kb+]

,1

l fdpslrql qxool vl rwwlhqh +bE? ' %cbE u ?1 Oh wudvirupdwh vrqr=

fcbE5 '[

?

%cbE?53? '[

?

%E? u E?3 b53? '[

?

%E?

#

3[&'f

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[?

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53?

~ fE5`

n

P

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3[&'f

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fE5`3&

+bE? ' %cbE u ? < tbE5 ' fcbE5* '

3[&'f

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fE5* `

3&

Od wudvirupdwd gl Irxulhu=

tbE/ '

3[&'f

i TE32Z

&b ufE

/

3

2Z&

4181 GHFLPD]LRQH H LQWHUSROD]LRQH 6:

0 1 2 M-1

λ

M-1

0

k

λ

...

...M

π2−1

2

:

:

22 ⋅−M

π

λπM

2−

)1(2 −− MM

π)1(

22−⋅−

MM

π

λπM

22 ⋅−

222 ⋅⋅−

M

π

M

M π2)1( ⋅−−M

π22 ⋅−

π2)1(2

M

M −−

πλ2

)1(

M

M −−

π2)1)(1(

M

MM −−−

0 0 0 0

0

0

0

0

...

...

...

Iljxud 4158= Wdehood ghjol vidvdphqwl qhood wudvirupdwd gl Irxulhu gl vhtxhqh srolidvh do ydulduh gho

ulwdugr gho shwwlqh gl fdpslrqdphqwr h ghood ulshwllrqh vshwwudoh n1

Lq sduwlfroduh vl rvvhuyd fkh vfhjolhqgr xq owur sdvvd0edvvr frq edqgd @P rjql vhtxhqd srolidvh

h* txdvl xq sdvvd0wxwwr ryyhur od ixqlrqh gl wudvihulphqwr h* rwwhqxwd d!dqfdqgr P uhsolfkh ghood

ixqlrqh gl wudvihulphqwr ruljlqdoh +@qrq ghflpdwd,/ rjql uhsolfd kd xqr vidvdphqwr rssruwxqr fkh

dqqxood oh uhsolfkh do gl ixrul ghood edqgd lq n@3 txdqgr wxwwh oh vhtxhqh srolidvh yhqjrqr vrppdwh1

Vh vl vfhjolhvvh gl vidvduh rssruwxqdphqwh oh glhuhqwl uhsolfkh vshwwudol lqgrwwh gdood ghflpdlrqh

vl srwuheehur vrppduh oh uhsolfkh lq prgr wdoh gd dyhuh xq owur sdvvd0edvvr prgxodwr +@owur

sdvvd0edqgd,/ or vidvdphqwr vl ulfdyd gdood wdehood 4158 +yhgl vhlrqh= Edqfkl gl owul sdvvd0edqgd,1

$! % & # #

D sduwluh gdood ghfrpsrvllrqh ghood ulvsrvwd doo*lpsxovr gl xq owur ILU lq vrwwrvhtxhqh ghflpdwh

+udssuhvhqwdlrqh srolidvh,= K+, @S3

b'f 3b Kb+] , h* srvvleloh lqglylgxduh xqd uhdoldlrqh

h!flhqwh +@ d edvvr frvwr frpsxwdlrqdoh, gho owur K+, txdqgr ë xvdwr frph owur dqwl0doldvlqj

qhood ghflpdlrqh1Vh vl dvvxph fkh od ulvsrvwd doo*lpsxovr gho owur deeld oxqjkhd Q @ O P / doorud

vl kd=

K+, @3[?'f

k+q,3? @3[b'f

u3[&'f

k+nP., 3E&nb @3[b'f

3bu3[&'f

k+nP . , 3&

~ MbE5

@3[b'f

3bKb+ ,

od vwuxwwxud h* txhood lq jxud 4159 +vlqlvwud,1 Rjql owur srolidvh kd ulwdugl 3 h txlqgl h* srvvleloh

wudvsruuh owur h ghflpdwruh +ulfrugdqgr gl wudvirupduh l ulwdugl, frph lqglfdwr lq jxud 41551 Lo frvwr

frpsxwdlrqdoh shu oh gxh vwuxwwxuh +qhoo*ruglqh, ë=

U@ @ iS P

@ iSQ UK @sSP

@ sS

=


fc

L=N/MM

z-1

z-1

z-1

Ho(zM)

H1(zM)

Hλ(zM)

HM-1(zM)

fc

z-1

z-1

z-1

L=N/M

Ho(z)

H1(z)

Hλ(z)

HM-1(z)

Mfc/M

Iljxud 4159= Vwuxwwxud srolidvh gl xq owur K+, frq ghflpdlrqh 4=P h vwuxwwxud d plqlpr frvwr

+ghflpdlrqh 4=P sulpd gho owur,1

O*lqvlhph ghjol P ulwdugl h ghl ghflpdwrul 4=P udssuhvhqwd xq frqyhuwlwruh vhulh2sdudoohor ryyhur

sulpd vl fdulfdqr wxwwh oh fhooh gl phpruld fruulvsrqghqwl dl ulwdugl hohphqwdul +3,/ srl txhvwhyhqjrqr fdpslrqdwh d lqwhuydoor whpsrudoh gl P fdpslrql1 Lq sudwlfd vl uhdold frq xqd phpruld

+exhu, frq fdulfdphqwr vhuldoh h vfdulfdphqwr sdudoohor> lq dowhuqdwlyd vl sxr* udssuhvhqwduh frq xq

frppxwdwruh d vsdrod lq jxud 415: +dwwhqlrqh dood vhtxhqldolwd* frq fxl yhqjrqr suhvhqwdwl l

fdpslrql lq xvflwd$,1

Lq frqfoxvlrqh od ghfrpsrvllrqh lq owul srolidvh frqgxfh d vwuxwwxuh qxphulfkh lq judgr gl

rwwlplduh lo frvwr frpsxwdlrqdoh txdqgr vl kd xq fdpelr ghood iuhtxhqd gl fdpslrqdphqwr1

Shu o*lqwhusrodlrqh +fdvfdwd gl lqwhusrodwruh h owur, h* srvvleloh xvduh xqd udssuhvhqwdlrqh wudvsrv0

wd gho owur qxphulfr1 Lq txhvwr fdvr vl srvvrqr ulshwhuh oh phghvlph frqvlghudlrql vyrowh shu od ghf0

lpdlrqh +ryyhur lo owur sxr* hvvhuh ghfrpsrvwr lq vhtxhqh srolidvh h srl vl viuxwwdqr oh sursulhwd*

gxdol 0 shu o*lqwhusrodwruh 0 ulvshwwr d txdqwr ylvwr lq jxud 4155,1 Lq jxud 415; vl ulsruwd od vwuxwwxud

dqdorjd +gxdoh, ghood jxud 41591 Dqfkh lq txhvwr fdvr vl kd xqd ulgxlrqh gho frvwr frpsxwdlrqdoh

lq txdqwr jol O owul srolidvh vrqr gl oxqjkhd Q2O h rshudqr dood iuhtxhqd ir1

Rvvhuydlrqh= Qhooh vwuxwwxuh srolidvh od ulvsrvwd doo*lpsxovr gl xq owur ILU ylhqh ghfrpsrvwd qhooh

vrwwrvhtxhqh ghflpdwh gl oxqjkhd ulgrwwd1 H* exrqd qrupd vfhjolhuh od oxqjkhd Q gho owur lq pr0

gr wdoh fkh rjql vhtxhqd ghflpdwd vld gl xjxdoh oxqjkhd/ lq dowuh sduroh Q h* xq pxowlsor gho idwwruh

gl vrwwrfdpslrqdphqwr +qhood ghflpdlrqh, r gho idwwruh gl vryudfdpslrqdphqwr +qhoo*lqwhusrodlrqh,1

Lq txhvwr fdvr rjql owur srolidvh Kb+, kd od vwhvvd oxqjkhd1

$$ ' ( ##

Vld gdwr xq owur sdvvd edvvr K+, frq edqgd E @ @P / lo owur sdvvd edqgd dood iuhtxhqd 2Z6

frq p @ 3> 4> = = = >P 4/ sxõ hvvhuh rwwhqxwr prgxodqgr +prowlsolfdqgr l frh!fhqwl gho owur shu xqd

4181 GHFLPD]LRQH H LQWHUSROD]LRQH 6<

M

x(n)

x(n-1)

x(n-2)

z-1

z-1

x(n-M+1)

x(n), x(n-M), x(n-2M), ...

x(n-1), x(n-M-1),...

x(n-2), x(n-M-2),...

x(n-M-1), x(n-2M+1),...

x(n), x(n-M), x(n-2M), ...

x(n-1), x(n-M-1),...

x(n-M-1), x(n-2M+1),...

x(n)

fs

fs/M

fs/M....

Iljxud 415:= Frqyhuwlwruh vhulh2sdudoohor= frppxwdwruh d vsdrod

vlqxvrlgh frpsohvvd, lo owur sdvvd edvvr K+,=

KE6+, @ K+ h32Z6

,>

qrwd fkh KEf+, @ K+,1 Shu rwwhqhuh xq edqfr gl owul sdvvd edqgd d sduwluh gdo surwrwlsr sdvvd

edvvr vduã qhfhvvdulr prgxoduh lo owur K+,/ o*xvflwd gdoo*p0hvlpr owur +r shu od p0hvlpd edqgd, h*

|E6+q, +yhgl jxud 415<,1 Lo frvwr gl txhvwd vwuxwwxud h* gl U@ @ Q P iS1

H* srvvleloh vfhjolhuh lo owur sdvvd edvvr lq prgr wdoh fkh prgxodwr d sdvvr 5@P ulhvfd d fr0

suluh wxwwd od edqgd frph shu xq vlvwhpd sdvvd0wxwwr +vl shqvl dg xq owur lghdoh frq edqgd @P,/

ryyldphqwh h* qhfhvvdulr vfhjolhuh dffxudwdphqwh lo owur K+,1 Od edqgd frpsohvvlyd rffxsdwd gdoodvhtxhqd |E6+q, h* 5@P / vduheeh suhihuleloh ghflpduh od vhtxhqd |E6+q, d 4=P lq prgr wdoh gd

qrq vsuhfduh fdpslrql lqxwlol +lo fdpslrqdphqwr d iuhtxhqd vxshulruh h* lqxwloh txdqgr vl glvsrqh

gl xq vhjqdoh d edqgd olplwdwd,/ oh vhtxhqh ghflpdwh |E6+qP, vl glfrqr pdvvlpdphqwh ghflpdwh

+ryyhur od pdvvlpd ghflpdlrqh dpplvvleloh gdood edqgd gho vhjqdoh,1 Lq txhvwr fdvr vl kd xqd

ulgxlrqh gho frvwr lq txdqwr rjql owur sxr* hvvhuh uhdoldwr frq xqd vwuxwwxud srolidvh/ lo frvwr

glyhqwd UK @ Q P iS@P @ Q iS1

Shu xqd uhdoldlrqh h!flhqwh gho edqfr gl owul ë srvvleloh ulfruuhuh dood vwuxwwxud srolidvh +shu

vvduh oh lghh qho vhjxlwr vl sxõ shqvduh dg xq owur sdvvd edvvr d phgld preloh= K+, @ 4 . 3 .32 . = = = . 3n @

S3?'f 3?/ vl qrwl fkh wdoh owur qrq kd xqd edqgd olplwdwd d @P,1 Gdood

udssuhvhqwdlrqh srolidvh gho owur vl kd=

K+, @3[b'f

3bKb+ ,>

shu lo vlqjror owur sdvvd edqgd=

KE6+, @ K+ h32Z6* , @3[b'f

+h32Z6* ,3b Kb+ h32Z6* ,

@3[b'f

hn2Z6b

3b Kb+ , @

3[b'f

Z6b 3b Kb+

,


Lfc

L

z-1

z-1

z-1

N/L

Ho(zL)

H1(zL)

Hλ(zL)

HL-1(zL)

......

fcLfc

L

z-1

z-1

z-1

N/L

Ho(z)

H1(z)

Hλ(z)

HL-1(z)

......

fc

Iljxud 415;= Vwuxwwxud srolidvh shu o*lqwhusrodlrqh +vlqlvwud, h vwuxwwxud d plqlpr frvwr

frpsxwdlrqdoh +ghvwud,1

Lo owur shu od p0hvlpd edqgd ë rwwhqxwr frpelqdqgr owul ghflpdwl Kb+ , frq vidvdphqwl rs0

sruwxql fkh whqgrqr d frpshqvduh jol vidvdphqwr lqwurgrwwl gdood ghflpdlrqh frq xq shwwlqh gl

fdpslrqdphqwr ulwdugdwr +yhgl jxud 4158,1 Yrohqgr ë srvvleloh ulvfulyhuh/ shu o*lqvlhph gl owul=

5999997

KEf+,

KE+,

KE2+,111

KE3+,

6:::::8 @

59999997

Z f Z f

= = = Z f

Z f Z

= = = Z3

Z f Z 2

Z2E3

1111 1 1

Z f Z3

ZE32

6::::::8

~ LGIW

5999997

KJ+ ,

3K+ ,

32K2+ ,

111

3nK3+ ,

6:::::8

gryh od pdwulfh +lqglfdwd vshvvr frph pdwulfh gl prgxodlrqh, udssuhvhqwd od LGIW ghooh vhtxhqh

owudwh gd xq edqfr gl owul srolidvh gho owur K+,1 H* rssruwxqr dyhuh xqd udssuhvhqwdlrqh d

eorffkl frph txhood lq jxud 4163 +vl qrwl fkh od LGIW ylhqh ulfdofrodwd vhtxhqldophqwh dg rjql qxryr

fdpslrqh ghood vhtxhqd +q,,1 Lo frvwr frpsxwdlrqdoh h*=

US @ iS ^Q

PP ~

edqfr gl owul

.R+LGIW ,` @

iS ^Q .P2` xvdqgr od GIW

iS ^Q . 5P orj2P ` xvdqgr od LIIW^rs2vhf`

+prowlsolfdlrql uhdol,1

Vh vfhjolhvvl K+, @S3

'f 3? doorud Kb+ , @ 4 +owul gd 4 vror fdpslrqh gl ydoruh xqlwdulr,/

lq txhvwr fdvr oh xvflwh |E6+q, udssuhvhqwdqr l fdpslrql ghood LGIW1Vh frqvlghudvvl oh xvflwh ghl owul sdvvd edqgd pdvvlpdphqwh ghflpdwh doorud vl dyuheeh or vfkhpd

htxlydohqwh lq jxud 4164 +lo ghflpdwruh qrq h* lq xhqdwr gdood pdwulfh gl prgxodlrqh lq txdqwr


...

0-B B 2π ω

|H(ω)|

0-B B

|H(0) (ω)| |H(1) (ω)| |H(2) (ω)| |H(M-1) (ω)| |H(0) (ω)|

M

π2M

π22 ⋅M

M π2)1( ⋅− 2π

H(0) (z)

H(1) (z)

H(M-1) (z)

x(n)

y(0) (n)

y(1) (n)

y(M-1) (n)

….

Iljxud 415<= Edqfr gl owul sdvvd edqgd

txhvwd qrq frqwlhqh ulwdugl h txlqgl rjql owur sxr* hvvhuh uhdoldwr uhwurfhghqgr od srvllrqh gho

ghflpdwruh P = 4,1 Lq txhvwr fdvr lo frvwr frpsxwdlrqdoh glyhqwd=

U_ @iS

P ^

Q

PP ~

edqfr gl owul

.R+LGIW ,` @

iS ^Q .P2`@P xvdqgr od GIW

iS ^Q . 5P orj2P `@P xvdqgr od LIIW

Shu vvduh oh lghh h* rssruwxqr frqiurqwduh lo frvwr frpsxwdlrqdoh gl xq edqfr gl owul pdvvl0

pdphqwh ghflpdwr +ghflpdlrqh 4 = P ë xjxdoh dood odujkhd gl edqgd gl rjql owur, h xq lpsoh0

phqwdlrqh srolidvh frq lo frvwr frpsxwdlrqdoh gl xq edqfr gl P owul gd Q fdpslrql +jxud 415<,=

U@ @ Q P iS1 Yrohqgr vl shqvl dg xq edqfr gl P @ 65 @ 5D owul gl oxqjkhd Q @ 97 @ 5S

+rjql owur srolidvh h* gl 5 fdpslrql, h iS @ 43 nK=

frvwr ^rshudlrql2vhf` iS @ 43 nK

U@ @ iS Q P @ iS 5 @ 537; iS 53 P rsv

UK @ iS Q @ iS 5S @ 97 iS 973 n rsv

US @ iS+Q . 5P orj2P, @ iS+5S . 5S 8, @ 693 iS 5=9 P rsv

U_ @ iS+5 . 5 orj2 5S, @ 47 iS 473 n rsv

Od frvwuxlrqh gl xq edqfr gl owul sxr* hvvhuh prwlydwd gdood qhfhvvlwd* gl vxgglylghuh xq vhjqdoh

lq edqgh gl iuhtxhqd h gl hoderuduh vhsdudwdphqwh rjql edqgd1 Gl vrolwr h* ulfklhvwr gl ulfrpelqduh

lo vhjqdoh lq prgr gd rwwhqhuh xq vhjqdoh rssruwxqdphqwh hoderudwr1 Vl prvwud txl gl vhjxlwr frph

ulfrpelqduh l glyhuvl frqwulexwl phgldqwh xqd vwuxwwxud gxdoh ghqrplqdwd edqfr gl vlqwhvl +od vwuxwwxud

fkh ghfrpsrqh lo vhjqdoh lq edqgh h* fkldpdwd edqfr gl dqdolvl, lq jxud 41651 D sduwluh gd xq owur

J+, vl rwwlhqh xq owur sdvvd edqgd shu prgxodlrqh= JE&+, @ J+Z3& ,/ shu n @ 3> 4> ===>P 41

O*xvflwd shu ulfrpelqdlrqh ghjol lqjuhvvl h*=

V+, @3[&'f

JE&+,\ E&+, @3[&'f

J+Z3& ,\ E&+, @

3[&'f

3[b'f

+Z3& ,3bJb+

Z3& ,\ E&+,

@3[b'f

3bJb+ ,

%3[&'f

Z&b \ E&+,

&


x(n)

z-1

z-1

z-1

Ho(zM)

H1(zM)

H2(zM)

HM-1(zM)

N/MM

IDFT

da

M c

ampi

oni

y(0)(n) ↔ H(0) (z) X(z)

y(1) (n) ↔ H(1) (z) X(z)

y(2) (n) ↔ H(2) (z) X(z)

(banda 0)

(banda 1)

(banda 2)

(banda M-1)

y(M-1) (n) ↔ H(M-1) (z) X(z)

….

….

Iljxud 4163= Edqfr gl owul srolidvh

ryyhur kd xqd vwuxwwxud vlploh +gxdoh, do edqfr gl dqdolvl lq jxud 41641

$) ' ( # #

O*hoderudlrqh ghl vhjqdol phgldqwh edqfkl gl owul sxr* hvvhuh vyrowd vxgglylghqgr xq vhjqdoh lq edqgh

vshwwudol d!dqfdwh +edqfr gl dqdolvl, r ulfrpelqdqgr oh edqgh vhsdudwh lq xq xqlfr vhjqdoh +edqfr

gl vlqwhvl,/ lq rjql fdvr l vhjqdol vrqr vhpsuh pdvvlpdphqwh ghflpdwl1 Lq jxud 4166 lo edqfr gl

dqdolvl suhfhgh lo edqfr gl vlqwhvl/ relhwwlyr h* txhoor gl vxgglylghuh lo vhjqdoh +q, qhooh vrwwredqgh[Ef+,>[E+,> ===>[E3+, h gl ulfrpelqduh lo vhjqdoh lq prgr gd ulrwwhqhuh a[+, * [+,1 Hv0

lvwrqr ghl fulwhul shu od surjhwwdlrqh ghl owul gl dqdolvl KE6+, h gl vlqwhvl JE6+, lq prgr gd

uldyhuh od ulfrvwuxlrqh shuihwwd +ryyhur a[+, @ [+,,1 Txdorud lo edqfr gl vlqwhvl suhfhgd txhoor gl

dqdolvl/ frph lq jxud 4167/ h* srvvleloh frpelqduh slx* vhjqdol [Ef+,>[E+,> ===>[E3+, owudwlfrq xq edqfr gl owul JE6+, h rwwhqhuh +vlqwhwlduh, xq xqlfr vhjqdoh fkh sxr* srl hvvhuh vhsdudwr

qhooh vxh frpsrqhqwl \ Ef+,> \ E+,> ===> \ E3+,/ ryyldphqwh vl ulfklhgh fkh \ E6+, * [E6+,1Vh ulvxowd KE6+,JE6+, @ 6

3g h KE6+,JE?+, @ 3 shu p 9@ q doorud qrq vl kd frqwulexwr gd

dowul vhjqdol vh qrq gdo vhjqdoh gl lqwhuhvvh +dvvhqd gl lqwhuihuhqd gd fdqdol dgldfhqwl r glvwruvlrqh gd

furvvwdon, h od ulfrvwuxlrqh gho vhjqdoh ulvxowd shuihwwd ryyhur ulvxowd \ E6+, @ [E6+, d phqr gl

xq ulwdugr gl N fdpslrql h xq idwwruh gl vfdod 61 Vh l owul KE6+, h JE6+, vrqr ghl sdvvd edqgd

doorud lo edqfr gl vlqwhvl suryyhgh d fuhduh xq vhjqdoh d pxowlsodlrqh gl iuhtxhqd +IGP, phqwuh

edqfr gl dqdolvl suryyhgh dood ghpxowlsodlrqh1

Rvvhuydlrqh ^shu xqd glvfxvvlrqh slx* dssurirqglwd vl ulpdqgd dg dowul fruvl` =

Vh [Ef+,>[E+,> ===>[E3+, irvvhur l vhjqdol jhqhudwl gd P vrujhqwl r P xwhqwl +vl shqvl

dg xq vlvwhpd gl wudvplvvlrqh, od vwuxwwxud WPX[ lq jxud 4167 udssuhvhqwhuheeh xq vlvwhpd gl

dffhvvr d glylvlrqh gl iuhtxhqd +IGPD, d P xwhqwl +lq txhvwr fdvr l owul vrqr ghl sdvvd edqgd, r

d glylvlrqh gl frglfh +FGPD, +lq txhvwr fdvr l owur vrqr ghl sdvvd0wxwwr ruwrjrqdol wud gl orur lq

prgr gd plqlplduh od glvwruvlrqh gd furvvwdon,1 H* dqfkh srvvleloh sduwluh gd xq vhjqdoh pxowlsor d


M

x(n)

z-1

z-1

z-1

Ho(z)

H1(z)

H2(z)

HM-1(z)

N/M

IDFT

da

M c

ampi

oni

y (0) (nM)

y (1) (nM)

y (2) (nM)

y (M-1) (nM)

MM

M

M

M

fcfc/M

….

….

Iljxud 4164= Edqfr gl owul srolidvh pdvvlpdphqwh ghflpdwl1

glylvlrqh gl whpsr +WGP,/ ghfrpsruor qhooh vxh frpsrqhqwl lqglshqghqwl phgldqwh xqd frqyhuvlrqh

vhulh2sdudoohor/ wudvirupduor lq xq vhjqdoh d glylvlrqh gl iuhtxhqd phgldqwh xq edqfr gl vlqwhvl h

ulfrpsruor srl d ydooh lq xq vhjqdoh dqfrud WGP +frqyhuvlrqh sdudoohor2vhulh,/ od vwuxwwxud h* lq

jxud 41681 Vh l owul JE6+, vrqr ghl sdvvd edqgd rwwhqxwl gd xq owur surwrwlsr +s1h1/ J+, @S3?'f 3?, doorud lo edqfr gl vlqwhvl sxr* hvvhuh rwwhqxwr phgldqwh od LGIW frq od vwuxwwxud srolidvh

ylvwd suhfhghqwhphqwh/ lq prgr dqdorjr vl sxr* rwwhqhuh lo edqfr gl dqdolvl> txhvwl frqfhwwl vrqr dood

edvh ghood wudvplvvlrqh gl xq vhjqdoh vx slx* sruwdqwl ruwrjrqdol +RIGP,1

Shu xqd glvfxvvlrqh vxl edqfkl frpsrvwl gd gxh owul h vxood ulfrvwuxlrqh ghl vhjqdol vl ulpdqgd do

oleur gl whvwr1


M

G(0)(z)

G(1)(z)

G(M-1)(z)

s(n)

y(0) (nM)

y(1) (nM)

y(M-1) (nM)

….

M

z-1

z-1

z-1

N/M

Go(z)

G1(z)

Gλ(z)

GM-1(z)

......

IDFT

da

M c

ampi

oni

y(0) (nM)

y(1) (nM)

y(M-1) (nM)

y(λ) (nM)

......

s(n)

Iljxud 4165= Edqfr gl owul gl vlqwhvl1

H(0) (z)

H(1) (z)

H(M-1) (z)

….

M M

G(0)(z)

G(1)(z)

G(M-1)(z)

x(0) (n)

x(1) (n)

x(M-1) (n)

….

banco di sintesibanco di analisi

X(z) X(z)^

Iljxud 4166= Edqfr gl owul shu frglfd d vrwwredqgh +VEF,1


M

G(0)(z)

G(1)(z)

G(M-1)(z)

x(0) (n)

x(1) (n)

x(M-1) (n)

….

H(0) (z)

H(1) (z)

H(M-1) (z)

y(0) (n)

y(1) (n)

y(M-1) (n)

….

M

banco di analisibanco di sintesi

Iljxud 4167= Edqfr gl owul shu wudqvpxowlsohlqj +WPX[,1

banco di sintesiM

G(0)(z)

G(1)(z)

G(M-1)(z)

x(0) (n)

x(1) (n)

x(M-1) (n)

….

z-1

z-1

z-1

TDMH(0) (z)

H(1) (z)

H(M-1) (z)

y(0) (n)

y(1) (n)

y(M-1) (n)

….

M

banco di analisi

z-1

z-1

z-1

TDM

FDM

Iljxud 4168= Frqyhuvlrqh gl xq vhjqdoh WGP2IGP2WGP1


) #

) ##

Xqr vwlpdwruh ë lq judgr gl vwlpduh dofxql ghl sdudphwul gho prghoor gl jhqhudlrqh gho surfhvvr xqd

yrowd qrwd xqd uhdoldlrqh1 Shuwdqwr or vwlpdwruh ë xqd ixqlrqh pdwhpdwlfd gl xq fhuwr qxphur

+Q, gl ydorul dvvxqwl gd xqd yduldeloh fdvxdoh +YF,1 H* hylghqwh txlqgl fkh or vwlpdwruh ë hvvr vwhvvr

xqd YF1 Vl yxroh/ dg hvhpslr/ vwlpduh lo sdudphwur vfdoduh dyhqgr d glvsrvllrqh Q ydorul ghooh YF

@ i> ===> j ryyhur or vwlpdwruh ë udssuhvhqwdwr gdood ixqlrqh @ j+ , ixqlrqh txlqgl

ghjol Q ydorul dvvxqwl gdooh Q YF1 Od vfhowd ghood ixqlrqh j+1, ghyh hvvhuh wdoh gd surgxuuh xq

ydoruh lo slü ylflqr srvvleloh do ydoruh yhur gd vwlpduh/ ryyhur 1 Qrwd fkh or vwlpdwruh sxõ hvvhuh

udssuhvhqwdwr gd xqd ghqvlwã gl suredelolwã sw + , ixqlrqh ghooh Q YF h ghood ixqlrqh j+1,1 Fl

vrqr 6 sdudphwul fkh fdudwwhuldqr od erqwã gl xqr vwlpdwruh +lqwhvr frph od vfhowd ghood ixqlrqh

j+1, ,=

Sroduldlrqh Xqr vwlpdwruh ë qrq0sroduldwr vh Hij @ shu rjql ydoruh gl Q ryyhur uds0

suhvhqwd od fdsdflwã gl dssurvvlpduh lo ydoruh d suhvflqghuh gdo qxphur gl plvxuh +udssuhvhqwd od

jdudqld fkh or vwlpdwruh qrq ë dhwwr gd huurul vlvwhpdwlfl,1 Vh od frqgllrqh gl qrq0sroduldlrqh

ë yhulfdwd vror shu Q judqgh +frqgllrqh txlqgl phqr uhvwulwwlyd ghood suhfhghqwh, doorud ë xqr

vwlpdwruh gl dvlqwrwlfdphqwh qrq sroduldwr= olp<"Hij @ 1

Yduldqd H* lo sdudphwur fkh ghvfulyh od yduldelolwã ghoor vwlpdwruh txdqgr fdpeldqr jol Q ydorul

ghooh YF1 Vl kd txlqgl

yduij @ 2w

@ Hi+ Hij,2j

lo fxl ydoruh/ vrolwdphqwh/ ghfuhvfh doo*dxphqwduh ghooh plvxuh Q1 Xqr vwlpdwruh ë frqvlvwhqwh vh

sroduldlrqh h yduldqd whqgrqr d hur do fuhvfhuh gho qxphur ghooh rvvhuydlrql1

H!flhqd Od fdudwwhulvwlfd gl frqyhujhqd ghood yduldqd gl xqr vwlpdwruh shu Q whqghqwh dg

lqqlwr1 Xqr vwlpdwruh qrq0sroduldwr ë slü h!flhqwh gl xq dowur vwlpdwruh qrq0sroduldwr

& @ k+ , vh ulvxowd=

2w

? 2i>;Q

Qrwd fkh shu ydoxwduh o*h!flhqd gl xqr vwlpdwruh qrq0sroduldwr vl xvd jhqhudophqwh ulihuluvl dood

yduldqd plqlpd udjjlxqjleloh gd xqr vwlpdwruh qrq0sroduldwr1 Txhvwd yduldqd plqlpd ë lqglfdwd

frph olplwh lqihulruh gl Fudpìu0Udr +2-

,> od glvxjxdjoldqd gl Fudpìu0Udr ulvxowd 2w

2-

1 Lq dowuh sduroh/ txdoxqtxh vwlpdwruh qrq0sroduldwr kd xqd yduldqd fkh ë vhpsuh vxshulruh

do ydoruh ghqlwr gd 2-

h shu txhvwr wdoh olplwh/ qrq vhpsuh udjjlxqjleloh lq sudwlfd/ frvwlwxlvfh

xqr ghl sdudphwul shu od ydoxwdlrqh ghood erqwã ghoor vwlpdwruh +shu od ghqllrqh gl 2-

vl yhgd lo

vxffhvvlyr sdudjudir,1

) # "

Od ghqllrqh ghoo*dxwrfruuhodlrqh qho grplqlr gho whpsr frqwlqxr shu vhjqdol frpsohvvl ë=

u+, @ H^W+w,+w. ,` @ H^+w,

W+w ,`> u+, @ u

W+,

Shu ghqllrqh/ or vshwwur gl srwhqd V+$, ë=

V+$, @

]"

3"

u+,h3/

g

4191 VWLPD VSHWWUDOH 7:

Lqidwwl/ frqvlghudqgr xq lqwhuydoor gl whpsr gl rvvhuydlrqh sdul d W|J| =

[+$, @

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# 1&&402

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od vroxlrqh lqglfdwd frph +e, vrwwudh l elw phqr vljqlfdwlyl +ryyhur lo uxpruh gl txdqwldlrqh, frq

xq ulwdugr xqlwdulr +lo uxpruh gl txdqwldlrqh ylhqh lqwhjudwr qho owur,1 Lo frvwr ghood vroxlrqh +e,

ë udssuhvhqwdwr gd vrppdwrul lq dulwphwlfd d suhflvlrqh 5E elw shu vrppduh l elw phqr vljqlfdwlyl

+l elw phqr vljqlfdwlyl yhqjrqr idwwl ulflufroduh qho owur,1 Xqd vroxlrqh fkh xvd prowlsolfdwrul d

suhflvlrqh 5E elw vduheeh vydqwdjjlrvd lq txdqwr o*duhd gl vlolflr shu prowlsolfdwrul gl oxqjkhd 5E

ulvxowd txdguxsod ulvshwwr dl prowlsolfdwrul shu oxqjkhd E ^vh vl lqglfd frq D o*duhd gl xq vrppdwruh

elqdulr/ od prowlsolfdlrqh gl q elw lq sdudoohor ulfklhgh xq duhd flufd q2D> od yhulfd ë vhpsolfh/ edvwd

vfulyhuh lo surgrwwr wud gxh qxphul elqdul frph vrpph ulshwxwh gl surgrwwl d 4 elw +ixqlrqh orjlfd

DQG,`1

!8 ## 18&-&402

Vl frqvlghul xqd vhtxhqd iz?j gl fdpslrql lqglshqghqwl h jdxvvldql d yduldqd 2 @ 4> od iuhtxhqd

gl fdpslrqdphqwr ë iS @ 4333K= Vl ghvlghud owuduh sdvvd0edqgd od vhtxhqd frq xq owur LLU d

iuhtxhqd fhqwudoh 458 K h edqgd sdvvdqwh 8 K1 Shuwdqwr=

d, +sxqwl 6, Vl surjhwwl lo owur LLU d gxh srol=

K+, @D

4 . dc23 . d2c2

32

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prgr fkh od srwhqd gho vhjqdoh owudwr ? @ z? k? vld xqlwduld1

f, +sxqwl 6, Vl ghwhuplql od yduldlrqh srvvleloh gho jxdgdjqr lq fhqwur edqgd frqvhjxhqwh dg xqd

txdqwldlrqh ghl frh!flhqwl frq 43 elw1

g, +sxqwl 6, Vl ghwhuplql lo olyhoor gho uxpruh gl txdqwldlrqh lq xvflwd/ txdqwldqgr l vhjqdol

frq ; elw1

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c2 @ h


% @ 4 @8

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2 $ D @4s65=9

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fhqwur edqgd ë +phwrgr yhwwruldoh,=

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5% vlq!

ë srvvleloh dssurvvlpduh od srwhqd lq xvflwd frq lo jxdgdjqr gho owur hqwur od edqgd d 06gE +shu

fdofroduh od srwhqd gdood ghqvlwã vshwwudoh ë qhfhvvdulr prowlsolfduh shu od edqgd,=

2% * 5

2 mKfm2 i3_

iS@

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yhgl hvhuflldulr,=

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yduldlrqh gryxwd dood txdqwldlrqh vduã=

d @dRR

5f@ 5

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s5@5 od yduldlrqh vduheeh vwdwd Kf@Kf * 57=;(,1

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ghoo*xvflwd +?,1 Od vfhowd gho jxdgdjqr D vhpsolfd od vroxlrqh gho sureohpd +xq ydoruh xqlwdulr gl

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gl srwhqd gho uxpruh eldqfr lq lqjuhvvr,/ xqd yrowd fdofrodwr lo olyhoor ghood srwhqd gl txdqwldlrqh

lq lqjuhvvr txhvwr ë lo ydoruh fkh vl ulwuryd lq xvflwd d fxl vl ghyh vrppduh od txdqwldlrqh ghoo*xvflwd

gl sdul srwhqd +l gxh surfhvvl vrqr lqfruuhodwl,1 Ylvwd od srwhqd xqlwduld lq xvflwd/ lo olyhoor 45; ghyh

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ghl vhjqdol ë=

d^ @6

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2? @ 5

d2^

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phqwuh lo VQU ë flufd 7317gE1

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Vl frqvlghul od vwuxwwxud shu od ghflpdlrqh lq jxud=

x(n) x’(n) y(n)H (z)

D=4

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$J

$ @5

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sdvvdqwh @7 = 6@7, iruqhqgr xqd udssuhvhqwdlrqh judfd ghood ulvsrvwd k+q,>

e, ^sxqwl 6` vl qhvwul od ulvsrvwd doo*lpsxovr gho owur lq prgr fkh lo pdvvlpr orer odwhudoh deeld

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srolidvh h vl ydoxwl lo frvwr frpsxwdlrqdoh plqlpr +rshudlrql2fdpslrqh,1

Ilqhvwud

+Q fdpslrql,odujkhd orer sulqflsdoh

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2q, @ i4> m>4>m> ===j1

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ghood odujhd gl edqgd$J @ gJ$, udssuhvhqwd xqd vroxlrqh shu lo fdpslrqdphqwr gl vhjqdol sdvvd

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O*dqgdphqwr ghoor vshwwur ë udssuhvhqwdwr lq jxud 51481

ω0 2π

Y(ω)

ω2π0

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ghflpdwd,/ shu od sduwh lppdjlqduld frqwulexlvfrqr jol dowul gxh owul +od vwuxwwxud frpsohwd ë lq jxud

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fldvfxqr dood iuhtxhqd iS@G @ iS@7= U @ 48iS rs12vhf1

x(n) xo(n)4

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z-1

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! # "$ 14&!&882

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do sxqwr +e, lq prgr gd ulxvduh od phghvlpd vwuxwwxud d phqr gl plqlph prglfkh/ txdolB Qrwd fkh

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j+q, @ k+q, frv+5

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hur dood iuhtxhqd i @ 483K=

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8

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frq vwuxwwxuh wudvsrvwh,1 Oh vlqjroh vhtxhqh srolidvh vrqr=

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J ^@$ J <

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f, +8 sxqwl, Vl yxroh vwlpduh/ d sduwluh gdo shulrgrjudppd/ od srwhqd gho uxpruh gl txdqwldlrqhf2^= l, vfulyhuh o*hvsuhvvlrqh gl wdoh vwlpdwruh ydoxwdqgr l vrol fdpslrql xwlol gho shulrgrjudppd/ ll,

ydoxwduh od yduldqd ghood vwlpd f2^1 +Vl dffhwwdqr dqfkh vroxlrql vxe0rwwlph sxufkë udjlrqhyrol,1

g, +7 sxqwl, Ulshwhuh o*dqdolvl gho sxqwr f, whqhqgr frqwr fkh lo uxpruh gl txdqwldlrqh ë shulrglfr1

d, Od srwhqd gho uxpruh gl txdqwldlrqh ulvxowd= 2

^@

2 7 532 1 Lpsrqhqgr od glvxj0

xdjoldqd ë srvvleloh fdofroduh lo qxphur gl elw h od yduldqd gho uxpruh dgglwlyr=

43 orjf

D2@5

2^

55$ E 6=69$ E @ 7>2

^@

532

6@ 4@:9;

e, Lo ydoruh phglr gho shulrgrjudppd QRQ ulvxowd sroduldwr gdood vlqxvrlgh +lq txdqwr txhvwd kd

xqd qhvwud gl rvvhuydlrqh Q@65 fkh ë xq pxowlsor lqwhur gho shulrgr sdul d ; fdpslrql> vl ë dvvxqwr/

shu vhpsolflwã/ W @ 4,=

H^V +$&,` @ 2^.Q

7^+n 7, . +n 5;,`>

frq $& @ 5n@Q 1 Od yduldqd ulvxowd txlqgl sdul d +hvvhqgr Q sdul dg xq qxphur lqwhur gl flfol/ lo

shulrgrjudppd fdofrodwr dood iuhtxhqd ghood vlqxvrlgh ë xqd yduldeloh fdvxdoh shu lo vror hhwwr gho

uxpruh gl txdqwldlrqh r dqdorjdphqwh/ lo shulrgrjudppd ghood vlqxvrlgh ë ghwhuplqlvwlfr,=

ydu^V +$&,` @ e^+4 . +n, . +n Q@5,,

f, Gdood ulvsrvwd +e, l fdpslrql xwlol shu od vwlpd ghood srwhqd gho uxpruh gl txdqwldlrqh vrqr l

vrol fdpslrql lqglshqghqwl gho shulrgrjudppd wudqqh lo fdpslrqh n @ 7 ryyhur= n @ i3> 4> 5> 6> 8> ==> 49jh qrq frqvlghudqgr oh iuhtxhqh qhjdwlyh lq txdqwr fruuhodwh dooh iuhtxhqh srvlwlyh1 Od vwlpd ghood

srwhqd gho uxpruh gl txdqwldlrqh gdo shulrgrjudppd ë rwwhqxwd frph phgld gl 49 fdpslrql

lqglshqghqwl i3> 4> 5> 6> 8> ==> 49j

f2^@

[&'tfcc2ccDccS

V +$&,@49

od fxl yduldqd ë=

ydu^f2^`@e

^ ^+Q@5 5, . 5 5` @492 @

4;

492e^

61431 TXDQWL]]D]LRQH GL XQ VHJQDOH VLQXVRLGDOH +43282<:, 494

Xqr vwlpdwruh slü dffxudwr dyuheeh gryxwr whqhuh frqwr fkh n @ 3>Q@5 kdqqr yduldqd grssld

h txlqgl vduheeh vwdwr srvvleloh djluh gl frqvhjxhqd1 47 fhooh i4> 5> 6> 8> ==> 48j kdqqr yduldqd e^h

gxh fhooh i3> 49j kdqqr yduldqd grssld 5e^gd fxl vhjxh fkh kr frpsohvvlydphqwh 48 fhooh d yduldqd

e^h txlqgl od yduldqd wrwdoh ghoor vwlpdwruh glyhqwd e

^@48 ^shu xqd vroxlrqh slü uljrurvd/ pd

QRQ suhylvwd qho surjudppd/ vl ulpdqgd dood glvfxvvlrqh grsr od vrox1 Hv18`1 Txhvwd vroxlrqh

ë frqjuxhqwh frq lo idwwr fkh kr frpsohvvlydphqwh Q@65 fdpslrql +@judgl gl olehuwã, gl fxl 5 vrqr

xwloldwl shu vwlpduh prgxor h idvh ghood vlqxvrlgh h l uhvwdqwl Q05@63 vhuyrqr shu ydoxwduh od yduldqd

gho uxpruh> qrwd fkh txhvwl kdqqr yduldqd 5e^gd fxl vhjxh 5e

^@63=

g, Vh vl frqvlghud lo uxpruh gl txdqwldlrqh/ txhvwr ë shulrglfr1 Lq txhvwr fdvr od vlqxvrlgh kd

shulrgr ; fdpslrql +vrwwrpxowlsor gl Q @ 65, h txlqgl od shulrglflwã gho uxpruh gl txdqwldlrqh ë

dqfk*hvvd sdul d ; fdpslrql1 Lqrowuh lo txdqwldwruh udssuhvhqwd xqd qrq olqhdulwã glvsdul h txlqgl/

lq suhvhqd gl xqd vrod vlqxvrlgh/ frqwlhqh oh vroh duprqlfkh glvsdul1 Lo uxpruh gl txdqwldlrqh

vduã txlqgl orfdoldwr lq n @ 7 h od fhood fruulvsrqghqwh dood whud duprqlfd lq n @ 451 Od vwlpd

ghood yduldqd gl uxpruh sxr hvvhuh frvï wurydwd gd xqd vrod fhood gho shulrgrjudppd +n @ 45, frq

frqvhjxhqwh yduldqd flufd e^+lo flufd ë ohjdwr doo*dvvxqlrqh gl uxpruh jdxvvldqr frq fxl ë ulfdydwd od

yduldqd gho shulrgrjudppd,1 Lq dowuh sduroh/ lq xqd vlqxvrlgh v? gl ; fdpslrql gl shulrgr frpsohvvlyr

yl vrqr 5 vrol fdpslrql lqglshqghqwl gho uxpruh gl txdqwldlrqh +ë ryylr vh vl shqvd fkh od vlqxvrlgh

ë frpsohwdphqwh ghqlwd gd xq vror txdgudqwh h txlqgl 5 fdpslrql, h txlqgl od yduldqd glyhqwd

5e^ 4@51Glvfxvvlrqh +sxqwr f,

Od vroxlrqh fruuhwwd +ulsruwdwd gl vhjxlwr VROR shu frpsohwhd pd QRQ lqfoxvd qho

surjudppd gho fruvr, suhyhgh gl shvduh lq prgr glhuhqwh l fdpslrql d yduldqd grssld +wlslfd0

phqwh shvduh n @ 3> Q@5 shu xq idwwruh 4@5, ylqfrodqgr or vwlpdwruh dg hvvhuh qrq0sroduldwr h d

plqlpd yduldqd1 Lqidwwl vh vl lqglfd frq & lo jhqhulfr frh!flhqwh +qrq qrwr,/ lo qxryr vwlpdwruh

glyhqwd= f2^ @[&MW

& V +$&,

gryh L @i3> 4> 5> 6> 8> ==> 49j1H* rud qhfhvvdulr lpsruuh fkh or vwlpdwruh vld qrq0sroduldwr +H^f2^ `@2^,h d plqlpd yduldqd shu wuryduh o*lqvlhph ghl Q@5 @ 49 frh!flhqwl=

H^f2^` @

[&MW

& H^V +$&,`@2

^

[&MW

& $[&MW

& @ 4

ydu^f2^` @ e^

3C [&M'tc2ccDccD

2& . 52f . 52*2

4DOd plqlpldlrqh ghood yduldqd frq lo ylqfror ghood qrq sroduldlrqh ghoor vwlpdwruh vl ulvroyh frq

lo phwrgr ghl prowlsolfdwrul gl Odjudqjh +yhgl fruvr Dqdolvl LL,=

plqbck&

;?=3C [

&M'tc2ccDccD

2& . 52f . 52*2

4D.

#[&MW

& 4

$<@>dqqxoodqgr oh +Q25.4, ghulydwh sduldol +judglhqwh, ulvshwwr dooh +Q25.4, lqfrjqlwh >f> ===> > D> ===> *2

vl ulfdydqr oh htxdlrql shu wuryduh or vwlpdwruh1 Gd txhvwd vwudgd vhjxh sursulr txdqwr vduheeh vwdwr

udjlqhyroh iduh=

f @ *2 @ &@5> & @ 5@+Q 5, @ 4@48> ;n 9@ 3> Q@5

Od yduldqd ghoor vwlpdwruh rwwlpr glyhqwd/ lq txhvwr fdvr/ ydu^f2^ `@e^ 2&+Q 5,@5 @e^@481


. ## ##: 12 1&-&4-2

Xq surfhvvr +qW ,> jdxvvldqr/ eldqfr qr dood iuhtxhqd it @ 4@+5W , @ 8PK/ frq srwhqd Sr @ 43

pZ ylhqh owudwr gdo vlvwhpd LLU=

K+, @4

4 5 frv! 3 . 2 32> frq @ 3=<9;9>

!

5W@ 733 nK

Lq xvflwd/ vl vrppd xq glvwxuer eldqfr/ g+qW ,>gl srwhqd S_ @ 4 Z1 Vld | @ k . g lo vhjqdoh

ulvxowdqwh1

d, ^5 Sxqwl` Vl wuryl od edqgd d 06gE gho owur +lq K,1 Vl udssuhvhqwl o*dqgdphqwr txdolwdwlyr

ghood ghqvlwã vshwwudoh gl srwhqd gho surfhvvr |+qW ,> ydoxwdqgrod qxphulfdphqwh shu i@3> 733 nK>8 PK/ +vl xwloll/ dg hvhpslr lo phwrgr ghl yhwwrul,1

e, ^8 Sxqwl` Vl ghwhuplql od srwhqd gho surfhvvr |+qW,/ vl fdofrolqr l sulpl 7 fdpslrql ghoo*dxwr0

fruuhodlrqh u++nW , h vl udssuhvhqwl o*dqgdphqwr txdolwdwlyr gl u++nW , shu rjql ydoruh gl n1

f, ^8 Sxqwl` Yrohqgr vwlpduh od srwhqd gho vhjqdoh +nW ,k+nW , vl lqglylgxlqr oh fhooh xwlol ghoshulrgrjudppd h vl ghwhuplql od yduldqd ghood vwlpd dvvxphqgr gl glvsruuh gl 433 vrwwr0vhtxhqh gl

|+nW , gd 833 fdpslrql fldvfxqd1

d, Od edqgd d 0 6 gE gho owur ë

i @5+4 ,

5W*

ir

433@ 433 nK

Od ghqvlwã vshwwudoh gl srwhqd ë vrppd ghl gxh frqwulexwl lqfruuhodwl=

V++i, @ W2% mK+ @ hs+m5i,,m2 . W2_

gryh

K+, @4

D2+,@

4

4 5 frv! 3 . 2 32

L ydorul lq frqwlqxd +@4 , h do Q|txlvw +@4, vrqr l vhjxhqwl=

V++i @ 3, * W2%

4

5+4 frv!,

2

. W2_ @ W2_

586

2%2_

. 4

* 3=68 433S ^Z@K`

V++i @ 8 PK, * W2%

4

5+4 . frv!,

2

. W2_* W2

_@ 3=4 433S ^Z@K`

phqwuh dood iuhtxhqd fhqwudoh i@733 nK/ frq lo phwrgr ghl yhwwrul vl rwwlhqh=

V++i @ 317 PK, * W2%

4

5 vlq!+4 ,

2

. W2_ * 7=5 433S ^Z@K`

e, Od srwhqd gho surfhvvr |+qW, ë od vrppd ghood srwhqd gl g+qW, +2_@ 4Z , h ghoo*xvflwd gho

vlvwhpd DU+5,1 Dssolfdqgr od ulfruvlrqh gl Ohylqvrq do srolqrplr D2+, vl rwwhqjrqr l frh!flhqwl gl

ul hvvlrqh=

f2 @ 2 @ 3=<6; f @

5 frv!

4 . 2@ 3=<9;

Od srwhqd lq xvflwd do vlvwhpd DU+5, ë

u+3, @2%

+4 f2,+4 f22,@ 4662% @ 4=66 Z

61441 SURFHVVR SDVVD0EDQGD DU+5, +442:2<:, 496

h gxqtxh od srwhqd gho surfhvvr |+qW, ë od vrppd ghl gxh whuplql= 5166 Z1 O*dxwrfruuhodlrqh gho

surfhvvr vl fdofrod gdooh ht1 \0Z +shu txhvwd yld vl srwhyd fdofroduh dqfkh od srwhqd,=

u+q, 5 frv! u+q 4, . 2 u+q 5, @ 2%+q,> ;q

qrwd fkh xqd yrowd fdofrodwl u+3,/ u+4, h u+5, phgldqwh oh ht1 \0Z ë srvvleloh wuryduh o*dxwrfruuhodlrqh

shu rjql ulwdugr q1 Od ghqvlwã vshwwudoh gl srwhqd whrulfd gho surfhvvr h od ixqlrqh gl dxwrfruuhodlrqh

vrqr udssuhvhqwdwh lq jxud 6171 Od srvllrqh ghl gxh srol ë lq hs+m!, h o*dqgdphqwr fkh fl vl

dvshwwd ë xq*rvfloodlrqh vprudwd gd xq hvsrqhqldoh udssuhvhqwdwr gd xq whuplqh

& @ 43& *Lf 4 @4

43ffeu&

dg hvhpslr yd d 4243 shu n * 4@3=347 * :3> or vprudphqwr dyylhqh gxqtxh qhoo*ruglqh gho fhqwlqdlrgl fdpslrql1 O*dqgdphqwr frvlqxvrlgdoh +vprudwr,/ kd iuhtxhqd qrupdoldwd 317243 flfol2fdpslrqh1

Dg hvhpslr lo sulpr hur shu od ixqlrqh frvhqr ë d 427 gl flfor/ txlqgl do fdpslrqh 43231727@91581

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-110

-100

-90

-80

-70

-60

-50

dB W

/Hz

freq. (MHz)

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

1.5

2

2.5

k

Iljxud 617= Ghqvlwd* vshwwudoh gl srwhqd gho surfhvvr sdvvd0edqgd +vrsud, h dxwrfruuhodlrqh +vrwwr,1

f, Lo ydoruh phglr ghoor vwlpdwruh shulrgrjudppd ë udssuhvhqwdwr gdood frqyroxlrqh ghood ghqvlwã

vshwwudoh gl srwhqd frq od ixqlrqh +wudvirupdwd ghood qhvwud gl Eduwohww,=

Z+i, @4

Q

vlq+iQW ,

vlq+iW ,

2

h vxffhvvlyr fdpslrqdphqwr lq i& @ n@WQ= Shu P @ 433 vrwwr0vhtxhqh gl oxqjkhd Q @ 833 od

ulvroxlrqh ghood vlqjrod fhood gho shulrgrjudppd ulvxowd sdul d ir@Q @ i @ 53nK= Lq txhvwr fdvr


od ulvroxlrqh ghood vlqjrod fhood ë lqihulruh dood odujkhd gl edqgd ghood ghqvlwã vshwwudoh gl srwhqd

gho surfhvvr frorudwr h txlqgl od frqyroxlrqh frq Z+i, lqwurgxfh xqd sroduldlrqh wudvfxudeloh/

od yduldqd ë sursrulrqdoh dood ghqvlwã vshwwudoh gl srwhqd ^+qW , k+qW , h g+qW , vrqr surfhvvl

lqglshqghqwl`=

H^ aV++i&,` @ V++i,Z+i,ms's& * W2% mK+hs+m5i&,,m2 . W2

_

ydu^ aV++i&,` *

kW2% mK+hs+m5i&,,m

2l2

.W2

_

2

Od yduldqd ghood vwlpd ghood ghqvlwã vshwwudoh gl srwhqd sxõ hvvhuh ulgrwwd hhwwxdqgr od phgld gl

P @ 433 shulrgrjudppl

aV+c +i&, @4

P

[6'

aVE6+ +i&,>

ydu^ aV+c +i&,` @4

P

kW2% mK+hs+m5i&,,m

2l2

.4

P

W2_

2shu n 9@ 3> Q@5

gryh vl ë lqglfdwr frq aVE6+ +i&, lo shulrgrjudppd ghood p0hvlpd vrwwr0vhtxhqd1 Xqd vroxlrqh ds0

survvlpdwd +xqd vroxlrqh uljrurvd ë prowr slü oxqjd, ghood vwlpd aS ghood srwhqd gho surfhvvr

+qW , k+qW , ë rwwhqxwd gdood ghqvlwã vshwwudoh gl srwhqd rwwhqxwd phgldqwh lo shulrgrjudppd/

vrppdqgr l frqwulexwl ghooh fhooh qhoo*lqwruqr ghood iuhtxhqd 733nK1 Lo qxphur gl fhooh +lqglshq0

ghqwl, ë flufd sdul d 8 hg ë ghwhuplqdwr gdo udssruwr wud od odujkhd gl edqgd d 06gE +433nK, h od

ulvroxlrqh gl xqd vlqjrod fhood +i @53nK,1 Shuwdqwr or vwlpdwruh ghood srwhqd ulvxowd=

aS @ 5i 2fn2[

&'2f32

aV+c +i&,>

od vwlpd ë sroduldwd gdood suhvhqd gho surfhvvr g+qW ,=

H^ aS ` @ 5i W2%

2fn2[&'2f32

mK+hs+m5i&,,m2 . 43i W2_>

phqwuh od yduldqd ulvxowd +dvvxphqgr fkh qhoo*lqwruqr gl 733nK od yduldqd ghood phgld gl P

shulrgrjudppl vld dssurvvlpdeloh frq lo ydoruh do fhqwur edqgd,=

ydu^ aS ` * 7 +i,2 8 ydu^ aV+c +i2f,`

Qrwd fkh od srwhqd vl srwhyd fdofroduh lqfoxghqgr xq qxphur pdjjlruh gl fhooh +yhgl jxud 617, rssxuh

vl srwhyd vwlpduh lo olyhoor gho uxpruh gdooh vroh fhooh lqwhuhvvdwh h vrwwuduuh txhvwr gdood vwlpd +shu

ryylduh dood sroduldlrqh ghood vwlpd,1

. ## ##: ? 1!&&4-2

Vld

|+q, @ +q, k+q, .z+q,

61451 SURFHVVR SDVVD0EDQGD . UXPRUH +57242<:, 498

xq surfhvvr glvfuhwr rwwhqxwr owudqgr lo surfhvvr jdxvvldqr eldqfr/ +q, +gl srwhqd 4 Z, phgldqwh

lo vlvwhpd k+q, frq ixqlrqh gl wudvihulphqwr=

K+, @4

4 5 frv 3 . 232

hg djjlxqjhqgr lo glvwxuer z+q,/ jdxvvldqr/ eldqfr hg lqfruuhodwr frq +q, h frq srwhqd 43 Z1 Vl

dvvxpd @ 3=< h @ @91d, +7 sxqwl, Vl udssuhvhqwl txdolwdwlydphqwh lo judfr ghood ghqvlwã vshwwudoh gl srwhqd gho surfhvvr

|+q,/ ghwhuplqdqgrqh l ydorul pdvvlpl h plqlpl +d txdoh iuhtxhqd vl wurydqr B,1e, +7 sxqwl, Vl vxssrqjdqr glvsrqlelol Q@65 fdpslrql gl |+q, h vl yrjold xwlolduh od whfqlfd gho

shulrgrjudppd shu vwlpduqh or vshwwur gl srwhqd vxgglylghqgr od vhtxhqd lq O vrwwrvhtxhqh1 Vl

glfd vh/ shu O @ ; dssdlrqr gxh slffkl vshwwudol glvwlqwl rssxuh xqr vror/ h lq txdoh srvllrqh n gho

shulrgrjudppd +3 n 65@O 4,1 Vl vwlpl hqwur txdoh lqwhuydoor sxõ ydulduh o*dpslhd ghl slffkl/dg hvhpslr frq xqd suredelolwã gho :3 (1

f, +7 sxqwl, Vl ghwhuplql od ixqlrqh gl wudvihulphqwr gl xqr vwlpdwruh gho vhjqdoh +q,/ d sduwluh

gdo vhjqdoh |+q,/ phgldqwh plqlpldlrqh ghoo*huuruh txdgudwlfr phglr qho grplqlr ghooh iuhtxhqh1

Vl suhflvl vh wdoh vwlpdwruh ë fdxvdoh/ dqwlfdxvdoh r qhvvxqr ghl gxh1

d, Od ghqvlwã vshwwudoh gl srwhqd gho surfhvvr |+q, ë rwwhqleloh 0 gdwd o*lqfruuhodlrqh wud +q, h

z+q, 0 frph vrppd ghooh ghqvlwã vshwwudol gl srwhqd ghl surfhvvl z+q, h +q, k+q,=

\r+i, @ [r+i, mK+i,m2 .Zr+i,

@2%ir

4

+4 hw3,+4 h3w3,

2

5'e2Zs*sr

.2ir

@ 2%

4

+4 h3E2Zs3w,+4 h3E2Zsnw,

2

. 2

dyhqgr dvvxqwr iuhtxhqd gl fdpslrqdphqwr ir @ 4/ hg lqglfdqgr frq 2% @ 4Z> 2 @ 43Z oh srwhqh

ghl surfhvvl +q, h z+q,1Vl wudwwd gxqtxh gl xqd vrppd gl xqd ixqlrqh sdvvdedvvr +frq srol lq hs+m, hg xqd frvwdqwh1

L slffkl vl rwwhqjrqr lq fruulsrqghqd ghl srol/ +5i @ , h ydojrqr=

\r+i,ms'w*2Z @ 2%

4

4

2 4

4 h32w

2

. 2

@2%

+4 ,2+4 . 2 5 frv 5,. 2 * 4432% . 2 @ 453

phqwuh lo plqlpr vl rwwlhqh dood iuhtxhqd gl Q|txlvw/ +od slü glvwdqwh gd hqwudpel l srol, h ydoh=

\r+,m5'3 @ 2%

4

4 . 5 frv . 2

2

. 2

*

2%7+4 . frv ,2

. 2 @2%<

. 2 * 43=4

O*dqgdphqwr ghoor vshwwur ë udssuhvhqwdwr lq jxud 6181

e, Shu ulvroyhuh lo txhvlwr do sxqwr +e, rffruuh frqvlghuduh fkh lo ydoruh phglr ghoor vwlpdwruh

shulrgrjudppd ë od ghqvlwã vshwwudoh gl srwhqd gho surfhvvr frqyroxwd frq od wudvirupdwd ghood


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

20

40

60

80

100

120

140Densità spettrale di potenza

Iljxud 618= Ghqvlwd* vshwwudoh gl srwhqd gho surfhvvr sdvvd0edqgd1

qhvwud uhwwdqjroduh do txdgudwr=

ZK+i, @4

Q@O

vlq+iQ@O,

vlq+i,

2

Lo frqwulexwr gho surfhvvr owudwr sxõ hvvhuh dssurvvlpdwr frph gxh lpsxovl fhqwudwl dooh iuhtxhqh

if @ @+5, @ 4@45 lqidwwl od edqgd sdvvdqwh ghl gxh slffkl ë i @ +4 ,@ @ 3=36/ lqihulruhdood edqgd ghood qhvwud gho shulrgrjudppd1 O*duhd gl fldvfxq lpsxovr ì sdul d phwã ghood srwhqd

wrwdoh/ rvvld

4

5

][r+i, mK+i,m2 gi @

2%5

]mK+i,m2 gi *

2%5 443 3=36 @ 4=92%

Lo ulvxowdwr ghood frqyroxlrqh gl xq lpsxovr frq od ixqlrqh ZK+i, ì od vwhvvd ixqlrqh wudvodwd d

iuhtxhqd if h vfdodwd shu od srwhqd ghoo*lpsxovr1 L gxh lpsxovl frqwulexlvfrqr qhoo*ruljlqh frq xq

shvr=

4=9

72%

vlq+7i,

vlq+i,

2s'*2

@ 7=7

fldvfxqr/ h qhood fhood n @ 4 frq xq shvr

3=7

vlq+7i,

vlq+i,

2s'E*e3*2

@ 4=5

61451 SURFHVVR SDVVD0EDQGD . UXPRUH +57242<:, 49:

shu hhwwr gho slffr d iuhtxhqd 4245 +lo slü ylflqr,/ h

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vlq+7i,

vlq+i,

2s'E*en*2

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shu hhwwr gho slffr slü orqwdqr/ lo frqwulexwr wrwdoh qhood fhood 4 ì flufd 429 gho frqwulexwr wrwdoh qhood

fhood 31

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et@H^e\ `sO

od glvwulexlrqh ë wlsr "2> vh dssurvvlpldpr frph xqd jdxvvldqd frq ydoruh phglr H^e\ `/ rwwhqldpr

xq lqwhuydoor sdul d =f, Vh ghqldpr e[+, @ D+,\ +,

od wudvirupdwd ghood vwlpd gl +q,/ rwwhqxwd phgldqwh or vwlpdwruh olqhduh frq wudvirupdwd D+,/ od

plqlpldlrqh ghoo*huuruh txdgudwlfr phglr qho grplqlr ghooh iuhtxhqh sruwd d +vl rphwwh gl lqglfduh

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H

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Od frpsrqhqwh ulfruvlyd gho owur D+, ë d idvh hur h vlpphwulfd +qrq fdxvdoh,/ lo owur frpsohvvlyr

qrq ë txlqgl fdxvdoh1


. ## ( 1&4&4-2

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S% @ 43pZ > S+ @ 73pZ +vl dvvxpd xqd uhvlvwhqd gl ulihulphqwr gl 4 rkp, yhqjrqr fdpslrqdwl

dood iuhtxhqd gl 4@W @ 5PK h qh yhqjrqr ulfdydwh oh vhtxhqh +qW , h |+qW ,1

Od vrod vhtxhqd +qW , ylhqh rud owudwd gdo vlvwhpd LLU=

K+, @;

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5W@ 534 nK + @ 5 =4338,

Lq xvflwd/ vl vrppd do vhjqdoh ? k? lo surfhvvr |? +glvwxuer,/ h vld

? @ ? k? . |?

od vhtxhqd ulvxowdqwh1

d, ^7 sxqwl` Vl udssuhvhqwl o*dqgdphqwr txdolwdwlyr ghood ghqvlwã vshwwudoh gl srwhqd gho vhjqdoh

+fdudwwhuldqgr od edqgd d 06gE, h gho glvwxuer1 Vl ghwhuplql od srwhqd gho vhjqdoh +ryyhur ? k?,h vl wuryl lo udssruwr vhjqdoh2glvwxuer lq gE1

e, ^7 sxqwl` Yrohqgr rud vwlpduh od srwhqd gho vhjqdoh ? k? gdood plvxud gl ?/ vl lqglylgxlqr oh

fhooh xwlol gho shulrgrjudppd qhooh txdol od sroduldlrqh lqwurgrwwd gdo glvwxuer |? ë wudvfxudeloh1

Vl ghqlvfd gdo shulrgrjudppd xqr vwlpdwruh ghood srwhqd gho vhjqdoh h vl fdudwwhull wdoh vwlpdwruh

lq whuplql gl sroduldlrqh h yduldqd dvvxphqgr gl glvsruuh gl xqd uhdoldlrqh gl 43e fdpslrql

i?j vxgglylvd lq 43 vrwwr0vhtxhqh glvjlxqwh gd 43 fdpslrql fldvfxqd +vl ulfklhgh gl jlxvwlfduh/

hyhqwxdophqwh frq l ydorul qxphulfl/ oh dssurvvlpdlrql lqwurgrwwh,1

f, ^7 sxqwl ` Vl ulshwd o*hvhufllr e, frqvlghudqgr shu lo owur o*htxlydohqwh sdvvd0edvvr ryyhur= K+, @

;@+4 3, h wudvfxudqgr lo uxpruh= ? @ ?k?1 Vl vwlpl od srwhqd gho surfhvvr ? sulpd phgldqwh

lo shulrgrjudppd +frph gd hvhufllr e, h vl frqiurqwl wdoh vwlpd frq or vwlpdwruh gl srwhqd gl xqd

vhtxhqd edvdwr vxood phgld ghl txdgudwl ghl fdpslrql1 Txdoh ghl gxh vwlpdwrul ulvxowd hvvhuh slü

dwwhqgleloh B

Vl ulfrugd fkh lo qxphur gl fdpslrql lqglshqghqwl shu Q fdpslrql gl xq surfhvvr DU+4, ë=

Qe^ * Q4 2

4 . 2

Oh gxh ghqvlwã vshwwudol=

V%+i, @ 2% W @ 8pZ@PK> V++i, @ 2+ W @ 53pZ@PK

d, Od ghqvlwã vshwwudoh gho vhjqdoh kd xq slffr +ulvrqdqd, lq fruulvsrqghqd ghood iuhtxhqd gl

534nK1 Shu wudfflduh fruuhwwdphqwh lo judfr ghooh ghqvlwã vshwwudol gl srwhqd ë qhfhvvdulr fdofroduh

dofxql ydorul vljqlfdwlyl1 H* lqrowuh lpsruwdqwh fdofroduh lo ydoruh ghood ghqvlwã vshwwudoh gl srwhqd

dood ulvrqdqd shu yhulfduh vh txhvwr ë r phqr vxshulruh dood ghqvlwã vshwwudoh gho uxpruh |?/ vhfrqgr

lo phwrgr judfr l srol vrqr d * =4 5/ od igw glyhqwd=

K+i @ 534nK, *;

+4 , 5* 45:=6

+lo sulpr whuplqh d ghqrplqdwruh ë gdwr gdo sror d 534nK phqwuh lo vhfrqgr whuplqh udssuhvhqwd

xqd dssurvvlpdlrqh ghood frugd rwwhqxwd gdood frqjlxqjhqwh frq lo sror d 0534nK,1 H* hylghqwh

61461 SURFHVVR ELDQFR ILOWUDWR H UXPRUH ELDQFR +452<2<:, 49<

fkh 8pZ@PK mK+i @ 534nK,m2 * ;4Z@PK 53pZ@PK1 Lo ydoruh plqlpr ghood ghqvlwã

vshwwudoh gl srwhqd vl kd dood iuhtxhqd gl Q|txlvw +vhpsuh judfdphqwh,=

K+i @ 4PK, *;

+4 . =<8, +4 . =<8,* 5=4$ 8pZ@PK 5=42 * 55=4pZ@PK

Shuwdqwr od ghqvlwã vshwwudoh gl srwhqd gho vhjqdoh ë vhpsuh vxshulruh dood ghqvlwã vshwwudoh gl srwhq0

d gho uxpruh wudqqh fkh qhoo*lqwruqr ghood iuhtxhqd gl Q|txlvw gryh hqwudpel dvvxprqr ydorul

frqiurqwdelol +yhgl jxud 619,1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5101

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103

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Iljxud 619= Ghqvlwd* vshwwudoh gl srwhqd vhjqdoh h xqd uhdoldlrqh +Q @ 4333, gho0

od ghqvlwd* vshwwudoh gho surfhvvr ? ^surjudppd Pdwode= wkhwd@5-sl-53425333> ukr@1<8>

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^pZ2PK`*,`1

Od edqgd d 06gE ulvxowd=

i3_ @ +4 ,@W * 64=;6nK

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uf+4 f2,+4 f

22, @ S%

4:3 FDSLWROR 61 VWLPD VSHWWUDOH

gdood ulfruvlrqh gl Ohylqvrq vl wuryd=

f @

5 frv

4 . 2* = ;39

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shu fxl vrvwlwxhqgr vrsud vl ulfdyd ^yhgl dqfkh oleur gl whvwr/ uhodlrqh +:14<,`

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4 2

4

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VQU @ 43 orjf+;2ufS+

, * 57gE

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3_ ghood ghqvlwã vshwwudohgl srwhqd gho vhjqdoh ulvxowd vxshulruh d i h txlqgl lo shulrgrjudppd ydoxwdwr qhoo*lqwruqr ghoodulvrqdqd qrq ë xqr vwlpdwruh sroduldwr d fdxvd ghood olplwdwhd gl Q +ryyhur od odujkhd gho orersulqflsdoh ghood wudvirupdwd ghood qhvwud gl Eduwohww/ 5NK/ ë prowr lqihulruh ulvshwwr dood odujkhdgl edqgd ghood ghqvlwã vshwwudoh gl srwhqd gho surfhvvr sdvvd edqgd/ 65NK/ vl sxõ txlqgl shqvduhfkh od frqyroxlrqh uhvwlwxlvfd dqfrud od vwhvvd ghqvlwã vshwwudoh gl srwhqd gho surfhvvr sdvvd edq0gd,1 Lqrowuh/ od ghqvlwã vshwwudoh gl srwhqd gho surfhvvr ? k? +vhjqdoh, qrq ë sroduldwr ghoodsuhvhqd gho uxpruh lq txdqwr/ qhoo*lqwruqr ghood ulvrqdqd/ od ghqvlwã vshwwudoh gl srwhqd gho ux0pruh +53pZ2PK, ë 069gE ulvshwwr d txhood gho vhjqdoh +;4Z2PK, +sroduldlrqh h yduldqd vrqrwudvfxudelol lq txdqwr ? 314(,1 Oh fhooh gho shulrgrjudppd lqwhuhvvdwh gdood ulvrqdqd ulvxowdqr

i3_

i* 49 fhooh frpsohvvlyh

ryyhur txhooh hqwur od edqgd d 06gE froorfdwh qhoo*lqwruqr ghood fhood dood iuhtxhqd gl ulvrqdqd

n @ Q534

5333@ 433=8$ fhooh ulvrqdqd n@433> 434

Vh vl lqglfd frq VE65 +n, @ m]E6+n,m2@Q lo shulrgrjudppd uhodwlyr dood p0hvlpd vrwwr0vhtxhqd gho

surfhvvr ? ^od vhtxhqd E6? @ ?nE63`/ or vwlpdwruh ghood srwhqd gho vhjqdoh ulvxowd +shu wuryduh

od srwhqd ë qhfhvvdulr lqwhjuduh l frqwulexwl uhodwlyl dooh i A 3 hi ? 3r slü vhpsolfhphqwh/ ylvwr fkh

VE65 +n, qhooh gxh edqgh QRQ vrqr lqglshqghqwl/ ë vx!flhqwh lqwurgxuuh od prowlsolfdlrqh shu 5,=

aSre?@,e @4

43

f[

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VE65 +n,i

&

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glprvwudwr ryyhur= H^VE65 +n,` * H^V5+n,`1 Shu ydoxwduh od sroduldlrqh ë srvvleloh dssurvvlpduh od

srwhqd hqwur od edqgd d 06gE frq lo ydoruh pdvvlpr +H^V6+433=8,` @ ;4Z@PK, h vl wudvfxudqr joldowul ydorul/ or vwlpdwruh uhvwlwxlvfh xq ydoruh gl srwhqd glyhuvd gd <1;Z=

H^ aSre?@,e` * 5 49H^VE65 +433=8,`i @ 65 ;4pZ@nK 5nK @ 8=4;Z

61461 SURFHVVR ELDQFR ILOWUDWR H UXPRUH ELDQFR +452<2<:, 4:4

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Od yduldqd ghood vwlpd ulvxowd=

ydu^ aSre?@,e` @4

43 7 49 ydu^VE6

5 +n,i ` * 9=7H2^VE65 +433=8,ì2 * 49=; 43epZ 2

qydu^ aSre?@,e`

r*2* 73<=;pZ=

f, Od ghqvlwã vshwwudoh ë gl wlsr sdvvd edvvr +htxlydohqwh sdvvd edvvr gho owur sdvvd edqgd, hg ëudssuhvhqwdwd gd xq DU+4, gl fxl ë qrwd o*dxwrfruuhodlrqh=

u& @;2S%4 2

& @ 9=9Z &=

Od iuhtxhqd gl wdjolr d 06gE ë i3_ @ +4 ,@5W * 49nK1 Od ghqvlwã vshwwudoh gl srwhqdgho surfhvvr ? @ ? k? ë lqglfdwd frq V5+i,/ lo shulrgrjudppd qhoo*lqwruqr ghood iuhtxhqd i @ 3

^qrwd fkh rud oh frqvlghudlrql qhoo*lqwruqr ghood iuhtxhqd gl ulvrqdqd 534nK +yhgl e, ghyrqr hvvhuhulshwxuh shu i @ 3` ulvxowd=

H^VE65 +i,` @ V5+i, @

S%

4@WmK+i,m2 @

S%

QimK+i,m2 *

S%

QimK+3,m2

gryh K+3, @ ;@+4 , @ 4931 Or vwlpdwruh ghood srwhqd gho vhjqdoh ulvxowd +qrwd fkh od ghqvlwãvshwwudoh gl srwhqd ë elodwhud h txlqgl shu wuryduh od srwhqd ë qhfhvvdulr vrppuh l frqwulexwl shun A 3 h n ? 3 0 l frqwulexwl qhooh gxh edqgh QRQ vrqr lqglshqghqwl,=

aSre?@,e @4

43

f[6'

%H[

&'3H

VE65 +n,i

&@

4

43

f[6'

i

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H[&'

VE65 +n,

&=

Vl ydoxwd rud od sroduldlrqh h od yduldqd ghoor ghoor vwlpdwruh1 Shu ydoxwduh od sroduldlrqhgl aSre?@,e ë srvvleloh dssurvvlpduh od srwhqd hqwur i3_ frq lo ydoruh pdvvlpr +V5+i @ 3, @

45;Z@PK, h wudvfxuduh jol dowul ydorul/ or vwlpdwruh ë sroduldwr lq txdqwr uhvwlwxlvfh xq ydoruh glsrwhqd glyhuvd gd 919Z=

H^ aSre?@,e` * 4:H^VE65 +3,ì @

4: S%Q

mK+3,m2 @4: S% ;2

Q+4 ,2@ 7=68Z

Od yduldqd ghood vwlpd ulvxowd=

ydu^ aSre?@,e` @4

43

+ydu^VE6

5 +3,i ` .

H[&'

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,*

4

43 +; 7 . 5,H2^VE6

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@67

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% @ 5=56 43S 2% * 3=556Z 2

$

qydu^ aSre?@,e`

r*2* 7:5pZ

Vl ulfrugd fkh shu fdofroduh od yduldqd vl ë xwloldwd od sursulhwã ydu^VE65 +n,` @ H2^V

E65 +n,` ^4 .

+n, . +n Q@5,`/ lo fdpslrqh n@3 kd yduldqd grssld h l fdpslrql n 9@ 3 qrq vrqr lqglshqghqwl lq


txdqwr VE65 +n, @ V

E65 +n,1 +Qhood vwlpd ghood srwhqd aSre?@,e vl gryuheeh shvduh lq prgr glyhuvr

lo fdpslrqh n @ 3 frvï frph looxvwudwr qhood glvfxvvlrqh do whpd gho 43282<:,1 Ylvwr fkh or vwlpd0wruh ë sroduldwr/ ë srvvleloh fdofroduh o*huuruh txdgudwlfr phglr H^+ aSre?@,e uf,

2` @ydu^ aSre?@,e` .

+H^ aSre?@,e` uf,2* 8=5Z 2

ydu^ aSre?@,e`= Vh od vrppd irvvh vwdwd hvwhvd d WXWWH oh fhooh gho sh0ulrgrjudppd +n@3/4/111/Q25, doorud or vwlpdwruh qrq vduheeh vwdwr sroduldwr= od edqgd htxlydohqwhgl uxpruh shu xq prghoor dg xq vror sror +s1h1 uhwh UF d fxl vl ulpdqgd shu xowhulrul ghwwdjol, ëudssuhvhqwdwd gdood edqgd d 06gE prowlsolfdwd shu @5/ od srwhqd glyhqwd 7=68Z @5 @ 9=;6Z

frlqflghqwh +d phqr gl dssurvvlpdlrql, frq lo ydoruh hhwwlyr 9=9Z=

Or vwlpdwruh ghood srwhqd gho surfhvvr ? gluhwwdphqwh gdooh rvvhuydlrql ulvxowd rud=

Sre?@,e @

S?'

2?

Q>

or vwlpdwruh qrq ë sroduldwr +yhgl oleur gl whvwr,1 Vh ë gdwr lo qxphur htxlydohqwh gl fdpslrqllqglshqghqwl Qe^ frq fxl vl ulgxfh od yduldqd qhood vwlpd ghood srwhqd gl xq surfhvvr DU+4, Qe^ *

Q+4 2,@+4 . 2,> l fdpslrql frpsohvvlyl d glvsrvllrqh vrqr Q @ 43e gd fxl Qe^ * 845=7; h txlqgl

ydu^ Sre?@,e` @+uf,

2

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845=7;Z 2 @ ;=8 4332Z 2

$

qydu^ Sre?@,e`

r*2* 5<5pZ=

Lq frqfoxvlrqh od vwlpd gd suhihuluh lq dvvhqd gl uxpruh ulvxowd Sre?@,e lq txdqwr ë pdjjlruh lo

qxphur gl judgl gl olehuwã41 aSre?@,e ë gd suhihuluh +qrqrvwdqwh od sroduldlrqh, qrq dsshqd odghqvlwã vshwwudoh gho uxpruh glyhqwd frqiurqwdeloh frq txhood gho vhjqdoh1

! ## # # 1&&4-2

Xq surfhvvr v+w, ë vrppd gl gxh surfhvvl uhdol +w, h |+w, jdxvvldql h lqglshqghqwl/ d ydoruh phglrqxoor h ghqvlwã vshwwudoh gl srwhqd=

V%+i, @

433S shu mi m 83NK

3 dowuryh= shu mi m A 83NK

V++i, @

S shu 83NK mi m 433NK

3 dowuryh= shu mi m ? 83NK h mi m A 433NK>

S @ 43pZ@NK +vl dvvxpd xqd uhvlvwhqd gl ulihulphqwr gl 4rkp,1 Od vhtxhqd v? @ v+qW , ërwwhqxwd fdpslrqdqgr lo surfhvvr v+w, dood iuhtxhqd 4@W @ 533NK/ vl kd txlqgl= v? @ ? . |?1

d, ^8 sxqwl` Vl fdofrolqr l sulpl 6 fdpslrql ~uf> u> u2 ghoo*dxwrfruuhodlrqh gho surfhvvr v?1e, ^7 sxqwl ` Vl wuryl gd ~uf> u> u2 l frh!flhqwl ghl gxh owul gl suhgllrqh D+, h D2+, gl oxqjkhd

ulvshwwlydphqwh 4 h 51f, ^6 sxqwl ` Vl udssuhvhqwl od ghqvlwã vshwwudoh gl srwhqd ghoo*huuruh gl suhgllrqh shu l gxh fdvl

^ryyhur shu D+, h D2+,` gdood ghqvlwã vshwwudoh gl v? +shu xqd udssuhvhqwdlrqh txdqwlwdwlyd vlfdofrol od ghqvlwã vshwwudoh shu dofxqh iuhtxhqh slü vhpsolfl,1 Vl fdofrolqr oh srwhqh ghoo*huuruh glsuhgllrqh vhpsuh shu l gxh owul D+, h D2+,1

4O*hvhufllr vl srwhyd ulvroyhuh dqfkh rvvhuydqgr fkh lo qxphur gl judgl gl olehuwd* qho shulrgrjudppd h* 4:3 +43

vrwwr0vhtxhqh lqglshqghqwl/ 4: fhooh rssxuh 4 fhood shu n@3 h ; fhooh fldvfxqd frq 5 judgl gl olehuwd*, shu fxl od yduldqd

vl ulgxfh gl *.f phqwuh e^ s D2e Od yduldqd ghood vwlpd gl srwhqd gdo shulrgrjudppd ulvxowd D2e*.f s

yrowh vxshulruh ulvshwwr dood vwlpd gluhwwd gd fxl vhjxh= ydud re?@,eo ' D2e*.ffydud re?@,eo ' f2DS` 21

61471 VRPPD GL GXH SURFHVVL VSHWWUDOPHQWH VHSDUDWL +542442<:, 4:6

O*dxwrfruuhodlrqh gho surfhvvr vl rwwlhqh/ gdwd o*lqglshqghqd vwdwlvwlfd/ gdood vrppd ghooh dxwr0

fruuhodlrql ghl vlqjrol surfhvvl=

ur+w, @ u%+w, . u++w,

u%+w, @ 433S 433NK vlq+ 433NK w,

+ 433NK w,

u++w, @ 5S 83NK vlq+ 83NK w,

+ 83NK w, frv+5 :8NK w,

O*dxwrfruuhodlrqh ghl vhjqdol fdpslrqdwl=

u%+q, @ 433Z vlq+q@5,

q@5

u++q, @ 4Z vlq+q@7,

q@7 frv+6@7q,

txlqgl l sulpl fdpslrql ulvxowdqr +lq Zdww,=

uf @ 434

u @433

@5.

vlq+@7,

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0 frq gxh surfhvvl gl srwhqd glhuhqwh H^d2` @ 43H^mDm2`1h, ^sxqwl 5 ` Vl dvvxpd fkh ? udssuhvhqwl lo vhjqdoh uhjlvwudwr gd xqd vfklhud gl 43 vhqvrul od fxl

vsdldwxud h* sdul d phwd* oxqjkhd g*rqgd / vl wurylqr oh gluhlrql gl duulyr ghooh gxh rqgh sldqh1

d,

u& @ H^W??n&` @ H^d2` .H^mDm2` hs+m$Jn, @ 2^4 . hs+m$Jn,`

e, Lo sror gl xq DU+4, vdud* lqwhuphglr d @4 h @hs+m$J,/ gdl fdofrol vl frqihupd=

dc @ f @ u

uf@ 4 . hs+m$J,

5

f, Lo fdvr DU+5, h* edqdoh h qrq ulfklhgh fdofrol lq txdqwr l gxh srol vrqr lq @4 h @hs+m$J,1

Yrohqgr sursulr iduh l fdofrol vl ulvroyh \Z u& . dc2u&3 . d2c2u&32 @ 2-E2& lq irupd pdwulfldoh=

u uf uWu2 u uf

57 4

dc2d2c2

68 @

57 3

3

3

68$

uf uWu uf

dc2d2c2

@

uu2

$

61631 VWLPD VSHWWUDOH DU H PD +46252<;, 53<

dc2d2c2

@

5 4 . hs+m$J,

4 . hs+m$J, 5

3

4 . hs+m$J,

4 . hs+m5$J,

@

h/J 4

h/J

Sror DU+5, = D2+, @ 3

= 4 +4 . hs+m$J,, 3 . hs+m$J,

32 @ 3$ vroxlrql= i @ 4j > @ h/J

g, Lq suhvhqd gl xq glvwxuer eldqfr oh iuhtxhqh gho sror +shu DU+4,, r ghl gxh srol +shu DU+5,,

ulpdqjrqr lqdowhudwh phqwuh fdpeld od srvllrqh udgldoh +vl dyylflqdqr doo*ruljlqh gl xqd txdqwlwd*

sursrulrqdoh do udssruwr vhjqdoh2uxpruh/ yhgl qrwh gl dssurirqglphqwr,1 Vh oh srwhqh ghl gxh

surfhvvl vrqr glyhuvh +43 yrowh pdjjlruh, pd vhqd uxpruh= shu lo fdvr DU+4, lo sror vdud* vsrvwdwr

yhuvr od vlqxvrlgh d srwhqd pdjjlruh +vhpsuh vxo fhufklr xqlwdulr, phqwuh shu DU+5, oh srvllrql ghl

srol vrqr oh vwhvvh/ lqidwwl

dc2d2c2

@

44 43 . hs+m$J,

43 . hs+m$J, 44

3

43 . hs+m$J,

43 . hs+m5$J,

@

h3/J . 4h/J

h/J

Sror DU+5, = D2+, @ 3$ 4 h3/J . 4

h/J3 . h/J32 @ 3$ vroxlrql= i @ 4j > @ h/J

=

h, od uhodlrqh wud GRD +, h sxovdlrql vsdldol +lq txhvwr fdvr lqglfdwh frq od ohwwhud x, h*=

5g

vlq @ x

vh lo fdpslrqdphqwr vsdldoh h* g @ @5 vl wuryd/ shu oh sxovdlrql qrupdoldwh= x @ vlq gd fxl/

shu oh gxh iuhtxhqh vsdldol 3 h @9 =

x @ 3$ @ 3J

x @ @9$ @ vlq3+4@9, * <= 9J

8 # 5 1&&402

Vl frqvlghul lo surfhvvr uhdoh +q, rwwhqxwr ulvroyhqgr o*htxdlrqh dooh glhuhqh qlwh=

+q, @ d+q 4, d2+q 5, . +q,

gryh=

d @ 4> d2 @ 3=3;>

H^+q,` @ 3> H^+q,+q. n,` @ u5+n,$ iu5+3,> u5+4,> u5+5,> u5+6,> ===j @ i8@7> t> 3> 3> ===j=

d, ^7 sxqwl` Vl fdofrolqr l frh!flhqwl gho suhglwwruh olqhduh gl ruglqh 4 h od srwhqd ghoo*huuruh gl

suhgllrqh shu lo surfhvvr +q,/ vxssrqhqgr t @ 31

d, ^7 sxqwl` Vl wurylqr o*ruglqh hg l sdudphwul gho prghoor PD fkh uhdold od vhtxhqd +q, d sduwluh

gd xqd vhtxhqd eldqfd z+q, gl srwhqd xqlwduld/ vxssrqhqgr t @ 4@51

f, ^7 sxqwl` Vl ulfdofrolqr l frh!flhqwl gho suhglwwruh olqhduh gl ruglqh 4 h od srwhqd ghoo*huuruh gl

suhgllrqh shu lo surfhvvr +q,/ vxssrqhqgr txhvwd yrowd t @ 4@51

Vl qrwd fkh shu t @ 3 i+q,j ë xq surfhvvr eldqfr/ shu t 9@ 3 i+q,j ë xq surfhvvr PD+4,1


d, Lo surfhvvr frpsohvvlyr ë xq DU+5, +frq l srol lq s @ 3=5/ s2 @ 3=;, ë rwwhqxwr gdo uxpruh

eldqfr d srwhqd 827/ o*dxwrfruuhodlrqh vl fdofrod phgldqwh \0Z=57 uf u u2u uf uu2 u uf

6857 4

dd2

68 @

57 8@7

3

3

68$

57 4 d d2d 4 . d2 3

d2 d 4

6857 ufuu2

68 @

57 8@7

3

3

68

57 ufuu2

68 *

57 ;= ;5

;= 49

:= 79

68

gd fxl vl ulfdyd lo suhglwwruh gl ruglqh 4 h od srwhqd ghoo*huuruh gl suhgllrqh=

@ u

uf@ = <58

uf+4 2, @ ;= ;5+4 = <582, * 4= 5:=

Qrwd fkh od vroxlrqh vl srwhyd ulfdyduh phgldqwh od ulfruvlrqh gl Ohylqvrq suhvwdqgr xq sõ gl dwwhq0

lrqh1 Xq suhglwwruh olqhduh gl ruglqh 5 ë lq judgr gl veldqfduh frpsohwdphqwh lo surfhvvr DU+5, h od

srwhqd ghoo*huuruh gl suhgllrqh +2032, ë hvdwwdphqwh 8271 Gd txhvwd vwudgd vl ulfdyd lo suhglwwruh gl

ruglqh 4 D+, h od srwhqd ghoo*huuruh gl suhgllrqh +203,=

f2 @ d2 @ =3;> f @ d@+4 . f2, @ d@+4 . d2, @ = <58> $ D+, @ 4 . f3

2032 @ uf+4 f2,+4 f22, @ 8@7$ uf @

8@7

+4 f2,+4 f22,* ;= ;5

203 @ uf+4 f2, * 4= 59

e, Prghoor PD+4, K+, @ eJ . e3frq srwhqd gl hfflwdlrqh/ dg hvhpslr/ xqlwduld=

U5+, @ K+,K+4@, @ +e2J. e2, . eJe+ . 3,

e2J. e2 @ 8@7

eJe @ 4@5$ gxh vroxlrql=

eJ @ 4@5> e @ 4$ J @ 5

eJ @ 4> e @ 4@5$ J @ 4@5

fruulvsrqghqwl dood vroxlrqh d plqlpd h d pdvvlpd idvh1

f, Lo surfhvvr frpsohvvlyr sxõ hvvhuh ylvwr frph xq surfhvvr DU+5, hfflwdwr gdo uxpruh frorudwr

PD+4, +lo surfhvvr ë DUPD,1 O*dxwrfruuhodlrqh gho surfhvvr DU+5, sxõ hvvhuh fdofrodwd dvvxphqgr/

dg hvhpslr/ o*hfflwdlrqh gho surfhvvr DU+5, d srwhqd xqlwduld +txhvwr ë idwwr vror shu vhpsolfduh l

frqwl vfdodqgr l ydorul ghoo*dxwrfruuhodlrqh gl 728 h xwloldqgr srl o*dxwrfruuhodlrqh U5+,,=57 4 d d2d 4 . d2 3

d2 d 4

6857 uf

uu2

68 @

57 4

3

3

68$

57 uf

uu2

68 @ +7@8,

57 ufuu2

68 @

57 := 387;

9= 8655

8= <9:;

68

iu%j @ iuj iu5j @ iuj i4@5> 8@7> 4@5ju%cf @ +4@5,u . +8@7,uf . +4@5,u * 48= 68

u%c @ +4@5,uf . +8@7,u . +4@5,u2 * 47= 9:

@ u%c

u%cf@ = <88

Od srwhqd ghoo*huuruh gl suhgllrqh=

u%cf+4 2, @ 48= 68+4 = <882, * 4= 68

61641 VWLPD VSHWWUDOH DU H PD +43282<:, 544

# 5 18&$&4-2

Vld gdwr xq surfhvvr ? @ z?.5z?3.z?32 rwwhqxwr gd xq uxpruh eldqfr z? d ydoruh phglr qxoor

h srwhqd xqlwduld=

d, +5 sxqwl, Fdofroduh l sulpl 6 fdpslrql ^uf> u/u2` ghoo*dxwrfruuhodlrqh gho surfhvvr ?/

e, +7 sxqwl, Vl hhwwxl xqd vwlpd vshwwudoh sdudphwulfd gho surfhvvr PD gl ruglqh 41

f, +6 sxqwl, Vl hhwwxl xqd vwlpd vshwwudoh sdudphwulfd gho surfhvvr DU gl ruglqh 41

d, O*dxwrfruuhodlrqh gho surfhvvr vl ulfdyd gdood ghqllrqh h ulvxowd= iu32> u3> uf> u> u2j @

i4> 7> 9> 7> 4j=e, Od vwlpd PD+4, gho surfhvvr ulfklhgh o*xvr ghl sulpl fdpslrql gxh fdpslrql ghoo*dxwrfruuh0

odlrqh gho surfhvvr rssruwxqdphqwh wurqfdwd frq xqd qhvwud d wudvirupdwd srvlwlyd1 Vl xvd od qhv0

wud wuldqjroduh i3> 4@5> 4> 4@5> 3j gd fxl vhjxh od vhtxhqd ghoo*dxwrfruuhodlrqh shu od vwlpd PD+4,=

i3> 5> 9> 5> 3j= Shu od vwlpd PD+4, ë qhfhvvdulr ulvroyhuh lo vlvwhpd=+2E

+4 . e2, @ uf @ 9

2E

e @ 4@5 u @ 5$ ulvroyhqgr ulvshwwr d e

h vfhjolhqgr od vroxlrqh d plq1 idvh=

e @ +6s8,@5 * 3=6;5

2E @ 5@ e @ 6 .

s8 * 8=569

f, Lo prghoor DU+4, vl ulfdyd gdooh ht1 \0Z ulvroyhqgr ulvshwwr dg dh 2-E

=

uf uu uf

4

d

@

2-E

3

$ d @ u

uf@ 5@6>2

-E @ uf . du @ 9 ;@6 * 6=66

# 1-&&82

Xqd vhtxhqd ë frvð frvwlwxlwd=

2?n6 @ x?> 2?nn6 @ x?

L fdpslrql ghood vhtxhqd ix?j vrqr jdxvvldql/ lqglshqghqwl d phgld qxood h yduldqd xqlwduld1d, +sxqwl 7, Vl fdofrol od ixqlrqh gl dxwrfruuhodlrqh ghood vhtxhqd ix?j do ydulduh gho fdpslrqh

gl ulihulphqwr p1 Txdorud o*dxwrfruuhodlrqh qrq vld vwdlrqduld/ vl fdofrol o*dxwrfruuhodlrqh phglddvvxphqgr fdvxdoh lo fdpslrqh gl ulihulphqwr p gdqgr hjxdol suredelolwã d fkh vld sdul r glvsdul1

e, +sxqwl 7, Vl ghwhuplql od vwlpd rwwlpd gl ? xwloldqgr vror ?3=

e, +sxqwl 7, Vl ghwhuplql od vwlpd rwwlpd gl ? xwloldqgr vld ?3 fkh ?32=

Vl vxjjhulvfh gl frqvlghuduh l gxh fdvl hohphqwdul= p @ 3 +p sdul, h p @ 4 +p glvsdul,1O*hvhufllr ulfklhgh gl ydoxwduh o*dxwrfruuhodlrqh phgld shu xq surfhvvr ? qrq vwdlrqdulr/ lqidwwl

txhvwr ë rwwhqxwr shu vryudfdpslrqdphqwr 4=5 h owudjjlr frq pdqwhqlwruh1 Sulpd gl iduh lqxwlol

fdofrol ë frqyhqlhqwh hvdplqduh l fdpslrql do ydulduh gl p/ vl qrwl fkh vl kdqqr fdpslrql xjxdol d

frsslh lqidwwl 2?n6 @ 2?n6n/ txhoor fkh fdpeld do ydulduh gl p ë od frssld gl fdpslrql xjxdol

lqidwwl1

p @ 3 p @ 4

f @ @ xf @ 2 @ xf2 @ @ x @ e @ xe @ D @ x2 D @ S @ x2


Whqhqgr frqwr gl txhvwr vl fdofrod o*dxwrfruuhodlrqh gd ghqllrqh shu l gxh fdvl=

H^&&n`$

;AAAAAAAA?AAAAAAAA=

p sdul=

2

c @ 3>;n2

c @ 4> n sdul

c @ 4> n glvsdul

3c @ 4> n glvsdul

c @ 4> n sdul

3 mcm A 4>;n

p glvsdul=

2

c @ 3>;n3

c @ 4> n sdul

c @ 4> n glvsdul

2

c @ 4> n glvsdul

c @ 4> n sdul

3 mcm A 4>;n

txlqgl o*dxwrfruuhodlrqh glshqgh gd n h lo surfhvvr qrq ë vwdlrqdulr1 Od fdvxdolwã gl p uhqgh fdvxdoh

od vfhowd gho fdpslrqh n gd fxl vl ulfdyd o*dxwrfruhodlrqh phgld vrppdqgr shu uljkh h shvdqgr shu

425 lq frqwulexwl +htxlsuredelolwã ghl sdul h glvsdul,=

u @ H^H^&&n`mn` @;?=

2

c @ 32

2

c @ 4

3 mcm A 4

Lo surfhvvr ë txlqgl dvvlplodeloh +lq phgld, dg xq PD+4,1 Od vwlpd rwwlpd glyhqwd rud edqdoh lqidwwl

qho sulpr fdvr ë ulfklhvwr lo suhglwwruh gl ruglqh 4 h qho vhfrqgr fdvr ë ulfklhvwr lo suhglwwruh gl ruglqh

5 +pd vl rvvhuyl fkh lo fdpslrqh ? ë vhpsuh lqfruuhodwr frq lo fdpslrqh ?32 h txlqgl ulpdqh lo vror

suhglwwruh gl ruglqh 4,=

a? @ dc?3 $ dc @u

uf@ 4@5=

# 5 1!&)&4-2

Vl frqvlghul od vhtxhqd ?? @ z? =<z?3 . |?

|? @ x? . =<|?3

Vh oh vhtxhqd x?> z? vrqr eldqfkh/ lqglshqghqwl wud orur/ h frq srwhqd xqlwduld=

5d, +7 sxqwl, vl fdofrolqr xq qxphur vx!flhqwh gl fdpslrql ghooh dxwrfruuhodlrql sulpd gl |? h srl

gl ?>

5e, +5 sxqwl, lo suhglwwruh olqhduh gl ruglqh 5 ghood vhtxhqd ?>

5f, +7 sxqwl, or vwlpdwruh olqhduh rwwlpr ghood vhtxhqd x? d sduwluh gd wxwwd od vhtxhqd ? +vl

rshul qho grplqlr ghooh iuhtxhqh,>

5g, +7 sxqwl, or vwlpdwruh olqhduh rwwlpr gho fdpslrqh z? xvdqgr l 6 fdpslrql ?3> ?> ?nd, Fdofroldpr dqlwxwwr oh fryduldqh wud oh yduldelol gl lqwhuhvvh=

? @ z? =<z?3 . |?

|? @ x? . =<|?357 u++3, u++4, u++5,

u++4, u++3, u++4,

u++5, u++4, u++3,

6857 4

=<3

68 @

57 2

3

3

68

61661 VWLPD VSHWWUDOH DU H PD +47292<:, 546

u++3, @2

=4<@ 8=592

> u++4, @ 7=:62

> u++n, @ u++3,+=<,

&

u%+3, @ 4=;42. 8=592

> u%+4, @ =<2

. 7=:62

> u%+5, @ 7=592

e, L frh!flhqwl gho suhglwwruh olqhduh srvvrqr hvvhuh ydoxwdwl phgldqwh oh ht1 \0Z=u%+3, u%+4,

u%+4, u%+3,

dc2d2c2

@

u%+4,

u%+5,

rssxuh gdood ulfruvlrqh gl Ohylqvrq=

f @ u%+4,

u%+3,@ dc

f2 @ u%+5, . dcu%+4,

u%+3, . dcu%+4,@ d2c2

dc2 @ dc+4 . f2,

f, Shu or vwlpdwruh qho grplqlr ghooh iuhtxhqh gho fdpslrqh x? vl kd +shu D+, @ 4 3=<3,=

[+, @Z +,D+, .X+,

D+,

h or vwlpdwruh glyhqwd=

^X+, @ K+,[+,> K+, @H ^X[W`

H ^[[W`

K+, @2DW 4

2mDm2 . 2@mDm2@

D

TmDme . 4@

4 =<3

T ^4=;4 =<+ . 3,`2 . 4> T @

22

gryh vl h* xvdwd od qrwdlrqh= D @ D+,/ DW @ DW+4@,/ mDm2 @ D+,DW+4@,1 Vl rvvhuyl fkh od vroxlrqhK+, frqvwd gl xq whuplqh fdxvdoh h gl xq whuplqh dqwlfdxvdoh1

Vh irvvh vwdwr ulfklhvwr or vwlpdwruh ghood vhtxhqd az? +lq dqdorjld do sxqwr g, doorud vduheeh vwdwr=

_Z +, @ E+,[+,> E+, @H ^Z[W`

H ^[[W`

E+, @DW2

2mDm2 . 2@mDm2@

DWmDm2mDme .T

@+4 =<,

4=;4 =<+ . 3,

^4=;4 =<+ . 3,`2 .T

> T @22

g, Or vwlpdwruh rwwlpr vl fdofrod frph= az? @S

&'3 e&?3& 1 O*ruwrjrqdolwã ghoo*huuruh gl

suhgllrqh %? @ z? az? iruqlvfh oh htxdlrql xwlol shu od vwlpd gho suhglwwruh +lq irupd vfdoduh r

pdwulfldoh,=

H^%??36` @ 3> p @ 4> 3> 4[

&'3

e&H^?3&?36` @ H^z??36`> p @ 4> 3> 4

57 u%+3, u%+4, u%+5,u%+4, u%+3, u%+4,u%+5, u%+4, u%+3,

6857 e3efe

68 @

57 H^z??n`

H^z??`H^z??3`

68


L frh!flhqwl glshqgrqr txlqgl gdoo*dxwrfruuhodlrqh ghood vhtxhqd ? h gdood fruuhodlrqh wud od

vhtxhqd ? h od vhtxhqd z? =

H^?z?` @ 2> H^?nz?` @ 3=<2> H^?3z?` @ 3

h txlqgl +qxphulfdphqwh,=57 e3efe

68 @

57 := 3: 6= ;6 7=59

6= ;6 := 3: 6= ;67=59 6= ;6 := 3:

68357 3=<4

3

68 *

57 = 5<;= 5<4

=35 5

68

Frqiurqwr wud od vroxlrqh lq grplqlr iuhtxhqd E+, +yhgl sxqwr f, h lq grplqlr whpsl dsshqd

fdofrodwd +qrwd fkh= m @s4,=

Es ^ @ hs+m$,` @+4 =< hs+m$,, ^4=;4 4=; frv$`

^4=;4 4=; frv$`2 . 4

$ mEs +$,m @4=;4 4=; frv$

+4=;4 4=; frv$,2 . 4m4 =< hs+m$,m

E|+ @ hs+m$,, @ = 5<; hs+m$, . = 5<4 . =35 5 hs+m$,$ mE|+$,m @ m= 5<; hs+m$, . = 5<4 . =355 hs+m$,m

Vl vduheeh srwxwr vyrojhuh lo phghvlpr sureohpd qho grplqlr whpsl frq xq vlvwhpd dgdwwdwlyr

+qrq hud ulfklhvwr gdo sureohpd pd ë rssruwxqr yhghuh frph vyrojhuor,1 Lq txhvwr fdvr/ lqglfdqgr

e @ ^ e3 ef e `A lo yhwwruh fkh udssuhvhqwd oh lqfrjqlwh gho sureohpd/ od uhodlrqh gl djjlruqdphqwr

gho vlvwhpd lwhudwlyr ë=

eE& @ e

E&3 . %?

57 ?n

??3

68

idfhqgr frlqflghuh oh lwhudlrql frq l fdpslrql fkh surjuhvvlydphqwh yhqjrqr dftxlvlwl +ryyhur o*lqglfh

n fkh udssuhvhqwd o*lwhudlrqh frlqflgh frq o*lqglfh q gho fdpslrqh,=597 e

E?3

eE?f

eE?

6:8 @

597 e

E?33

eE?3f

eE?3

6:8. %?

57 ?n

??3

68

gryh

%? @ z? az? @ z? az? @ z? [

&'3

eE?&?3& @ z? +eE&,A

57 ?n

??3

68

Od vfhowd gl ë ohjdwd djol dxwrydorul ghood pdwulfh gl fryduldqd

U @ H^

57 ?n

??3

68 ?n ? ?3

` @

57 := 3: 6= ;6 7=59

6= ;6 := 3: 6= ;67=59 6= ;6 := 3:

68 @ HH

A

H @

57 = 8;7 6:= 896 37= 8;7 6:

= 6<; 46= ;59 76= 6<; 46

= :3: 444= :4; : 433H

= :3: 44

68

@

57 48= 35 3 3

3 6= 6:< 33 3 5= ;4

68

61671 VWLPD VSHWWUDOH GL XQ SURFHVVR FRORUDWR +4525234, 548

Od frqgllrqh hvwuhpd + 4@ , shu o*djjlruqdphqwr vhqd rvfloodlrql ë

4@ @4

4@ @

4

48= 35

Od frqyhujhqd ghoo*dojrulwpr lwhudwlyr shu od gluhlrqh q0hvlpd +ryyhur oxqjr o*dxwryhwwruh q0hvlpr,

ë ohjdwd do whuplqh irudqwh=

+4 ?,&

Od frqyhujhqd vl kd lq xq qxphur gl sdvvl Q wdoh gd ulgxuuh lo whuplqh irudqwh d olyhool wudvfxudelol/

shu vhpsolflwã vl ë vrolwl sruuh 42433=

+4 ?, @ 4@433$ Q @

5orjf+4 ?,

Txhvwd frqgllrqh ghyh hvvhuh yhulfdwd shu wxwwl jol dxwrydorul h txlqgl dqfkh shu o*dxwrydoruh slü

slffror +ryyhur shu lo ydoruh gl Q slü judqgh,=

Q @5

orjf+4 4?,

Xvdqgr lo ydoruh pdvvlpr gho sdvvr gl judglhqwh 4@ shu dffhohuduh lo slü srvvleloh od frqyhujhqd vl

kd=

Q4? @5

orjf+4 b4?

b4@

,@

5orjf+4 2H

Df2,* 55 sdvvl

Qrwd fkh od frqyhujhqd glshqgh gdo udssruwr wud o*dxwrydoruh pdvvlpr h o*dxwrydoruh plqlpr ghood

pdwulfh gl fryduldqd @ 4@ @4? +txhvwr ë lqglfdwr frph vsduvlwã ghjol dxwrydorul/ sursrulrqdoh

do udssruwr wud lo ydoruh pdvvlpr h lo ydoruh plqlpr ghood ghqvlwã vshwwudoh gl srwhqd gho surffhvvr/ lq

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u

v

θ 2Ωo

2Ωo

Iljxud 716= Vhjqdoh d uljkh +rqgd txdgud, owudwr qr dood whud duprqlfd h fdpslrqdwr frq

htxlyrfdlrqh1

!) 1 3 2 ## 10&)&402

Vl frqvlghul xq vhjqdoh +> |, vrppd gl gxh vlqxvrlgl 5G dyhqwl dpslhd xqlwduld h iuhtxhqd

vsdldoh 4@ +flfol2p, rulhqwdwh o*xqd vhfrqgr o*dvvh h o*dowud vhfrqgr o*dvvh |= vl wuryl o*lqwhuydoor gl

fdpslrqdphqwr S gl xq uhwlfror d edvh txdgudwd do olplwh ghoo*htxlyrfdlrqh=

d, ^5 sxqwl ` vl wuryl o*dqgdphqwr ghood wudvirupdwd gl Irxulhu vld gho vhjqdoh +> |, fkh gho vhjqdoh

fdpslrqdwr +p>q, @ +pS> qS, qho fdvr lq fxl +> |, udssuhvhqwl xqd vrppd gl gxh vlqxvrlgl>

vl wuryl srl o*dqgdphqwr ghood vhtxhqd +p>q,>

7191 FDPSLRQDPHQWR +DO OLPLWH GHOO*HTXLYRFD]LRQH, GL VLQXVRLGL +4;292<;, 57<

e, ^5 sxqwl ` vl wuryl o*dqgdphqwr ghood wudvirupdwd gl Irxulhu vld gho vhjqdoh 2+> |, fkh gho vhjqdoh

fdpslrqdwr 2+p>q, @ 2+pS> qS, qho fdvr lq fxl 2+> |, udssuhvhqwl xqd vrppd gl gxh frvl0

qxvrlgl gl dpslhd h iuhtxhqd xjxdoh do fdvr suhfhghqwh> vl wuryl srl o*dqgdphqwr ghood vhtxhqd

2+p>q,>

f, ^5 sxqwl ` vl vslhjklqr l ulvxowdwl ghl sxqwl 0d0 h 0e0 dqdoldqgr vld lo grplqlr vsdl fkh txhoor

ghooh iuhtxhqh vsdldol>

g, ^6 sxqwl ` vl uxrwlqr rssruwxqdphqwh oh frruglqdwh gho uhwlfror gl fdpslrqdphqwr +r vl uxrwl lo

vhjqdoh, h vl lqfuhphqwl o*lqwhuydoor gl fdpslrqdphqwr qr dg rwwhqhuh xqd urwdlrqh fkh sruwd S do

ydoruh pdvvlpr srvvleloh +ryyhur vhpsuh do olplwh ghoo*htxlyrfdlrqh,1 Wuryduh lq txhvwh frqgllrql

o*lqwhuydoor gl fdpslrqdphqwr S hg lo vhjqdoh fdpslrqdwr

+p>q, @ +p

S> qS,> l @ 4> 5 vld qho

fdvr ghood vrppd gl gxh vlqxvrlgl +l @ 4, fkh qho fdvr ghood vrppd gl gxh frvlqxvrlgl +l @ 5,1

g, ^6 sxqwl ` Glprvwuduh fkh/ qho qxryr vlvwhpd gl frruglqdwh/ lo vhjqdoh ë vhsdudeloh1 Vl ulfrugd fkh

vlq. vlq @ 5 vlq+.

5, frv+

5,=

Oh vlqxvrlgl h oh frvlqxvrlgl gl shulrgr yhqjrqr fdpslrqdwh frq 5 fdpslrql shu flfor +S @ @5,/

vh vhtxhqh vrqr + @ 5@,=

+> |, @ vlq+ , . vlq+ |,$ +p>q, @ vlq+p, . vlq+q, @ 3#$ ]+$%> $+,

]+x> y, @4

5m^+x > y, +x. > y, . +x> y , +x> y . ,`

]+$%> $+, @[6c?

]+$% 5p>$+ 5q, @ 3 +fdqfhoodlrqh uhsolfkh htxlyrfdwh,

2+> |, @ frv+ , . frv+ |,$ 2+p>q, @ frv+p, . frv+q, @ +4,6 . +4,? #$ ]2+$%> $+,

]2+x> y, @4

5^+x > y, . +x. > y, . +x> y , . +x> y . ,`

]2+$%> $+, @ +$% > $+, . +$% . >$+, . +$%> $+ , . +$%> $+ . , +vrppd uhsolfkh htxlyrfdwh,

Od vhtxhqd 2+p>q, kd dqgdphqwr

i2+p>q,j?c6 @

;AAAA?AAAA=

5 3 5 3 5

3 5 3 5 3

5 3 5 3 5

3 5 3 5 3

5 3 5 3 5

<AAAA@AAAA>

Uxrwdqgr lo vlvwhpd gl frruglqdwh +r l vhjqdol, o*rffxsdlrqh vshwwudoh ë

x4@ @ y4@ @5

s5@ @

s5$

S @

5

s5 @

s5

phqwuh oh vlqxvrlgl h frvlqxvrlgl uxrwdwh glyhqwdqr +yhgl jxud 717,=

+> |, @ vlq

s5+. |,

. vlq

s5+ |,

@ 5 vlq

s5 frv

s5|

2+> |, @ frv

s5+. |,

. frv

s5+ |,

@ 5 frv

s5 frv

s5|


u

u

v

Ω

ab

a* b*

2 2Ω /

2 2Ω /

Iljxud 717= Vlqxvrlgl +d @ e @ 4@5m, r frvlqxvrlgl +d @ e @ 4@5, uxrwdwh1 D ydooh gho fdpslrqdphqwr

or vshwwur uhsolfdwr vl vrppd +d . e . dW . e

W= shu oh vlqxvrlgl d . e . dW . e

W @ 3 phqwuh shu oh

frvlqxvrlgl d. e. dW . e

W @ 5,1

+p>q, @ 5 vlq s5 frv

s5|

%'6

S+'?

S

@ 5vlqp frvq @ 3

2+p>q, @ 5 frv s5 frv

s5|

%'6

S+'?

S

@ 5frvp frvq @ 5+4,6n?

!- # ## % 1$&&82

Xq vhjqdoh 5G ë frvwlwxlwr gdood vrppd gl gxh vlqxvrlgl oh fxl gluhlrql gl flolqgulflwã vrqr d 78f63f

frq o*dvvh h oh fxl iuhtxhqh vrqr gl 6 flfol2phwur1

4, +sxqwl 5, Vl vfulyd lo vhjqdoh 5G vld lq frruglqdwh fduwhvldqh fkh srodul1 Vl wudffl or vshwwur 5G

gl wdoh vhjqdoh lqglfdqgr fkldudphqwh od vfdod ghjol dvvl1

5, +sxqwl 5, Vl ghwhuplql od oxqjkhd O gho odwr edvh gl xq uhwlfror gl fdpslrqdphqwr d edvh

txdgudwd shu fxl qrq yl vld htxlyrfdlrqh1

6, +sxqwl 5, Vl ghwhuplql rulhqwdphqwr h oh gxh glphqvlrql O%> O+ gl xq uhwlfror gl fdpslrqdphqwr

d edvh uhwwdqjroduh shu fxl qrq yl vld htxlyrfdlrqh qho fdvr gho vhjqdoh gl fxl do sxqwr 4,1

7, +sxqwl 6.6, Vl frvwuxlvfd xq owur 5G fkh vrssulpd xqd ghooh gxh vlqxvrlgl vsdldol/ vld qho fdvr

gl fdpslrqdphqwr frq uhwlfror lq 5, fkh frq lo uhwlfror lq 6,1

Lo whpd udssuhvhqwd od vrppd gl gxh vlqxvrlgl vsdldol frq xjxdoh sxovdlrqh vsdldoh @ 56 @

9udg@p h dlpxwk d 48J h :8J1 Oh iuhtxhqh vsdldol vrqr=

x @ frv 48J @ 9 frv Z

2 @ 2s94 .

s6* 4;= 53:udg@p

y @ vlq 48J @ 9 vlq Z

2@

2s94

s6* 7= ;:;udg@p

x2 @ frv :8J @ 9 frv DZ2 @

2s94

s6* 7= ;:;udg@p

y2 @ vlq :8J @ 9 vlq DZ2

@ 2s94 .

s6* 4;= 53:udg@p

71;1 HTXLYRFD]LRQHGL VRPPD2SURGRWWRGL GXH VHJQDOL VLQXVRLGDOL 5G +4525234,584

Shu lo fdpslrqdphqwr vsdldoh d sdvvr O vl ghyh lpsruuh=

5

O 5pd+x> y> x2> y2, @ 6

s9

4 .

4

6

s6

O 5

6s94 .

s6 * = 4:5p

lq txdqwr x4@ @ y4@ +yhgl jxud 718,1 Od ghqvlwd* gho fdpslrqdphqwr h* gl 4@O2 * 66=; fdpslrql2p2=

u

vu’

v’

v’maxu’max

Ω

Iljxud 718= Gxh vlqxvrlgl vsdldol frq dlpxwk 78J 63J

Shu lo fdpslrqdphqwr frq xq vlvwhpd gl dvvl uxrwdwr d 78ghj +yhgl jxud 718, oh pdvvlph iuhtxhqh

vsdldol +h l uhodwlyl sdvvl gl fdpslrqdphqwr, ulvxowdqr rud

x

4@ @ frv 63J @ 9 frv

9@ 6

s6 * 49= 657udg@p$ O%

x4@

@4

6s6* = 4<5p

y

4@ @ vlq 63J @ 9 vlq

9@ 6 * <= 757udg@p$ O+

y4@

@4

6@ =66p

frq xqd ghqvlwd* gl 4@+O%O+, @ <s6 * 48= 8 fdpslrql2p2 +fkldudphqwh lo tv1 vhfrqgr fdvr lo

fdpslrqdphqwr h* slx* h!flhqwh,1

O*hvwudlrqh +r od vrssuhvvlrqh, gl xqd ghooh gxh vlqxvrlgl +yhgl jxud,= qho sulpr fdvr +dvvl +x> y,,h*

qhfhvvdulr xq owur 5G +s1h1/ xq owur d yhqwdjolr r xq owur sdvvd edvvr d edqgd sdvvdqwh frq vxssruwr

uhwwdqjroduh,/ qho vhfrqgr +dvvl +x> y,, fdvr h* vx!flhqwh xq owur 4G lq xqd ghooh gxh gluhlrql1

!0 #& # ## %1&&82

Gxh vlqxvrlgl kdqqr iuhtxhqh i @ 5> i2 @ 7 flfol@p h dlpxw @ 48J h 2 @ :8J1


4, +sxqwl 6, Vl wudffl or vshwwur 5G ghood vrppd ghooh gxh vlqxvrlgl +vl lqglfkl od vfdod ghjol dvvl,1

5, +sxqwl 6, Vl wudffl or vshwwur 5G gho surgrwwr ghooh gxh vlqxvrlgl +vl lqglfkl od vfdod ghjol dvvl,1

6, +sxqwl 6, Vl fdpslrql vhqd htxlyrfdlrqh od vrppd h lo surgrwwr ghooh gxh vlqxvrlgl frq xq

uhwlfror d edvh uhwwdqjroduh> lq txdoh ghl gxh fdvl vl kd fdpslrqdphqwr d plqlpd ghqvlwd*B

7, +sxqwl 6, Vl ulshwd lo sxqwr 6 dvvxphqgr gl uxrwduh rssruwxqdphqwh od vrppd h lo surgrwwr

lq prgr gd plqlplduh shu fldvfxq vhjqdoh od ghqvlwd* gho fdpslrqdphqwr1

6, +sxqwl 6, Vl fdofrol od ulvsrvwd doo*lpsxovr gl xq owur shu o*hvwudlrqh gdo surgrwwr fdofrodwr do

sxqwr 6 ghood vlqxvrlgh d shulrgr pdvvlpr1

Or vshwwur ghood vrppd h gho surgrwwr h* prvwudwr lq jxud 7191 Gd frqvlghudlrql jhrphwulfkh vl

kd=

vrppd = +x4@ > y4@ , @ i frv 48J> 5 frv 48Jjsurgrwwr = +x4@ > y4@ , @ i J frv+:8

J ,> J vlq+:8J . ,j

gryh

J @ s8

@ dufwdq4

5* 59= 8ghj

Lo fdpslrqdphqwr vdud*

x4@

|

y4@

h od ghqvlwd* gho fdpslrqdphqwr +lq qxphur gl fdpslrql2p2, h* =

G @4

|

x4@ y4@

2

qhl gxh fdvl h*

GrJ66@ 5 2+frv Z

2,2

2@

5 492

2+frv

45,2 @ 5<= ;8fdps@p

2

GRoJ_J||J 8 2 frv+ D

2 , vlq+ D

2 . ,

2@

8 492 frv+ D2 , vlq+ D

2 . ,

2@ 85fdps@p

2

Uxrwdqgr lo vlvwhpd gl dvvl gl :8ghj od ghqvlwd* ulpdqh lqyduldwd qhl gxh fdvl +vrppd h surgrwwr, hg

h* sdul d

G 5 2

2@ 65fdps@p

2

Dowuh urwdlrql vrqr srvvlelol/ shu txhvwr vl ulpdqgd do whpd gho 582342341

71<1 LQWHUSROD]LRQH GL XQD VLQXVRLGH 5G +542442<:, 586

u

v

Ω

2Ω

u

v

Ωο

somma prodotto

umax

vmax

umax

vmax

2ΩΩ

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Iljxud 719= Vrppd h surgrwwr gl gxh vlqxvrlgl frq dlpxwk 048ghj h :8ghj/ sxovdlrql vsdldol @

@ 7udg@p h 2 @ 5 @ ;udg@p ulvshwwlydphqwh1

!4 ## % 1&&4-2

Vl frqvlghul xq vhjqdoh 50G vlqxvrlgdoh uhdoh v+> |, @ frv+x.y|, fdpslrqdwr d sdvvr lq hqwudpeh

oh gluhlrql=

d, ^7 sxqwl` Vl ghqlvfd xq vlvwhpd olqhduh +lghdoh, shu wudvoduh lo vhjqdoh fdpslrqdwr v+q>p, gl

@7 qhood gluhlrqh h @6 qhood gluhlrqh | +txdol vrqr l ydorul pdvvlpl ghooh sxovdlrql +x> y,

dpplvvlelolB,1 Vl wuryl od ulvsrvwd doo*lpsxovr gho vlvwhpd lghdoh1

e, ^6 sxqwl` Vl dssurvvlpl lo vlvwhpd olqhduh do sxqwr d, phgldqwh xq owur vhsdudeloh1 Vl frqvlghul

frph owur xq lqwhusrodwruh d pdqwhqlphqwr qhood gluhlrqh h xq lqwhusrodwruh olqhduh qhood gluhlrqh

|1 Vl ghvfulyd lo owur h vh qh fdofrol od wudvirupdwd gl Irxulhu 50G1

f, ^8 sxqwl` Vl ydoxwlqr l olplwl ghoo*dssurvvlpdlrqh do sxqwr e, fdofrodqgr or vshwwur ghoo*huuruh do

ydulduh ghood sxovdlrqh vsdldoh +x> y,1 Vl fdofrol od srwhqd ghoo*huuruh gl wudvodlrqh1

Lo vhjqdoh fdpslrqdwr ulvxowd=

v?c6 @ v+q>p, @ frv+xq. yp,

oh sxovdlrql ghyrqr ulhqwuduh hqwur lo grplqlr hohphqwduh= mxm h mym 1

Shu od wudvodlrqh gho vhjqdoh fdpslrqdwr ë qhfhvvdulr xq owur +vhsdudeloh, od fxl ulvsrvwd doo*lpsxovr


ë lqqlwd4=

K+x> y, @ hs+mx@7, hs+my@6,' k?c6 @ kE%? kE+

6

kE%?

@vlq^+q 4@7,`

+q 4@7,> k

E+6

@vlq^+p 4@6,`

+p 4@6,

u/π∆

v/π∆

Errore di interpolazione

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Iljxud 71:= Dqgdphqwr ghoor vshwwur ghoo*huuruh gl wudvodlrqh V+x> y,1

O*dssurvvlpdlrqh vxjjhulwd qho whvwr fruulvsrqgh d=

j?c6 @ jE%? jE+

6

jE%?

@ jE%+q, @ uhfw^+@7,@`m

%'? @ ?

jE+6

@ jE++p, @ 4 m@6m@`m

%'6 @ 5@66 . 4@663

ryyhur qrq hhwwxd dofxqd rshudlrqh oxqjr hg lqwhusrod olqhduphqwh qhood gluhlrqh |1 Od wudviru0

pdwd gl Irxulhu +vsdlr0glvfuhwr, ulvxowd=

j?c6 @ 5@6?c6 . 4@6?c63 ' J+x> y, @ 5@6 . 4@6 hs+my, @4

6^4 . hs+my@5, frv+y@5,`

Lqglfdqgr frq v?c6 lo vhjqdoh ulwdugdwr/ txhvwr ulvxowd=

v?c6 @ v?c6 j?c6 @ mJ+x> y,m frv^xq. yp._J+x> y,`

od srwhqd gho vhjqdoh ulwdugdwr v?c6 ulvxowd mJ+x> y,m2@5 phqwuh od srwhqd gho vhjqdoh ë 4@51 O*huuruh

ë dqfk*hvvr vlqxvrlgdoh=

?c6 @ frv+x+q 4@7, . y+p 4@6,, v?c6 @ v?c6 +k?c6 j?c6,>

4Vl ulfrugd fkh ulwdugduh gl + qrq h* qhfhvvduldphqwh xq qxphur lqwhur, xq vhjqdoh fdpslrqdwr d sdvvr h*

htxlydohqwh dg lqwhusroduh lo vhjqdoh glvfuhwr/ s1h1 frq xq lqwhusrodwruh lghdoh +sdvvdjjlr glvfuhwr0frqwlqxr,/ ulwdugduh

lo vhjqdoh gl h lqqh ul0fdpslrqduh lo vhjqdoh ulwdugdwr1 Oh wuh rshudlrql srvvrqr hvvhuh vyrowh phgldqwh xq owur

htxlydohqwh rwwhqxwr fdpslrqdqgr od ulvsrvwd doo*lpsxovr ghoo*lqwhusrodwruh ulwdugdwr gl +od yhulfd h* vhpsolfh/ edvwd

ydoxwduh od ulvsrvwd doo*lpsxovr gho vlvwhpd frpsohvvlyr,1

71431 ILOWUR D YHQWDJOLR +442442<;, 588

od srwhqd ghoo*huuruh ë udssuhvhqwdwd gdood srwhqd gho vhjqdoh vlqxvrlgdoh gl dpslhd mK+x> y,J+x> y,m =

V0+x> y, @ mK+x> y,J+x> y,m2 @5 @ mhs^m+x@7 . y@6,` ^5@6 . 4@6 hs+my,`m2 @5

shu gduqh xqd udssuhvhqwdlrqh vl dvvxph xqlwdulr o*lqwhuydoor gl fdpslrqdphqwr + @ 4,=

V0+x> y, @ mhs^m+x@7 . y@6,` ^5@6 . 4@6 hs+my,`m2 @5

o*dqgdphqwr ë udssuhvhqwdwr qhooh jxuh 71:/ 71;1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

v/π∆

Errore di interpolazione (u=0)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

u/π∆

Errore di interpolazione (v=0)

Iljxud 71;= Huuruh V+x> y, shu vlqxvrlgl x @ 3 +vrsud, h y @ 3 +vrwwr,1

!8

Vl fdofrol od ulvsrvwd doo*lpsxovr gl xq owur lghdoh 50G fdudwwhuldwr gd xqd ixqlrqh gl wudvihulphqwr

d idvh hur lo fxl prgxor ë xqlwdulr qho sulpr h whur txdgudqwh + vl shqvl dg xq vlvwhpd wrprjudfr

gryh vldqr glvsrqlelol vror oh surlhlrql frqwhqxwh lq xq dqjror gl 78f ulvshwwr dood elvhwwulfh gho

sulpr h whur txdgudqwh,=

d, ^sxqwl 5` Frqvlghudqgr sulpd lo fdvr glvfuhwr vl wuryl od ixqlrqh gl wudvihulphqwr qho grplqlr

irqgdphqwdoh +x> y,1e, ^sxqwl 6` Vl fdpslrql od wudvirupdwd lq prgr fkh o*dqwlwudvirupdwd ulpdqjd uhdoh h fkh od ixqlrqh

vld ghqlwd vx xq uhwlfror 66>f, ^sxqwl 6` vl wuryl o*dqwlwudvirupdwd1

g, ^sxqwl 7` Vl wuryl od ulvsrvwd doo*lpsxovr qho fdvr gl vhjqdol frqwlqxl qhoor vsdlr viuxwwdqgr od

vhsdudelolwã1

Glvsrqhqgr gl phwã ghooh plvxuh ë htxlydohqwh d owuduh lo vhjqdoh frq xq owur d yhqwdjolr vxo sulpr

h whur txdgudqwh od fxl ulvsrvwd ë +yhgl oleur gl whvwr,

K+x> y, @4

5^4 . vljq+x, vljq+y,`' k+> |, @

4

5+> |,

4

54

4

|=


Shu lo fdvr glvfuhwr +lqglfdwr gdo shglfh g, ë qhfhvvdulr ulfrugduh l ydorul dvvxqwl gdood ixqlrqh vhjqr

+qho grplqlr irqgdphqwdoh,=

vljq_+!, @

;?=

4 ? ! ? 33 ! @ 3>4 3 ? ! ?

gd fxl vl kd=

K_+x> y, @4

5.

4

5vljq_+x, vljq_+y,

Fdpslrqdqgr od ixqlrqh gl wudvihulphqwr K_+x> y, qhooh srvllrql x& @ n 5@Q @ n 5@6 h

y @ c 5@Q @ c 5@6 vl kd=

K_+x&> y, @4

5&c . &c .

4

5&c .

4

5&c $ iK_+x&> y,j @

57

3 4@5 44@5 4@5 4@54 4@5 3

68

dqwlwudvirupdqgr +viuxwwdqgr od sursulhwã gl vhsdudelolwã ghood wudvirupdwd gl Irxulhu 5G shu lo whuplqh

vljq_+x, vljq_+y, ryyhur fdofrodqgr lo surgrwwr ghooh wudvirupdwh 4G,=

ik6c?j @4

56c? .

4

55m

6vlq+

5

6p,

5m

6vlq+

5

6q, @

4

56c?

5

<vlq+

5

6p, vlq+

5

6q,=

u

v

φ

u

v

u

v

u

vH-π/4(u,v) H -π/4(u,v/tanβ)

α=-π/4

Iljxud 71<= Wudvirupdlrql shu rwwhqhuh xq owur d yhqwdjolr gl dshuwxud dqjroduh h rulhqwdphqwr

txdovldvl d sduwluh gd xq owur d yhqwdjolr vxo sulpr0whur txdgudqwh1

Rvvhuydlrqh= Vh yrohvvlpr xq owur d yhqwdjolr frq rulhqwdphqwr h dshuwxud txdoxqtxh vduheeh

srvvleloh dssolfduh urwdlrql do owur d yhqwdjolr frq vxssruwr sulpr0whur txdgudqwh frph lqglfdwr

lq jxud 71</ o*lqvlhph ghooh urwdlrql sruwd dg xq owur d yhqwdjolr frq dshuwxud txdoxqtxh 5 h

lqfolqdlrqh jhqhulfd !1 Shu uxrwduh gl xq dqjror od ulvsrvwd gl xq owur K+x> y, ë qhfhvvdulr

dssolfduh o*lqvlhph ghooh wudvirupdlrql +oh wudvirupdlrql vrqr lqglfdwh frq shglfh, qhood vxffhvvlrqh

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∆


d, ^5 sxqwl ` vl glprvwul vh v+> |, udssuhvhqwd r phqr xq vhjqdoh vhsdudeloh>

e, ^6 sxqwl ` vl fdofrol od wudvirupdwd gl Irxulhu 5G gl v+> |,>

f, ^6 sxqwl ` vl glprvwul vh v2+> |, udssuhvhqwd r phqr xq vhjqdoh vhsdudeloh h vh qh fdofrol od

wudvirupdwd gl Irxulhu 5G>

g, ^5 sxqwl ` vl ydoxwl od frqyroxlrqh gl v+> |, h gl v2+> |, frq xq owur od fxl ulvsrvwd ë=

k+> |, @ uhfw+@5, uhfw+|@5, @

4 shu mm > m|m

3 dowuryh>

fdofroduh od frqyroxlrqh qho grplqlr ghooh iuhtxhqh1

h, ^5 sxqwl ` frph fdpeld od frqyroxlrqh do sxqwr g vh vl uxrwd lo owur gl 78ghj1B

Lo vhjqdoh v+> |, ë vhsdudeloh phqwuh lo vhjqdoh v2+> |, @ +4@5, . +4@5, v+> |, ë vrppd gl

vhjqdol vhsdudelol pd QRQ ë hvvr vwhvvr vhsdudeloh1 Shu prvwuduh od vhsdudelolwã ë frprgr +pd qrq ë

txhvwd o*xqlfd vroxlrqh, uliduvl dg xqd vhtxhqd d vfdffklhud=

v+p>q, @

4 4 4 4

4 4 4 4

4 4 4 4

@

111

4

4

4111 ~

s?E?

=== 4 4 4 4 === ~ s6E6

#$ V+x> y, @ I?+x, I6+y,

Lo vhjqdoh i+, fkh frpsrqh v+> |, @ i+, i+|, udssuhvhqwd xq*rqgd txdgud gl shulrgr 5 fkh

frqvwd vror gl duprqlfkh glvsdul +yhgl Whruld ghl Vhjqdol,=

i+, @7

frv

7

6frv 6 .

7

8frv 8 . ===> gryh @ @

ryyldphqwh od wudvirupdwd gl Irxulhu gl v2+> |, sxõ hvvhuh fdofrodwd door vwhvvr prgr ulfrugdqgr gl

djjlxqjhuh od frpsrqhqwh frqwlqxd +dwwhqlrqh/ o*hvfxuvlrqh gho vhjqdoh ë glphdwd lq txhvwr fdvr,1

Od frqyroxlrqh frq lo owur uhvwlwxlvfh lo ydoruh ghood frpsrqhqwh frqwlqxd/ od wudvirupdwd gl Irxulhu

5G gl k+> |, vl dqqxood vx wxwwl l ydorul gl x @ n@ h y @ n@ shu ;n 9@ 3 h txlqgl vl dqqxood

vxooh duprqlfkh glvsdul gl +shu v+> |, ë hur phqwuh shu v2+> |, ë 425,1 Uxrwdqgr gl 78ghj lo

owur +r l vhjqdol, vl rwwlhqh/ shu hqwudpel l fdvl/ xq vhjqdoh glhuhqwh frq frqwhqxwr grplqdqwh gl

edvvd0iuhtxhqd +lo owur k+> |, ë xq owur sdvvd0edvvr vhsdudeloh frq sxovdlrqh vsdldoh gl wdjolr sdul

d @,1

!0 # 1-&-&402

Vld gdwr xq owur 5G od fxl ulvsrvwd doo*lpsxovr ë udssuhvhqwdwd gdood pdwulfh gl fdpslrql lqglfdwd gl

vhjxlwr=

ik+p>q,j @4@5 4@5 4@5 4@54 4 4 44@5 4@5 4@5 4@5

+sdvvr gl fdpslrqdphqwr vsdldoh= @53fp,1

715;1 ILOWUR VHSDUDELOH +5:2:2<;, 5;8

d, ^5 sxqwl ` vl glprvwul fkh k+p>q, udssuhvhqwd xq owur vhsdudeloh ^ryyhur sxõ hvvhuh rwwhqxwd

frph surgrwwr gl gxh ulvsrvwh doo*lpsxovr= k+p>q, @ k+p, k2+q, `>e, ^5 sxqwl ` vl fdofrol od wudvirupdwd gl Irxulhu ghood ulvsrvwd k+p>q,>f, ^6 sxqwl ` vl fdofrol od wudvirupdwd gl Irxulhu ghood surlhlrqh gl k+p>q, shu ! @ 3 h ! @ 78ghj

gryh ! ë o*dqjror frq o*dvvh p>

g, ^6 sxqwl ` vl frqyroyd od ulvsrvwd k+p>q, frq xqd vhtxhqd Q Q fdpslrql +Q @ 63, h vl ydoxwllo frvwr frpsxwdlrqdoh ghood frqyroxlrqh +frvwr shu fdpslrqh xwloh,>

h, ^5 sxqwl ` vl ydoxwl lo frvwr frpsxwdlrqdoh fdofrodqgr od frqyroxlrqh +frq l phghvlpl sdudphwul

gho sxqwr g, phgldqwh IIW 5G +frvwd phqr ulvshwwr do fdvr gB,1

Lo owur ë vhsdudeloh=

k+p>q, @4@5 4@5 4@5 4@54 4 4 44@5 4@5 4@5 4@5

@4@544@5 ~ 2E?

4 4 4 4 ~ E6

#$ K+x> y, @ K+x, K2+y,

K+x, @ 5 ^frv+x@5, frv+6x@5,`

K2+y, @ 4 . frv+y,

Od wudvirupdwd ghood surlhlrqh vl fdofrod frph vhlrqh ghood wudvirupdwd gl Irxulhu=

K+ >!, @ K+x> y,m'lULt'lt?

! @ 3$ K+ , @ 7 ^frv+ , frv+5 ,`

! @ 78ghj$ K+ , @ 5kfrv+ @

s5, frv+5 @

s5,l k

4 . frv+ @s5,l

Shu l ghwwdjol vx frph ghulyduh lo frvwr frpsxwdlrqdoh vl ulpdqgd do whpd gho 582:2<:/ vl ulshwh

txl euhyhphqwh lo fdofror shu vxssruwr qrq txdgudwr gd 671 Od frqyroxlrqh vx Q uljkh frvwd

frpsohvvlydphqwh= 7Q+Q.704,/ vxooh frorqqh= 6+Q.604,+Q.704,> lo frvwr joredoh ë=

frvwr@7Q+Q . 6, . 6+Q . 6,+Q . 5, @ +Q . 6,+:Q . 9,>

lo frvwr shu fdpslrqh ë+Q . 6,+:Q . 9,

+Q . 6,+Q . 5,* : prow12fdpslrqh

ryyhur ë flufd od vrppd ghooh gxh frqyroxlrql 4G sulpd shu uljkh +7 prow12fdpslrqh, h srl shu

frorqqh +6 prow12fdpslrqh,1 Shu od IIW5G vl fhufd od srwhqd gl 5 lpphgldwdphqwh vxshulruh +lq

txhvwr fdvr O@97,/ lo frvwr glyhqwd=

5O2^5 . 9 orj2 O`

+Q . 6,+Q . 5,* 679 prow12fdpslrqh

Od frqyroxlrqh phgldqwh od IIW5G qrq ë vhpsuh od vroxlrqh slü frqyhqlhqwh txdqgr od vhtxhqd

gho owur ë vhsdudeloh1


!4 ##& # 18&-&402

Vld gdwr xq owur sdvvd0edvvr qhood gluhlrqh h sdvvd dowr qhood gluhlrqh | od fxl pdvfkhud K+x> y,ë udssuhvhqwdwd lq jxud +d vlqlvwud shu vhjqdoh 5G vsdlr0frqwlqxr/ d ghvwud lo grplqlr irqgdphqwdoh

shu lo vhjqdoh 5G fdpslrqdwr,

u

v

vo

uo

1

1

u

v

Ωo

1

1

Ωo

2Ωo

2Ωo

2vo

d, ^6 sxqwl ` vl glprvwul fkh od ixqlrqh gl wudvihulphqwr K+x> y, fruulvsrqgh dg xq owur vhsdudeloh

+ryyhur sxõ hvvhuh rwwhqxwd frph surgrwwr gl gxh ixqlrql gl wudvihulphqwr= K+x> y, @ K+x, K+y, ,>e, ^6 sxqwl ` vl fdofrol od ulvsrvwd doo*lpsxovr k+> |, gho owur vsdlr0frqwlqxr>

f, ^6 sxqwl ` vl fdofrol od ulvsrvwd doo*lpsxovr k+p>q, gho owur fdpslrqdwr +shu lo grplqlr irqgd0

phqwdoh lq jxud,>

g, ^6 sxqwl ` vl olplwl od ulvsrvwd k+p>q, lq prgr wdoh gd dyhuh xq frvwr shu fdpslrqh gl 43

prowlsolfdlrql h vrpph +vl dvvxpd gl gryhu owuduh xqd vhtxhqd 5G gl glphqvlrql prowr judqgl,1

K+x, @ uhfw+x@5xf,#$ k+, @xf

vlq+xf,

+xf,sdvvd0edvvr

K+y, @ uhfw+y@yf, ^+y 6yf@5, . +y . 6yf@5,`

#$ k+|, @yf

5vlq+yf|@5,

+yf|@5, 5 frv+

6yf5|, sdvvd0edqgd

K+x> y, @ K+x, K+y,#$ k+> |, @xfyf

2vlq+xf,

+xf,vlq+yf|@5,

+yf|@5,frv+

6yf5|,

Gdood jxud gho whvwr vl kd

f @ xf @ yf> S @ 7 f $ @ @5 f

k+p>q, @ 2f

2vlq+p@5,

p@5vlq+q@7,

q@7frv+

6

7q,

Vh vl glvsrqh gl xqd vhtxhqd gd P P fdpslrql h xq owur 50G vhsdudeloh gl vxssruwr Q Q +lo

ydoruh gl Q ë lqfrjqlwd, doorud lo frvwr shu fdpslrqh glyhqwd +yhgl whpd g*hvdph gho 582:2<: lq fxl vl

dvvxph P Q,=Q+5P .Q 4,

P .Q 4* 5Q $ Q @ 8=

71631 VHTXHQ]H VHSDUDELOL +6424233, 5;:

Lq dowhuqdwlyd shu ydoxwduh lo frvwr2fdpslrqh ë vx!flhqwh yhulfduh fkh/ shu xq owur vhsdudeloh/ od

frqyroxlrqh 5G sxõ hvvhuh uhdoldwd frq gxh frqyroxlrql 4G gd Q fdpslrql sulpd shu uljkh h srl

shu frorqqh/ lo frvwr ë gl Q rshudlrql2fdps1 shu rjql frqyroxlrqh 4G1

Lq uhdowã ë frqyhqlhqwh wurqfduh lo owur 5G k+p>q, lq prgr fkh lo wurqfdphqwr dyyhqjd s1h1 lq

fruulvsrqghqd gho sulpr hur vld qhood gluhlrqh fkh qhood gluhlrqh | shu fxl vl kd= gluh1 $p35eoJ @ 5> gluh1 | $ q35eoJ @ 71 Gd txhvwr dssduh frqyhqlhqwh vfhjolhuh xq owur gl vxssruwr

QQ2 lq fxl vl vfhjolh Q @ 6 @ 4.4.4 h Q2 @ : @ 6.4.6 lq txhvwr prgr lo frvwr ë Q.Q2 @ 431

!8 # 1&&882

Vl frqvlghul xqd glvwulexlrqh gl fdpslrql gl dpslhd xqlwduld/ d irupd gl furfh/ frq 6 fdpslrql

glvsrvwl vld oxqjr o*dvvh rulrqwdoh fkh yhuwlfdoh1

d, ^6.4 sxqwl` Vl wuryl od wudvirupdwd gl Irxulhu 5G gho vhjqdoh1 Vl wudwwd gl xq vhjqdoh vhsdudelohB

e, ^7 sxqwl` Vl ghwhuplql vld od ixqlrqh gl dxwrfruuhodlrqh fkh or vshwwur gl srwhqd gho vhjqdoh11

f, ^7 sxqwl` Vl wuryl od wudvirupdwd gl Irxulhu 5G glvfuhwd gho vhjqdoh1

Lo vhjqdoh +qhood vhtxhqd 5G vl dvvxpd xq sdvvr gl fdpslrqdphqwr xqlwdulr,

v+p>q, @

57 3 4 3

4 4 43 4 3

68' V+x> y, @ 4 . h

. h3 . h

. h3 @ 4 . 5 frvx. 5frv y

Ryyldphqwh qrq h* vhsdudeloh +vl qrwd gdood vhtxhqd fkh qrq sxr* hvvhuh rwwhqxwd gdo vror surgrwwr

gl gxh odph ixqlrql r gdood wudvirupdwd gl Irxulhu fkh qrq sxr* hvvhuh hvsuhvvd frph surgrwwr gl gxh

ixqlrql qhooh sxovdlrql qrupdoldwh x h y,1 O*dxwrfruuhodlrqh h*=

v+p>q, v+p>q, @

599997

3 3 4 3 33 5 5 5 34 5 8 5 43 5 5 5 33 3 4 3 3

6::::8' mV+x> y,m

2

Od GIW vl fdofrod fdpslrqdqgr od wudvirupdwd gl Irxulhu 5G=

V+n> n2, @ V+x> y,m'2Z&*'2Z&2*

@ 4. 5 frv5n6

. 5 frv5n6

! #:# 1&&882

Vl frqyroydqr gxh glvwulexlrql gl 6 fdpslrql gl dpslhd xqlwduld glvsrvwl oxqjr o*dvvh rulrqwdoh h

yhuwlfdoh=

i+> |, @ j%+> |, j++> |,

gryh =j%+> |, @ +> |, . +> |, . +.> |,j++> |, @ +> |, . +> | ,. +> | .,

5;; FDSLWROR 71 VHJQDOL H VLVWHPL 5G

+shu vhpsolflwd* vl sxr* sruuh @ 4,1d, ^5.4 sxqwl` Vl fdofrol od frqyroxlrqh h vl udssuhvhqwl judfdphqwh i+> |,1 Lq txhvwr fdvr i+> |,

h* xq vhjqdoh vhsdudelohB

e, ^6 sxqwl` Vl fdofrol od wudvirupdwd gl Irxulhu 5G gl i+> |,1f, ^6 sxqwl` Vl fdofrol i+>*, frph surlhlrqh gl i+> |, shu * @ 3> @7> @5> vl udssuhvhqwl judfd0

phqwh i+>*, qhl wuh fdvl1 Vl wuryl od wudvirupdwd i+ > *,1g, ^6 sxqwl` Vl fdofrol od surlhlrqh ghoo*dxwrfruuhodlrqh gl i+> |, shu * @ 3> @7> @51 Vl wuryl od

wudvirupdwd shu oh wuh surlhlrql1

& 'Od vhtxhqd 5G h* xq uhwwdqjror +ryyldphqwh vhsdudeloh,=

i+p> q, @

57 4 4 4

4 4 44 4 4

68' I +x> y, @ +4.h.h3,+4.h.h3, @ +4.5 frvx,+4.5frv y,

Oh surlhlrql vl ulfdydqr judfdphqwh=

i+> * @ 3, @ i+>* @ @5, @ 64 4 4

' I + > * @ 3, @ 6+4 . 5 frv ,

i+> * @ @7, @4 5 6 5 4

frq sdvvr wud l fdpslrql

s5@5

' I + > * @ @7, @ I +x @ s5@5> y @

s5@5, @ +4 . 5frv

s5@5,2

O*dxwrfruuhodlrqh gl xq sulvpd d edvh uhwwdqjroduh h* xqd ixqlrqh wuldqjror vhsdudeloh/ lo fdofror h*

vhpsolfh vh vl vyrojrqr sulpd oh dxwrfruuhodlrql 4G ghooh gxh odph1 Shu oh surlhlrql h oh wudvirupdwh h*

frqyhqlhqwh lo fdofror gluhwwdphqwh qhooh wudvirupdwh +oh surlhlrql vrqr oh dqwlwudvirupdwh, ulfrugdqgr

fkh od wudvirupdwd ghoo*dxwrfruuhodlrqh h* mI +x> y,m21

! . #" # # 18&)&4-2

d, ^6 sxqwl` Vl vfulyd o*htxdlrqh ghood ixqlrqh i+> |, d edvh uhwwdqjroduh +glphqvlrql ghood edvh=

57, prvwudwd lq jxud 714<= vl glprvwul fkh ë xqd ixqlrqh vhsdudeloh h vl wuryl od vxd wudvirupdwd

gl Irxulhu 5G1

e, ^6 sxqwl` Vl glprvwul fkh od surlhlrqh ghood frqyroxlrqh elglphqvlrqdoh gl gxh ixqlrql ë

frqyroxlrqh ghooh gxh surlhlrql>

f, ^7 sxqwl` vl glvhjqlqr h vl fdofrolqr oh surlhlrql ghood ixqlrqh i+> |, shu * @ 3> @7> @5xwloldqgr/ vh vl ulwlhqh xwloh/ txdqwr glprvwudwr lq e,1

g, ^8 sxqwl 0 Pdwode` Vxood edvh ghl ulvxowdwl ulfdydwl ulvsrqghqgr dooh grpdqgh suhfhghqwl/ vl

wuryl xq prgr h!flhqwh shu fdofroduh h glvhjqduh oh surlhlrql ghood ixqlrqh i+> |, do ydulduh gl

* @ ^3 = 3=38 = 4` +vl vxjjhulvfh gl sruuh @ 4,1d, Od ixqlrqh ë vhsdudeloh +frqyroxlrqh gl gxh odph ruwrjrqdol h gluhwwh vhfrqgr jol dvvl frrugl0

qdwl,=

i+> |, @ +4mm

,uhfw+@5, uhfw+|@7,#$ I +x> y, @

vlq+x@5,

x@5

2

7

vlq+5y,

5y

e, Od yhulfd ë vhpsolfh qho grplqlr wudvirupdwr1 Vld od ixqlrqh

i+> |, @ j+> |, j2+> |,#$ I +x> y, @ J+x> y,J2+x> y,

71651 SURLH]LRQL GL XQD WHQGD FDQDGHVH= XQ VHJQDOH VHSDUDELOH +53292<:, 5;<

x

y

z

2∆

4∆

1

Iljxud 714<= Whpd 532392<:

od vxd surlhlrqh ss +u> *, ryyhur od surlhlrqh ghood ixqlrqh i+> |, qhood gluhlrqh *=

ss +u> *, #$ Ss + > *, @ I +x> y,m'lULt)'lt?)

@ J+ frv*> vlq*,J2+ frv*> vlq*,

Ss + > *, @ S+ > *, S2+ > *,#$ ss +u>*, @ s+u>*, s2+u>*,

f, Vh od ixqlrqh ë vhsdudeloh/ frph lq txhvwr fdvr/ oh surlhlrql vl vhpsolfdqr +vl ulfrugd od

vfdodwxud ghood ixqlrqh2wudvirupdwd= j+dw,#$ J+i@d,@mdm ,=

i+> |, @ j%+, j++|,$ ss +u> *, @4

frv*j%+u@ frv*,

4

vlq*j++u@ vlq*,>

txhvwr ydoh shu * 9@ n@5/ n @ 3>4>5> === +shu wdol ydorul lo sureohpd glyhqwd edqdoh,1

Oh surlhlrql vrqr=

ss +u>* @ 3, @ 7+4 mum

, uhfw+u@5,

ss +u>* @ @7, @ 5+4 mums5

, uhfw+u

s5@7,

ss +u>* @ @5, @ uhfw+u@7,

g, Shu udssuhvhqwduh judfdphqwh oh surlhlrql do ydulduh ghoo*dqjror vl srqh @ 4 h vl fdofrod frph

dqwlwudvirupdwd qho grplqlr ghooh iuhtxhqh=

ss +u>*,#$ 7

vlq+ frv*@5,

frv*@5

2

vlq+5 vlq*,

5 vlq*

( Lo fdofror ë vyrowr qho grplqlr ghooh iuhtxhqh=Q@589>iuht@^0Q25=Q2504`243>

5<3 FDSLWROR 71 VHJQDOL H VLVWHPL 5G

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

r

Insieme delle proiezioni phi=0,...,pi

Iljxud 7153= Surlhlrql do ydulduh gl *

u@43-^0Q25=Q2504`2Q>dqjror@+sl,-^3=138=40138`>iru lskl@4=ohqjwk+dqjror,>skl@dqjror+lskl,>i@iuht-frv+skl,>i|@iuht-vlq+skl,>I@7-+vlqf+i,1a5,1-+vlqf+7-i|,,>i+lskl/=,@uhdo+wvkliw+lw+wvkliw+I,,,,>hqgjxuhsorw+u/i,odeho+*u*,>wlwoh+*Lqvlhph ghooh surlhlrql skl@3/111/sï,jxuhphvk+u/4;3-dqjror2sl/i,|odeho+*skl +ghj,*,>odeho+*u*,wlwoh+*Surlhlrql v+u/skl,*,

! . # # 1!&8&4-2

Vl frqvlghul xq sulvpd uhwwdqjroduh d edvh txdgudwd gl odwr G h dowhd K@4/ fhqwudwr qhoo*ruljlqh h

rulhqwdwr frph jol dvvl frruglqdwl1

d, +7 sxqwl, Vh qh wudffl lo vlqrjudppd frpsohwr1

e, +7 sxqwl, Vl ghwhuplqlqr oh vhlrql ghoor vshwwur elglphqvlrqdoh frph wudvirupdwh ghooh surlhlrql

vld qhood gluhlrqh ghoo*dvvh fkh qhood gluhlrqh d 78f frq hvvr1

71661 SURLH]LRQL GL XQ SULVPD D EDVH TXDGUDWD +572342<:, 5<4

-5

0

5

0

50

100

150

200-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

r

Proiezioni s(r,phi)

phi (deg)

Iljxud 7154= Surlhlrql do ydulduh gl *

f, +7 sxqwl, Vl yhulfkl frvd dffdgh qhood gluhlrqh d 63f frq o*dvvh ghooh +vl ulfrugl fkh xq wudshlr

ë od frqyroxlrqh gl gxh uhwwdqjrol,1

Or vshwwur 5G gho sulvpd uhwwdqjroduh

k+> |, @ 4> mm> m|m ? G

5

ë=

K+x> y, @

"]]3"

k+> |,hE%n+gg| @

](*2

3(*2

h%

g]

(*2

3(*2

h+

g|

K+x> y, @ K+3> 3, vlq+xG@5,

xG@5 vlq+yG@5,

yG@5

Oh wudvirupdwh ghooh surlhlrql vrqr oh vhlrql ghoor vshwwur= xvdqgr oh frruglqdwh srodul

x @ frv* > y @ vlq*

vl rwwlhqh shu * @ 3 od wudvirupdwd gl xq uhwwdqjror=

* @ 3>$ y @ 3$ S + , @ K+3> 3, vlqf+ G@5,

* @ 78f/ gl xq wuldqjror=

* @ 78f>$ x @ y @ frv 78f @ =:3: $ S + , @ K+3> 3, vlqf2+=:3: G@5,

* @ 63f/ gl xq wudshlr=

* @ 63f>$ x @ frv 63f> y @ vlq63f $ S + , @ K+3> 3, vlqf+ G@7,vlqf+=;99 G@5,

fkh ë dssxqwr xqd frqyroxlrqh gl gxh uhwwdqjrol1

Vl vxjjhulvfh gl fdofroduh oh surlhlrql shu xq sulvpd wudvodwr/ s1h1 k+ O%> | O+,/ r frq dvvl gl

oxqjkhd glyhuvd +yhgl whpd 57252<<,1


!! . 3 1-&)&442

Vl frqvlghul lo vhjqdoh 5G i+> |, frqyroxlrqh gl gxh odph uhwwdqjrodul glvsrvwh oxqjr jol dvvl frruglqdwl

h fhqwudwh qhoo*ruljlqh=

i+> |, @ uhfw+

G,+|, uhfw+ |

G,+,

d,= ^sxqwl 5`= Vl udssuhvhqwl judfdphqwh i+> |, h vh qh wuryl or vshwwur I +x> y,=e,= ^sxqwl 6`= Vl frqyroyd od i+> |, frq vh vwhvvd fdofrodqgr

j+> |, @ i+> |, i+> |,>

vl fdofrol or vshwwur J+x> y, h vl gld xqd udssuhvhqwdlrqh judfd gl j+> |, h gl J+x> y,=f,= ^sxqwl 7` Vl wurylqr oh surlhlrql gl j+> |, vxoo*dvvh h vx xq dvvh d 78f frq o*dvvh lqglfdqgroh

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y@^5;5=654 4=654 4=73`>iru m@4=74/v+=/m,@u+y++740m,-5.4=+740m,-5.654,/m,>hqgV5@w5+v,>jxuh> lpdjhvf+o/i/dev+wvkliw+V5,,,frorupds+mhw,> dlv+^015 15 3 43`,wlwoh+*Vshwwur IN vhqd htxlyrfdlrqh*,>odeho+*qxphur grqgd ^42p`*,> |odeho+*iuhtxhqd ^K`*,>

Immagine dei segnali raccolti dai sensori

distanza [m]

tem

po [s

]

0 50 100

0

2

4

6

8

10

12

14

16

Spettro FK

numero d'onda [1/m]

freq

uenz

a [H

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-0.2 0 0.2

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9

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Spettro FK senza equivocaz

numero d'onda [1/m]

freq

uenz

a [H

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-0.2 0 0.2

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Realizzazioni originali

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Realizzazioni filtrate intorno a 500 MHz

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tempo [ns]0 10 20 30 40 50

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Applicazioni di Elaborazione Numerica dei Segnali - Ph.D ...

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