The Propagation Series

Transcription

1 //009 Th Progtio Sris rst /0 Th Progtio Sris Q: You rlir sttd tht sigl flow grhs r hlful i (cout m ) thr wys. I ow udrstd th first wy: Wy - Sigl flow grhs rovid us with grhicl ms of solvig lrg systms of simultous qutios. But wht out wys d?? Wy W ll s th sigl flow grh c rovid us with rod m of th wv rogtio ths throughout microwv dvic or twork. Wy - Sigl flow grhs rovid us with quick d ccurt mthod for roximtig twork or dvic. Jim Stils Th Uiv. of Kss Dt. of EECS

2 //009 Th Progtio Sris rst /0 A: Cosidr th sfg low: Not tht od is th oly iddt od. This sigl flow grh is for rthr comlx sigl-ort (ort ) dvic. Sy w wish to dtrmi th wv mlitud xitig ort. I othr words, w sk: 0. = Γ i Usig our four rductio ruls, th sigl flow grh ov is simlifid to: Jim Stils Th Uiv. of Kss Dt. of EECS

3 //009 Th Progtio Sris rst /0 Q: Hy, od is ot coctd to ythig. Wht dos this m? A: It ms tht = 0 rgrdlss of th vlu of icidt wv. I.E.,: Γ i = = 0 I othr words, ort is mtchd lod! Q: But look t th origil sigl flow grh; it dos t look lik mtchd lod. How c th xitig wv t ort zro? A: A sigl flow grh rovids it of rogtio rod m through th dvic or twork. It llows us to udrstd oft i vry hysicl wy th rogtio of icidt wv oc it trs dvic. W ccomlish this y idtifyig from th sfg rogtio ths from iddt od to som othr od (.g., xitig od). Ths ths r simly squc of rchs (oitig i th corrct dirctio!) tht ld from th iddt od to this othr od. Ech th hs vlu tht is qul to th roduct of ch rch of th th. Jim Stils Th Uiv. of Kss Dt. of EECS

4 //009 Th Progtio Sris rst /0 Prhs this is st xlid with som xmls. O th tw iddt (icidt wv) od d (xitig wv) od is show low: W ll ritrrily cll this th, d its vlu: 0. ( ) ( ) ( ) = = 0. Aothr rogtio th (th 5, sy) is: Jim Stils Th Uiv. of Kss Dt. of EECS

5 //009 Th Progtio Sris rst 5/0 = 0.0 Q: Why r w doig this? 5 ( 0.5) ( 0.) ( 0.5) ( 0.8)( 0.5)( 0.8) ( 0.5) = = ( 0.5)( 0.)( 0.8) ( 0.5) A: Th xitig wv t ort (wv mlitud ) is simly th surositio of ll th rogtio ths from icidt od! Mthmticlly skig: = Γ = = i Q: Wo t thr wful lot of rogtio ths? A: Ys! As mttr of fct thr r ifiit umr of ths tht coct od d. Thrfor: = Γ i = = Q: Yiks! Dos this ifiit sris covrg? A: Not tht th sris rrsts fiit hysicl vlu (.g., Γ i ), so tht th ifiit sris must covrg to th corrct fiit vlu. Jim Stils Th Uiv. of Kss Dt. of EECS

6 //009 Th Progtio Sris rst 6/0 Q: I this xml w foud tht Γ i = 0. This ms tht th ifiit rogtio sris is likwis zro: Γ = = 0 Do w coclud from this tht ll rogtio ths r zro: i = 0????? A: Asolutly ot! Rmmr, w hv lrdy dtrmid tht = 0. d = 0.0 dfiitly ot zro-vlud! I fct for this xml, o of th rogtio ths r rcisly qul to zro! Q: But th why is: = 0??? A: Rmmr, th th vlus r comlx. A sum of o-zro comlx vlus c qul zro (s it rtly dos i this cs!). Jim Stils Th Uiv. of Kss Dt. of EECS

7 //009 Th Progtio Sris rst 7/0 Thus, rfctly rtiol wy of viwig this twork is to coclud tht thr r ifiit umr of o-zro wvs xitig ort : Γ = whr 0 i It ust so hs tht ths wvs cohrtly dd togthr to zro: i Γ = = 0 thy sstilly ccl ch othr out! Q: So, I ow rcit th fct tht sigl flow grhs: ) rovids grhicl mthod for solvig lir qutios d ) lso rovids mthod for hysiclly vlutig th wv rogtio ths through twork/dvic. But wht out hlful Wy : Wy - Sigl flow grhs rovid us with quick d ccurt mthod for roximtig twork or dvic.?? Jim Stils Th Uiv. of Kss Dt. of EECS

8 //009 Th Progtio Sris rst 8/0 A: Th rogtio sris of microwv twork is vry logous to Tylor Sris xsio: d f( x) f ( x ) = ( x ) dx = 0 x = Not tht thr likwis is ifiit umr of trms, yt th Tylor Sris is quit hlful i girig. Oft, w girs simly truct this ifiit sris, mkig it fiit o: Q: Yiks! Dos t this rsult i rror? N d f( x) f ( x ) ( x ) dx = 0 x = A: Asolutly! Th tructd sris is roximtio. W hv lss rror if mor trms r rtid; mor rror if fwr trms r rtid. Th trick is to rti th sigifict trms of th ifiit sris, d truct thos lss imortt isigifict trms. I this wy, w sk to form ccurt roximtio, usig th fwst umr of trms. Jim Stils Th Uiv. of Kss Dt. of EECS

9 //009 Th Progtio Sris rst 9/0 Q: But how do w kow which trms r sigifict, d which r ot? A: For Tylor Sris, w fid tht s th ordr icrss, th sigificc of th trm grlly (ut ot lwys!) dcrss. Q: But wht out our rogtio sris? How c w dtrmi which ths r sigifict i th sris? A: Almost lwys, th most sigifict ths i rogtio sris r th forwrd ths of sigl flow grh. forwrd th \ˈfoṙ wərdˈ äth\ ou A th through sigl flow grh tht sss through y giv od o mor th oc. A th tht sss through y od two tims (or mor) is thrfor ot forwrd th. I our xml, th is forwrd th. It sss through four ods s it trvls from od to od, ut it sss through ch of ths ods oly oc: Jim Stils Th Uiv. of Kss Dt. of EECS

10 //009 Th Progtio Sris rst 0/0 Altrtivly, th 5 is ot forwrd th: W s tht th 5 sss through six diffrt ods s it trvls from od to od. Howvr, it twic sss through four of ths ods. Th good ws out forwrd ths is tht thr r lwys fiit umr of thm. Agi, ths ths r tyiclly th most sigifict i th rogtio sris, so w c dtrmi roximt vlu for sfg ods y cosidrig oly ths forwrd ths i th rogtio sris: N = f whr rrsts th vlu of o of th N forwrd ths. f Jim Stils Th Uiv. of Kss Dt. of EECS

11 //009 Th Progtio Sris rst /0 Q: Is th th oly forwrd th i our xml sfg? A: No, thr r thr. Pth is th most dirct: = Of cours w lrdy hv idtifid th : = Jim Stils Th Uiv. of Kss Dt. of EECS

12 //009 Th Progtio Sris rst /0 Ad filly, th is th logst forwrd th: ( 0.5) ( 0.8)( 0.5)( 0.8) ( 0.5) = = = 0.08 ( 0.8) ( 0.5) Thus, roximt vlu of Γi is: Γ i = = f = = 0.06 Q: Hy wit! W dtrmid rlir tht Γ i = 0, ut ow your syig tht Γ i = Which is corrct?? Jim Stils Th Uiv. of Kss Dt. of EECS

13 //009 Th Progtio Sris rst /0 A: Th corrct swr is Γ i = 0. It ws dtrmid usig th four sfg rductio ruls o roximtios wr ivolvd! Covrsly, th vlu Γ i = 0.06 ws dtrmid usig tructd form of th rogtio sris th sris ws limitd to ust th thr most sigifict trms (i.., th forwrd ths). Th rsult is sir to oti, ut it is ust roximtio (th swrs will coti rror!). For xml, cosidr th rducd sigl flow grh (o roximtio rror): Exct SFG Comr this to th sm sfg, comutd usig oly th forwrd ths: Arox. SFG Jim Stils Th Uiv. of Kss Dt. of EECS

14 //009 Th Progtio Sris rst /0 No surris, th roximt sfg (usig forwrd ths oly) is ot th sm s th xct sfg (usig rductio ruls). Th roximt sfg cotis rror, ut ot this rror is ot too d. Th vlus of th roximt sfg r crtily clos to tht of th xct sfg. Q: Is thr y wy to imrov th ccurcy of this roximtio? A: Crtily. Th rror is rsult of tructig th ifiit rogtio sris. Not w svrly tructd th sris out of ifiit umr of trms, w rtid oly thr (th forwrd ths). If w rti mor trms, w will likly gt mor ccurt swr. Q: So why did ths roximt swrs tur out so wll, giv tht w oly usd thr trms? A: W rtid th thr most sigifict trms, w will fid tht th forwrd ths tyiclly hv th lrgst mgituds of ll rogtio ths. Q: Ay id wht th xt most sigifict trms r? A: Yu. Th forwrd ths r ll thos rogtio ths tht ss through y od o mor th o tim. Th xt most sigifict ths r lmost crtily thos ths tht ss through y od o mor th two tims. Jim Stils Th Uiv. of Kss Dt. of EECS

15 //009 Th Progtio Sris rst 5/0 Pth is xml of such th: 0. Thr r thr mor of ths ths (ssig through od o mor th two tims) s if you c fid thm! Aftr dtrmiig th vlus for ths ths, w c dd mor trms to our summtio (ow w hv sv trms!): Γ i = 7 = 05. ( ) ( 5 6 7) ( 0.06) ( ) = = = Jim Stils Th Uiv. of Kss Dt. of EECS

16 //009 Th Progtio Sris rst 6/0 Not this vlu is closr to th corrct vlu of zro th ws our rvious (usig oly thr trms) swr of As w dd mor trms to th summtio, this roximt swr will gt closr d closr to th corrct vlu of zro. Howvr, it will xctly zro (to ifiit umr of dciml oits) oly if w sum ifiit umr of trms! Q: Th sigificc of giv th sm to ivrsly roortiol to th umr of tims it sss through y od. Is this tru? If so, th why is it tru? A: It is tru (grlly skig)! A rogtio th tht trvls though od t tims is much lss likly to sigifict to th rogtio sris (i.., summtio) th th tht sss through y od o mor th (sy) four tims. Th rso for this is tht th sigificc of giv trm i summtio is ddt o its mgitud (i.., ). If th mgitud of trm is smll, it will hv fr lss ffct (i.., sigificc) o th sum th will trm whos mgitud is lrg. Q: You sm to syig tht ths trvlig through fwr ods hv lrgr mgituds th thos trvlig through my ods. Is tht tru? If so why? Jim Stils Th Uiv. of Kss Dt. of EECS

17 //009 Th Progtio Sris rst 7/0 A: K i mid tht microwv sfg rlts wv mlituds. Th rch vlus r thrfor lwys scttrig rmtrs. O imortt thig out scttrig rmtrs, thir mgituds (for ssiv dvics) r lwys lss th or qul to o! Sm Rcll th vlu of th is simly th roduct of ch rch tht forms th th. Th mor rchs (d thus ods), th mor trms i this roduct. Sic ch trm hs mgitud lss th o, th mgitud of roduct of my trms is much smllr th roduct of fw trms. For xml: 0.7 = 0. d = 0.08 I othr words, ths with mor rchs (i.., mor ods) will tyiclly hv smllr mgituds d so r lss sigifict i th rogtio sris. Not th i our xml trvld log o rch oly: = 0. Pth hs fiv rchs: Jim Stils Th Uiv. of Kss Dt. of EECS

18 //009 Th Progtio Sris rst 8/0 = 0. Pth sv rchs: = 0.08 Pth i rchs: = 0.0 Pth 5 lv rchs: 5 = 0.0 Pth 6 lv rchs: 6 = 0.0 Pth 7 thirt rchs: 7 = Hofully it is vidt tht th mgitud dimiishs s th th lgth icrss. Jim Stils Th Uiv. of Kss Dt. of EECS

19 //009 Th Progtio Sris rst 9/0 Q: So, dos this m tht w should do our four rductio ruls, d istd us tructd rogtio sris to vlut sigl flow grhs?? A: Asolutly ot! Rmmr, tructig th rogtio sris lwys rsults i som rror. This rror might sufficitly smll if w rti ough trms, ut kowig rcisly how my trms to rti is rolmtic. W fid tht i most css it is simly ot worth th ffort us th four rductio ruls istd (it s ot lik thy r rticulrly difficult!). Q: You sy tht i most css it is ot worth th ffort. Ar thr som css whr this roximtio is ctully usful?? A: Ys. A tructd rogtio sris (tyiclly usig oly th forwrd ths) is usd wh ths thr thigs r tru:. Th twork or dvic is comlx (lots of ods d rchs).. W c coclud from our kowldg of th dvic tht th forwrd ths r sufficit for ccurt roximtio (i.., th mgituds of ll othr ths i th sris r lmost crtily vry smll). Jim Stils Th Uiv. of Kss Dt. of EECS

20 //009 Th Progtio Sris rst 0/0. Th rch vlus r ot umric, ut istd r vrils tht r ddt o th hysicl rmtrs of th dvic (.g., chrctristic imdc or li lgth). Th rsult is tyiclly trctl mthmticl qutio tht rlts th dsig vrils (.g., Z 0 or ) of comlx dvic to scific dvic rmtr. For xml, w might us tructd rogtio sris to roximtly dtrmi som fuctio: Γi( Z,, Z, ) 0 0 If w dsir mtchd iut (i.., ( ) Γ i Z0,, Z0, = 0) w c solv this trctl dsig qutio for th (rly) ror vlus of Z0,, Z0,. W will us this tchiqu to grt ffct for dsigig multi-sctio mtchig tworks d multi-sctio could li coulrs. θ θ θ c siθ θ si c θ θ θ θ θ si c θ θ si c θ θ si c θ θ si c θ θ si c θ θ si c θ θ θ si c θ θ si c θ θ θ Jim Stils Th Uiv. of Kss Dt. of EECS θ θ θ si c θ θ si c θ θ θ Th sigl flow grh of thr-sctio could-li coulr.