Sistemi e Tecnologie della Comunicazione

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1 Sstm olog dll Comuzo Complmt : sr trsformt d Fourr Formul d prostfrs L formul d prostfrs sprmoo l vlor d so o d somm d gol prodott d s d gol gol, vvrs: ( α β ) ( α ) ( β ) ( α ) ( β ) ( α β ) ( α ) ( β ) ( α ) ( β ) ( α β ) ( α ) ( β ) ( α ) ( β ) ( α β ) ( α ) ( β ) ( α ) ( β ) ( α ) ( β ) ( α ) ( β ) ( α ) ( β ) ( α ) ( β ) [ ( α β ) ( α β )] [ ( α β ) ( α β )] [ ( α β ) ( α β )] [ ( α β ) ( α β )]

2 Fuzo prod d prodo S pr tutto l sguto /f Dt l fuzo: ( ft) o tro rltvo, s dmostr h prod d prodo : ( f ( t )) ( ft f) ft f f ( ft) ( ft ) l ultm uguglz quto, pr og golo s h: ( ) ( ) Itgrl utl () () (3) (4) ( ft) ( ft) ( ft ϕ) ( ft ϕ) pr ( ϕ) pr ( ϕ) pr

3 Dmostrzo Itgrl (): ( ft) Itgrl () pr : ( ) f pr : s f [ ( f) ( ) ] [ ( ) ] ( ft) s f f f ( ft) [ ( f) ( ) ] [ ( ) ] [ ] Dmostrzo (ot.) Itgrl (3): ( ft ϕ) [ ( ft) ( ϕ) ( ft) ( ϕ) ] ( ϕ) ( f ) ( ϕ) ( f ) Appldo rsultt dgl tgrl () () s h l dmostrzo. L tgrl (4) s dmostr modo logo, ppldo l formul d prostfrs po rsultt dgl tgrl () () 3

4 Altr tgrl utl (5) (6) (7) ( ft) ( kft) ( ft) ( kft) ( ft) ( kft) / /, k s k s k s k s k Dmostrzo Itgrl (5): ( ft) ( kft) [ ( ft kf ) ( ft kf )] ( ( k ) f ) ( ( k ) f ) Poh l tgrl () vl pr og, l tgrl (5) vl. Gl tgrl (6) (7) s dmostro modo logo, ppldo l formul d prostfrs d rsultt dgl trgl ()-(4), rorddo h pr (-k) l tgrl dl o o ullo! 4

5 Sr d Fourr Dt u qulss fuzo prod d prodo otu o drvt otu trtt lmtt, possbl srvrl om somm d s : v ( t ) ( ft) b ( ft) dov f / l frquz dll fuzo I offt dllo svluppo soo dt dll rlzo: b v ( t ) v ( t v ( t ) ) ( ft) ( ft) Dmostrzo d offt d Fourr Cofft : v ( t ) > b > ( ft) ( ft) pr gl tgrl () () qusto so! - tutt trm dll du sommtor soo ull, qud: v ( t ) v ( t ) 5

6 Dmostrzo d offt d Fourr Cofft k (pr offt b k l dmostrzo log): v ( t ) > ( kft) ( kft) ( ft) ( kft) b > ( ft) ( kft) l prmo dddo vl zro (pr l tgrl ()), d l trzo h (pr l tgrl (5), pr tutt gl ), qud v ( t ) ( kft) ( ( k ) ft) > pr l tgrl (6) soo ull trm dll somm o k, qud v ( t ) ( kft) k k v ( t ) ( kft) Esmpo Vdmo lo svluppo sr d Fourr dll fuzo Il lolo d offt : ( ϕ) v ( t ) A ft A ( ft ϕ) (vd tgrl (3)) A ( ft ϕ) ( ft) A[ ( ft) ( ϕ) ( ft) ( ϕ) ] ( ft) A ( ϕ) ( ft) ( ft) A ( ϕ) ( ft) ( ft) 6

7 Esmpo Il sodo dddo ullo (tgrl (5)), mtr l trzo ullo pr, qud pr A ( ϕ) A ( ϕ) b A ( ft ϕ) ( ft) A[ ( ft) ( ϕ) ( ft) ( ϕ) ] ( ft) A ( ϕ) ( ft) ( ft) A ( ϕ) ( ft) ( ft) Il prmo dddo smpr ullo, l sodo ullo pr, qud b pr b A A ( ϕ) ( ϕ) Esmpo S h qud lo svluppo: A ( ft ϕ) ( ft) b ( ft) A ( ϕ) ( ft) A ( ϕ) ( ft) A ( ft ϕ) A( ( ft) ( ϕ) ( ft) ( ϕ) ) S puo ossrvr h lo svluppo sr d Fourr dll fuzo d smpo od qusto so o l formul d prostfrs (o potv ssr ltrmt!) 7

8 8 Not sull formul d Eulro L formul d Eulro dfs l fuzo spozl d spot mmgro: D qust formul s dduoo l rlzo: ( ) ( ) ( ) ( ) ( ) ( ) Svluppo d Fourr form omplss ( ) ( ) ) ( ft ft ft ft ft ft b b b ft b ft t v dfmo: b b

9 Svluppo d Fourr form omplss possmo qud srvr: v ( t ) ft ft poh possmo srvr (sosttudo d l trzo dddo): s h: ft v ( t ) ft ft Svluppo d Fourr form omplss dfdo f: bbmo l sprsso fl: v ( t ) ft 9

10 Rlzo tr offt I bs ll dfzo s h: b ; ; b d u l rlzo vrs: ( ); b Im( ) ; R smpl f dmostrr h pr og : ft v ( t) Esmpo: od qudr Esgumo lo svluppo form omplss dll fuzo od qudr prod d prodo : v ( t) A A pr k < t< k; pr k < t< k; I offt dllo svluppo soo dt d k Ζ k Ζ ft ft A A

11 Esmpo: od qudr pr s h: A A A A pr s h: A A f f ft ft [ ] [ ] f f ( ( )) f Esmpo: od qudr ssdo (rordmo h /f): f ( ) ( ) s h: A A f f ( ) ( ( ) ( ) ) ( ( ) ) A qud: A pr pr pr dspr

12 Esmpo: od qudr sprmdo l rsultto trm d b: ; ; b 4A pr pr pr dspr possmo qud srvr lo svluppo dll od qudr om: v ( t) 4A dspr ( ft)